non-linear analysis of the bond and crack …
TRANSCRIPT
NON-LINEAR ANALYSIS OF THE BOND AND CRACK DISTRIBUTION
IN REINFORCED CONCRETE MEMBERS
by
Fariborz Labib, M.Sc.(Eng.),
A Thesis submitted for the degree of
Doctor of Philosophy
in the Faculty of Engineering
of the University of London
Imperial College of Science and Technology, London.
September, 1976.
2
ABSTRACT
The behaviour of the reinforced concrete tension and
bending members is studied analytically using a two-dimensional
plane stress finite element model.
The formation of internal cracks, primary and horizontal
cracks, the types of crack propagation and the interaction between
these cracks are described for concentric and eccentric tension
members. The shapes of the internal cracks and their influence on
the shape of the primary cracks are presented. The variation of
bond stress, steel stress and slip is given. The analytical
results of these members were in good agreement with the corresponding
experimental data
The formation of flexural cracks and the variation of the crack
width over the depth and along the flexural span is obtained for a
partially prestressed beam. A suggestion is presented which relates
the variation of the crack width to the sequential propagation of
the cracks. The analytical results compared favourably with the
experimental data.
A non-linear reversal bond-slip relationship was assumed
for the above members.
Assuming perfect bond a prestressed box girder was analysed
and compared with experimental results.
An automatic computer program was developed to analyse the
above members with non-linear materials and/or bond characteristics.
3
ACKNOWLEDGEMENTS
The author thanks Professor A.J. Harris for the
opportunity to undertake this study in the Concrete Structures
and Technology Section of the Department of Civil Engineering
at Imperial College
The author wishes to thank Dr. A.D. Edwards for
supervising the project. His concern and valuable advice
throughout the research work is gratefully acknowledged.
The author thanks Miss C.D.M. Collins who so carefully
typed the thesis.
4
CONTENTS
Page
ABSTRACT
2
ACKNOWLEDGEMENTS
3
CONTENTS
4
NOTATIONS
12
CHAPTER 1 INTRODUCTION
1.1 Nature of the problem 14
1.2 Objective of the present research
16
CHAPTER 2 A REVIEW OF PAST WORK ON REINFORCED CONCRETE
2.1 Behaviour of concrete 18
2.1.1 Uniaxial behaviour 18
2.1.2 Biaxial behaviour 19
2.2 Behaviour of steel
21
2.3 Bond in reinforced concrete 21
2.3.1 Investigation of bond in beams 22
and pull-out specimen
2.3.2 Rehm's investigation 24
2.3.3 Recent investigations on bond
25
2.4 A brief review of cracking in reinforced
26
concrete structures
2.4.1 Cracking in reinforced concrete 27
under tension or bending
2.4.1.1 Investigation based on 27
bond-slip hypothesis
2,4.1.2 Broms investigation 27
2.4.1.3 Statistical investigations 30
5
Page
2.4.1.4 Investigations by Cement 31
and Concrete Association
2.4.1.5 Goto's investigation 33
2.4.1.6 Investigation of long- 34
term cracking
2.5 A brief review of non-linear techniques 35
2.6 A review of the finite element method 36
2.6.1 The development of the method 36
2.6.2 Finite element study of cracking 37
in reinforced concrete members
CHAPTER 3 MATERIAL PROPERTIES AND NUMERICAL TECHNIQUES FOR
MATERIAL NON-LINEARITY
3.1 Introduction 41
3.2 Behaviour of concrete 41
3.2.1 Microcracking and structure of 41
concrete
3.2.2 Behaviour of concrete under biaxial 44
state of stress
3.2.3 The idealised constitutive relation 46
of concrete in biaxial state of stress
3.3 The constitutive relationship for reinforcing 55
bars
3.4 Bond between reinforcement and concrete 58
3.4.1 Nature of bond 58
3.4.2 Mechanism of bond in plain bars 59
3.4.3 Mechanism of bond in deformed bars 60
3.4.4 The main parameters affecting the bond
61
resistance of deformed bars
6
Page
3.4.5 A brief description of some 63
experimental bond tests
3.4.6 Idealised bond-slip relation 64
adopted for the analysis
3.5 Numerical techniques for material non- 68
linearities
3.5.1 Statement of the problem 68
3.5.2 Incremental or stepwise procedure 69
3.5.3 Iterative procedure 71
3.5.3.1 Constant stiffness process 74
3.5.3.2 Variable stiffness process 74
3.5.4 Mixed procedure 75
3.5.5 Comparison of the basic procedures 75
3.5.6 Non-linear techniques adopted for 76
the analysis
CHAPTER 4 FINITE ELEMENT FORMULATIONS
4.1 Introduction 89
4.2 The finite element displacement procedure 90
4.3 Formulations of the element stiffness matrix 92
4.4 The finite element types selected for the 95
members
4.5 Stiffness properties of the elements 97
4.5.1 Bar elements 97
4.5.2 Linkage elements 98
4.5.3 Rectangular plate elements 100
4.5.4 Rectangular plate elements with 103
constant shear
4.5.5 Composite elements 104
7
Page
4.6 Transformation matrices
4.7 Cracking and crushing of concrete elements
4.8 Calculation of stresses and unbalanced
forces due to the material non-linearities
CHAPTER 5 BEHAVIOUR OF CONCENTRICALLY LOADED TENSION
MEMBERS
5.1 Objective of the analysis 120
5.2 An assessment of the behaviour of 121
concentrically loaded rectangular tension
members
5.3 Description of the members selected for 123
the analysis
5.4 Loading conditions of the members 124
5.5 Behaviour of short tension member T-RC3 124
5.5.1 Behaviour of the member before 124
cracking
5.5.2 Behaviour of the member after cracking 125
and comparison with experimental data
5.6 Behaviour of long tension member TRC2-3 130
5.6.1 Behaviour of the member prior to 130
cracking
5.6.2 Formation of cracks and post-cracking 131
behaviour of the member
5.6.2.1 Formation of the first 132
primary crack
5.6.2.2 Formation of the second 132
primary crack
8
Page
5.6.2.3, Width of the cracks at 133
later stages of loading
5.6.3 Analytical results and comparison 134
with the experimental data
5.7 Concluding remarks 137
CHAPTER 6 BEHAVIOUR OF ECCENTRICALLY LOADED TENSION MEMBERS
6.1 Objective of the analysis 171
6.2 Description of the members 171
6.3 Condition of loading 172
6.4 Behaviour of short tension members 172
6.4.1 Short member S1 (TRE3) 172
6.4.2 Crack formation in members S2 and 174
S3 and comparison with other short
members
6.5 Behaviour of long eccentric tension members 176
6.5.1 Behaviour of the members before 176
cracking
6.5.1.1 Long member Ll 176
6.5.1.2 Long member L3 176
6.5.2 Post-cracking behaviour of the 177
members
6.5.2.1 Formation of cracks in 177
member Ll
6.5.2.2 Formation of cracks in 178
member L3
6.5.2.3 Distribution of steel 179
stress and elongation of
the members
9
Page
6.6 Concluding remarks 180
6.7 Experimental and analytical widths of the 181
cracks
CHAPTER 7 ANALYSIS OF FLEXURAL MEMBERS
7.1 Analysis of a partially prestressed box 203
beam
7.1.1 Objective of the analysis 203
7.1.2 Modified analytical model of the 204
beam
7.1.3 Description of the experimental 204
member and idealisation
7.1.4 Condition of loading 206
7.1.5 Analytical results of the beam 207
and comparison with experimental
data
7.1.5.1 Application of prestressing 207
force
7.1.5.2 Formation, width and 207
spacing of flexural cracks
7.1.5.3 Distribution of steel stress, 210
bond and slip
7.1.5.4 Deflection of the beam 212
7.1.5.5 Comparison of crack width 212
and spacing
10
Page
7.1.5.6 Analytical results of 213
the crack width and spacing
of the elements of equal
size along the flexural
span
7.1.6 Concluding remarks 214
7.2 Two-dimensional analysis of a prestressed 216
box beam
7.2.1 Objective 216
7.2.2 A brief description of the 216
experimental investigation
7.2.3 Analytical model of the box beam 217
7.2.4 Material properties of the box beam 218
7.2.5 Condition of loading 219
7.2.6 Analysis of the beam and comparison 219
with experimental results
7.2.7 Concluding remarks 223
CHAPTER 8 COMPUTER PROGRAM
8.1 Introduction 259
8.2 Failure criterion of materials 260
8.3 Convergence criteria of a solution 260
8.4 Description of the method of analysis 263
8.5 Failure of the structure 264
8.6 Sequence of the operation 265
8.7 A brief description of the subroutines 267
8.8 Computer time for the execution of the program 270
8.9 Input and output of the program 271
11
CHAPTER 9 CONCLUSIONS
Page
9.1 General conclusions 273
9.2 Conclusions from the analytical results 273
9.3 Recommendations for future work 277
REFERENCES 278
APPENDICES 1, 2, 3, 4, 5 292
12
NOTATION
a Length of an element
As Cross sectional area of a bar
b Height of an element - breadth of flange
in an I-beam
B ] Element strain matrix
D Bar diameter
D Elasticity matrix
[ D Elasticity matrix in global directions
[ D ]pi Elasticity matrix in principal directions
E Elastic modulus
Esec Secant modulus
[ E Elasticity matrix
fc Concrete compressive strength
ft
Concrete tensile strength
[ f Displacement vector
[ F Vector of nodal forces
G Shear modulus
h Overall height of an I-beam
[ K Stiffness matrix
L Length
M Bending moment
M Displacement function
[ R ] Transformation matrix
S Crack spacing
t Thickness of an element
te Effective cover thickness
13
C v I Vector of nodal displacements
W Crack width
a Ratio of stresses
(3 Shear retention factor
Concrete strength in uniaxial compression P
Y Ratio of strains
E S ] Vector of nodal displacements
6 Strain
C Strain at peak stress in biaxial compression P
[ 6 ] Strain vector
C 6 ]x Strain vector in global directions
[ c ip Strain vector in principal directions
p Poisson's ratio
a Stress
a Peak stress in biaxial compression P
[ a ] Stress vector
[(IL Stress vector in global directions
C aJ, Stress vector in principal directions
T Bond strength
Th Bond stress along the reinforcement
Tv Bond stress perpendicular to the reinforcement
14.
CHAPTER 1
INTRODUCTION
1.1 NATURE OF THE PROBLEM
With advances in technology and the emergence of new
materials (special concretes and high-strength steel reinforcing
bars) a new generation of problems appeared in the behaviour of
reinforced concrete structures. The evaluation of the response
of these structures requires a thorough understanding of the
structural characteristics of concrete. Particularly important
among these are:
(1) The nature of the hetrogeneous.concrete behaviour
and the effect of state of stress on the concrete
strength.
(2) Bond between concrete and steel reinforcement.
(3) The initiation and propagation of cracking.
(4) The effect of temperatures, humidity and loading
history.
Moreover, all of the above factors are not independent,
they are closely inter-related.
The combination of high compressive strength and low tensile
strength is an additional feature of concrete. Today, cracks
in reinforced concrete structures are an acceptable feature under
reasonable stress conditions. Provided that the individual cracks
and deformation are not excessively large, reinforced concrete
structures can perform satisfactorily with respect to load carrying
capacity. Wide cracks are objectionable not only for aesthetic
15
reasons but are usually associated with high steel stress and
permit the entrance of water or aggressive solutions which might
corrode the steel. To ensure adequate serviceability of the
structure, in addition to its strength, the control of cracking is
equally considered as a limit state requirement.
According to experimental data the width of cracks is
subjected to relatively large scatter, also the different
characteristics of each investigator's specimen and the indirect
effect of the above mentioned variables, has resulted in differing
conclusions. The results have been expressed in terms of average
crack width. Some included only primary cracks, some others
considered all the observed cracks. Since the sum of all the
widths is more or less determinate, the width of the cracks is
inversely proportional to the number of cracks which are encouraged
to form. Many investigations deal with the matter of ensuring
that the cracks are numerous but narrow.
In devising methods for effectively dealing with cracks,
the basic mechanism of cracking must be understood. Most investigators
believe that the understanding of the cracking mechanism of a
simple uniaxial tension member loaded through a single reinforcing
bar, can secure valuable information regarding the more important
cases (flexural members and tension members with several reinforcing bars).
The use of classical theory and conventional methods for
the analysis of reinforced concrete members has been proved to be
inadequate. While these methods impose numbers of limitations,
their results are concerned with the deformational behaviour on
the macroscopic scale.
16
With advances on the high speed digital computer the
methods of analysis which can give comprehensive information
about the behaviour of a stucture use matrix algebra. The matrix
methods are based on the concept of substituting the actual
continuous structure by a mathematical model made up from structural
elements with the known elastic properties expressable in matrix
form.When these elements are fitted together according to a set of
rules derived from the theory of elasticity, they provide the
properties of the actual structural system.
The important extension of matrix methods introduced as
finite element method, is to use two or three dimensional structural
elements for the representation of a continuum. This method is
able to use elements of various sizes, types and shapes, arbitrary
geometry, support and loading conditions. With the recent
development in the method, solutions can be obtained for any
rationally conceived constitutive law of the material behaviour.
1.2 OBJECTIVE OF THE PRESENT RESEARCH
Many attempts have been made to investigate experimentally
as well as analytically the behaviour of reinforced concrete members.
Despite all these efforts a complete understanding of the problem
still evades research workers. One of the influential factors
on the behaviour of these members is the interaction between
concrete and the reinforcement. This interaction (bond) is
clearly interrelated with the other factors such as concrete
strength, surface property of the steel, loading and support
conditions, progressive cracking, etc. Due to the intricate
nature of the problem, the observation of an experiment generally
indicates the overall effects of these factors. On the analytical
17
side, using the finite element method, only a few investigators
have attempted to include the effect of bond. Due to the low
strength of concrete in tension and high tensile strength of
the deformed bars, the formation of cracks adjacent to the
tension reinforcement is inevitable. The assumption of perfect
bond neglects the existing interaction between the two materials
and its effect on the progressive cracking and the structural
behaviour.
The present work attempts to predict the progressive
cracking, bond distribution and the behaviour of the reinforced
concrete uniaxial and bending members with a presentation of
this interaction. Such an interaction is continuously affected
by the formation of cracks which may cause a reversal movement of
the surrounding concrete and may separate the concrete from the
steel. A rational analytical bond-slip relationship should be
capable of dealing with such a behaviour. The recent development
of the initial stress method has been advantageously employed to
express the above behaviour and the other non-linearities
(constitutive relationship of the materials and cracking of concrete).
Such an analytical model may lead one to observe the crack
initiation, propagation and the effect of progressive cracking
in the behaviour of the structure. It can also give an insight
into the variations and the shapes of the cracks.
18
CHAPTER 2
A REVIEW OF THE PAST WORKS ON REINFORCED CONCRETE
2.1 BEHAVIOUR OF CONCRETE
2.1.1 Uniaxial Behaviour
The experimental investigations into the behaviour of concrete
under uniaxial compression have resulted in a typical shape of stress-
strain diagram. The diagram starts out with a nearly linear portion
up to 30% of ultimate load called the proportional limit, beyond which
the curve deviates gradually from the straight line, reaches a peak
and then has a descending part. Since the material of most aggregate
particles exhibits a reasonably linear stress-strain relationship,
the departure from linearity is primarily due to the presence of
microcracks at the interface between the aggregates and the cement
paste(1) . Thus the shape of the stress-strain relation and the final
failure of concrete is due to progressive internal microcracking.
The summary of the investigations into the compressive stress-strain
response under uniaxial monotonic and cyclic loadings is given in
Ref.(2). Various mathematical expressions have been presented for
uniaxial behaviour of concrete. In the equation suggested by
Desayi et al.(3) the ratio of E/E (E is the initial tangent modulus
sec
and Esec is the secant modulus) is fixed. An improved version was
proposed by Saenz(4) which takes account of the varying secant modulus.
E/E
varies from nearly 4 for normal concrete with ultimate strength sec
of 1000 p.s.i. (7 N/mm2) to about 1.3 for high quality concrete of
10,000,p.s.i. (70 N/mm2). It is also known that the curvature of
19
the diagram may differ considerably due to the properties of the mix,
for instance, the lower the cement content the more curved is the
diagram. The other factors influencing the shape of the stress-
strain diagram are, the type of aggregate, age, testing condition,
rate of loading etc. The limit of validity of the various simplified
formulae proposed is dependent upon approximations made.
2.1.2 Biaxial Behaviour
The investigations into the biaxial behaviour of concrete
are usually divided into groups based on the type of the specimen
used. A review of the investigations on the type of the specimen
is given by Kupfer et al.('). The problem facing the investigator
is how to induce the desired state of stresses or strains into the
particular specimen. The concrete cubes or thin square plates used
for studies of the biaxial compressive strength are confined along
the loaded faces due to the friction between the bearing platens of
the testing machine and the concrete. Biaxial compressive stresses
can also be generated by subjecting a cylindrical specimen to
hydrostatic pressure in the radial direction and pressure in the
vertical direction. Hollow cylinders subjected to torsion and axial
compression were investigated by Bresler and Pister(6) , Goode and
Helmy(7). McHenry and Karni(8) tested hollow cylinders under
axial compression and internal hydrostatic pressure. As the tests
were conducted'on a variety of specimens with different types of
concrete and subjected to various loading conditions and machine
testing effects, the overall results cannot express a consistent
behaviour. Comparisons of the various test results(9) indicate
that, in general, the strength in uniaxial compression is about
8-10 times larger than the strength in uniaxial tension and the
20
strength in biaxial compression is about 10-25 per cent greater
than the strength in uniaxial compression. Kupfer et al. (5,10)
presented relatively comprehensive strain measurements of various
concretes under predefined ratios of biaxial stress. Concrete
specimens of 20 x 20 x 5 mm were tested by the new test appratus
developed by Hilsdorf(11) Restraint of the test specimen
was avoided by using brush-like load bearing platens. A series of
stress-strain curves in biaxial compression, tension compression and
biaxial tension were reported. Buyukorturk et al.(12) investigated
the biaxial behaviour of concrete in an experimental model consisting
of preselected circular discs of aggregates with radius r set in a
square array with average clear distance of 0.2r and 0.5r. They
also used the test arrangement originally proposed by Hilsdorf(11)
Formation and propagation of microcracks which are a significant
cause of non-linear response were followed by x-ray in the uniaxial
loading. A plane stress finite element model representing the
experimental model was used for analytical study. Liu et al.(13)
tested real concrete specimens and idealised models (aggregate discs
of three different sizes randomly embedded in a mortar matrix)
under the same test arrangement used by previous investigators(5,12)
Based on the results of the real concrete specimens a general stress
strain relationship for concrete in uniaxial and biaxial compression
was proposed which also agreed with test results obtained by Kupfer(5).
The proposed relationship(14) which satisfies the boundary conditions
at zero stress and peak stress, also includes the ratio of stresses
in the two orthogonal directions. The relation reduces to the
equations proposed by Saenz(4)
and Desayi(3) under special cases in the
21
uniaxial direction. An empirical failure envelope for concrete
under biaxial compression was also presented(14)
2.2 BEHAVIOUR OF STEEL
The typical stress-strain curve for deformed bars used in
reinforced concrete is obtained from the steel bar loaded monotonically
in tension. The curve usually exhibits an initial linear elastic
portion, a yield plateau, a strain hardening and a final stage until
rupture occurs. The stress at yield point, i.e. yield strength,
is an important property of the reinforcement. For bars with ill-
defined yield point the yield strength is generally taken as the
stress corresponding to a particular offset strain. The stress-strain
curves for steel in tension and compression are assumed to be
identical. The characteristics of prestressing wires are similarly
defined by its load-elongation relationship.
2.3 BOND IN REINFORCED CONCRETE
The bond between steel and concrete has always been a complex
problem. Early research workers realised that the safety of a
reinforced concrete structure should not depend on the end anchorage
and the reinforcement should be bonded throughout its entire length.
Gilkey et al.(15) confirmed that the bond stress is proportional
to compressive strength of concrete. With higher concrete strength,
however, there was a consistent reduction in the ratio of bond
resistance to compressive stress for increasing concrete strength.
22
2.3.1 Investigation of Bond in Beams and Pull-out Specimens
The bond performance of various reinforcing bars,embedded
in concrete of different strengths, has been traditionally determined
from beams and pull-out tests. The bond strength in these tests
was expressed as the average bond stress along the member. The
peak bond stress is, however, well in excess of the average value.
Clark(16,17)
conducted comprehensive research into beams and pull-out
specimens to secure information on the effect of size of the bar,
type of deformation on the bar and strength of concrete. The bars
were categorised in terms of the ratio of shearing to bearing area
on the surface of the bar. A series of bond-slip curves (for free
end and loaded end) were plotted which led-to the development of
reinforcing bars having bond quality far superior to the others.
He also noticed that the bond strength was affected by the changes
in the position of the bar. The average bond strength was
significantly higher when the bars were nearer to the bottom than
they were nearer to the top of the specimen. The loss of bond
strength in top bars due to the settlement of concrete was about
30 to 40%. Mains(18) measured steel strains at close intervals
along the embedded bars by means of electric resistance strain gauges
and showed the variation of steel stress between two flexural cracks.
The strain gauges were mounted internally in a hollow rectangle
formed by sawing the bar longitudinally and tack welding the bar.
Mathey and Watstein(19) in their investigations of bond in beams
and pull-out specimens observed that:
(1) The bond strength decreased with an increase in the
length of embedment for a bar of given size. The bond value
also decreased with an increase in the bar diameter for a given
23
length diameter ratio.
(2) Two criteria of failure were used to define "critical"
bond stresses, a loaded end slip of 0.01 in.and a free end slip of
0.002 in.
(3) The loaded end slip in the beam and pull-out specimens
increased with the stress in the bar and was essentially independent
of the length of embedment for a given size bar.
The last observation was later confirmed by Ferguson et al.(20)
from their pull-out tests on different embedments. They indicated
that bond stress tends to concentrate near the loaded end, thus when
the average bond stress is calculated over a long length its value
is small compared to the maximum value. They also observed that
the slip at the free and loaded end can be considerably affected by
the position of the bar when cast.
Increased concrete cover has been found to produce some increased
resistance against splitting. However, the improved bond performance
is not proportional to the additional cover thickness. Ferguson and
Thompson(21)
observed that for large size bars the beneficial effect
is not very significant.
Perry and Thompson(22) used a technique similar to Main's
for measuring the steel strain. Their observation on bond stress
in eccentric pull-out specimens indicated that the point of maximum
bond stress moved away from the loaded end as the force in the bar
increased, and the location of maximum bond stress for the same
force in the bars moved closer to the loaded end as the concrete
strength was increased.
24
In the investigation of bond from pull-out tests, as the
bar is pulled from the surrounding concrete, the transverse
compression which has a beneficial effect on bond strength is
induced against the bar. The performance of bond in these tests,
therefore, is not typical of the situation encountered in practice.
Various forms of bond test specimens were thus proposed for the
Above reason. The conventional consideration of bond is examined
and discussed in a report by Ferguson(23) •
2.3.2 Rehm's Investigation
Previous to Rehm, bond tests all had an embedded length
several times larger than the bar diameter. Rehm(24)
restricted the
bond length of the bar in pull-out tests to a very short distance,
generally equal to the diameter of the bar,in order to establish
what may be called "the fundamental law of bond". He investigated
numbers of commercially available ribbed bars, but his main objective
was to observe the behaviour of concrete between the ribs machined
on plain bars. He found that the pressure under the ribs was many
times higher than the cube strength of concrete. Rehm related
several aspects of the bond problem to the geometric parameter a/c
where a is the height of the ribs and c is the spacing between the
ribs. When the ribs are high and spaced too closely, the shearing
strength of the mortar and the bar will govern the behaviour and the
bar will pull out. When the rib spacing is larger than approximately
ten times the rib's height , the partially crushed concrete may form
a wedge in front of the rib and failure is normally by splitting of
the surrounding concrete. His results reveal interesting facts
about the state of stress in the concrete surrounding the steel.
25
The lateral restraint of the specimen increases substantially the
shearing strength of mortar but this cannot be realised in all
reinforced concrete structures. Rehm's conclusion that the
slip resistance was proportional to concrete strength possibly
only applies to the ideal condition of the tests.
2.3.3 Recent Investigations on Bond
Lutz and Gergely(25)
examined the action of the bonding
forces and the associated slip and cracking of bars with various
surface properties. They found that bars having ribs with a steep
face slip mainly by compressing the concrete in front of the bar
rib. The concrete is crushed and a concrete wedge forms in front
of the rib. Bars with flat ribs, however, slip with the ribs
sliding relative to the concrete. They observed that near a
transverse crack bonding forces cause large circumferential tensile
stresses and radial (splitting) cracks form there. Also radial
tensile stresses destroy the contact near the crack and allow
separation and slip of the bar. These large radial tensile stresses
near the crack were confirmed by a finite element analysis(73)
Bersler and Bertero(26)
investigated the behaviour of
reinforced concrete under repeated loading. A circular notch
mid-way along a reinforced concrete cylinder acted as a crack initiator.
Using carefully instrumented experiment procedures they observed the
deterioration of bond resistance under repeated loads. An axisymetric
finite element model was developed in which a thin layer around the
reinforcement was assumed to have the properties of a soft material
(boundary layer) simulating the steel-concrete interface. The model
was analysed to interpret qualitatively the physical behaviour
of the material.
26
Nilson(27)
adopted Main's technique for measuring the
steel stress and placed internal strain gauges,in concrete at
discrete locations longitudinally along the deformed bar of uniaxial
tension specimens. With both the steel stress and concrete stress
distributions known, the relative displacement between the steel
and concrete was indirectly calculated. His results were
presented as a series of curves relating local bond stress to local
slip. His results indicated that the bond strength of a point
along the member decreases towards the end of the specimen or by analogy
towards a cracked face. The reliability of his results is open to
question due to the following:
(1) The concrete strain gauges were located far from the
interface of the steel and concrete (12.5 mm from the steel surface)
where the transfer of stress between concrete and steel develops.
(2) Splitting cracks have been frequently seen to form near the
end of the uniaxial specimens reinforced with deformed bars(28)
Before the formation of these cracks a totally different state of bond
stress existed there. The measurements of Nilson however show that
the bond strength near the end is always very low.
(3) The presence of the strain gauges are possibly affected
by the formation of internal cracks within the member (the presence
of these internal cracks is clearly shown by Goto(28)
2.4 A BRIEF REVIEW OF CRACKING IN CONCRETE STRUCTURES
Causes of cracking in concrete are numerous. Comprehensive
reviews of principal causes of cracking are available elsewhere(29,30)
27
2.4.1 Cracking in Reinforced Concrete under Tension or Bending
2.4.1.1 Investigations based on bond-slip hypothesis:
As a result of the more prevalent use of high strength steel,
extensive investigations have been carried out concerning the crack
width and spacing in reinforced concrete members. Various semi-
theoretical and experimental equations have been developed
for which the crack width calculation was based on the bond-slip
hypothesis. Watstein(35) found that the width of cracks for the
most efficient type of bar was less than 50% of those of plain
bars at the steel stress of 40,000 p.s.i. The crack widths observed
for various bars varied approximately with the spacing of the cracks.
In Clark's test(31) of flexural members the average width of the
cracks was proportional to the increase of steel stress beyond that
causing initial cracking, and average spacing of cracks decreased
rapidly with an increase in steel stress beyond that causing the
first crack. Watstein and Mathey(34) noticed that the crack width
increased with distance from the reinforcement in tests of axially
reinforced tension specimens. Hognestad(32) from his tests on beams
of rectangular cross-section confirmed the results obtained by Clark(31)
He indicated that a wide experimental scatter inherently exists in
crack width and spacing of flexural members. The crack width was
proportional to the bar diameter for plain bars, but less dependent
on bar diameter for modern American deformed bars.
2.4.1.2 Broms investigation:
Broms proposed a cracking mechanism(36,37) based on an elastic
analysis of concrete stresses and considered the redistribution of
stresses that occur when a new crack forms and alters the geometry of
28
the member. He conducted an extensive experimental
(38,39,40,41,42,43) investigation on reinforced concrete tension
and flexural members and characterised the type of crack which
formed in them (primary, secondary, etc.). Based on his elastic
analysis and experimental data, he developed a simple method of
calculating the crack width and spacing. The summary of his
experimental observation is as follows:
(1) Primary tensile cracks were observed on the surface
of the tension and flexural members. These cracks transversed the
total section of the tension members and extended to the neutral
axis of the flexural members. Secondary tensile cracks were
observed(40) when the stress in the reinforcement reached approximately
20,000 to 30,000 p.s.i. (140-210 N/mm2). These cracks were confined
to the vicinity of the reinforcement.
(2) The average crack spacing Save was found to increase
approximately linearly with increasing distance from the reinforcement(42,43)
as given by the equation
S = 2t ave e
where te is an effective cover thickness. This effective cover thickness
depends on the location where the crack spacing is desired and the
arrangement of reinforcement. When a single reinforcing bar is used
te = t, where t is the distance from the steel at which the spacing
is required. The average crack width at the level of the steel can
be calculated by the equation
Wave
= Save
. s
29
where es
is the average steel strain.
(3) Longitudinal tensile cracks originated at existing
primary or secondary cracks and spread along the reinforcement.
Bifurcated transverse cracks were observed in the flexural members
at high load levels.
(4) At low stress levels in the reinforcement in uniaxial
concentric tension members of rectangular cross-section whose height
was approximately 2.5 times the width, the width of the primary cracks
all round the specimen was approximately the same, At steel stresses
exceeding 30,000 p.s.i. the crack widths at the top face were two
to three times the widths at the side face at the level of the
reinforcement. The width of secondary cracks reached a maximum at
the level of the reinforcement in contrast to the primary cracks.
(5) Measurement of the total elongation between the load
points of the tension members confirmed that,the average steel stress
can be closely calculated on the basis of a fully cracked section(40)
This assumes that no load is carried by the concrete shell surrounding
the reinforcement. The analysis of the available test data indicates
that for tension specimens the critical steel stress (stress level at
which the part of the total tensile force carried by the concrete is
small) corresponding to steel percentage of 4,2 and 1, are about
15,000, 30,000 and 60,000 p.s.i. respectively.
(6) Experimental technique for investigation of internal cracks:
The formation of internal cracks was investigated through the injection
of resin into some of the short tension members(41). The surface
strains were measured by means of strain gauges of 1 in. length. The
strain gauges were placed in the direction of and perpendicular to
the reinforcement. In general the effects of a crack were not registered
30
by a particular strain gauge until a tensile crack approached and
passed the level of the strain gauge considered(40)
. A reversal
of strain in a strain gauge indicated that one or several internal
secondary cracks had formed near the gauge(39)
The failure or
reversal strain of a gauge took place at a relatively low load
level(39)
The presence of the internal cracks was confirmed
when the specimens were cut open. It should be noted, however,
that the internal cracks were only visible at later stages of
loading when their width had developed sufficiently. Broms
suggested that the concrete strain at the level of the reinforcement
was negligible and thus the summation of the crack widths at this
level should equal the extension of the reinforcement(40)
. The total
width of the visible cracks, however, indicated that additional
secondary cracks were present at the level of the steel. He
suggested that they were so small (less than 0.001 in.) that resin
did not penetrate into them and on unloading they closed and could
not be observed when the specimen was cut open and examined by means of
a hand microscope(40)
2.4.1.3 Statistical investigations:
Gergely and Lutz(44)
have subjected the data from the previous
investigations to statistical analysis to determine the importance
of the variables involved. Many combinations of variables were tried
and it was very difficult to obtain an equation that fitted all sets
of data. The following major conclusions were reached regarding the
factors affecting the crack width.
31
(1) The steel stress is the most important factor.
(2) The thickness of the concrete cover is an important
factor but not the only geometric consideration.
(3) The bar diameter is not a major factor.
(4) The size of the bottom face crack width is influenced
by the amount of strain gradient from the level of the steel to
the tension face of the beam.
(5) The area of concrete surrounding each reinforcing bar
is also an important geometric variable.
They proposed two equations for the most probable maximum
crack width at the level of reinforcement and on the bottom face of
the beam.
Nawy(45)
investigated the cracking characteristics of 27
beams. The crack width data at the reinforcement level obtained
by himself, Hognestad(32), Karr-Matlock(33)
and Base et al.(46)
were plotted against the variables of the equations suggested by
Karr-Matlock(33) , Gergely-Lutz(44)
and Base et al.(46)
. The scatter
of data about the best line fit in all the three plots was very
considerable.
2.4.1.4 Investigations by Cement and Concrete Association :
Base et al.(46) proposed a fundamentally different approach, the
"no slip theory" in which they assumed that for the range of crack
widths normally permitted in reinforced concrete, there is no slip
of steel relative to the concrete. The crack is therefore assumed
to have zero width at the surface of the reinforcing bar and to
increase in width as the surface of the member is approached. The
results of beam tests led to the following conclusions:
32
(1) The type of reinforcing steel has little influence
on the surface crack width. Beams reinforced with plain bars
gave average crack widths about 20% greater than beams reinforced
with deformed bars.
(2) Variation of bar diameter has no effect on cracking.
(3) Variation of steel percentage within the range tested
had no significant influence on cracking.
(4) Crack width and crack spacing were both found to be
linearly related to the distance from the point where the crack was
measured to the surface of the nearest longitudinal reinforcing bar.
(5) Crack width was found to be proportional to the measured
surface strain at the level where the cracks were measured.
They suggested the following formula for prediction of
crack widths:
W = KCe
where C = distance from the point where crack width is required
to the surface of the nearest reinforcing bar
C = surface strain at the level where crack width is being
calculated
K = Constant
Beeby(47)
extended the study to cover reinforced concrete
slabs spanning in one direction and proposed a new hypothesis
for cracking behaviour of reinforced concrete members. This
hypothesis may be briefly summarised as follows.
33
The actual crack pattern at a given point results from the
interaction of two basic patterns.
(1) A crack pattern controlled by the initial crack height,
h0. The only influence that the reinforcement has on this pattern
is to control the crack height. The crack spacing and crack width
are linearly proportional to h0.
(2) A crack pattern controlled by the nearness of the reinforcing
bar where a predominantly linear relationship (no slip theory) is
predicted between crack width and distance from the nearest reinforcing
bar,C. However slip or internal cracking occurring before the crack
pattern has fully developed, modify this linear relationship. They
will result in the crack having some width at the surface of the bar
and cause larger crack spacing and widths. In other words, the effect
of slip or internal cracking is to modify the C controlled crack pattern
towards the h0 controlled crack pattern.
Directly over the reinforcement, pattern 2 dominates but,
with increasing distance from the bars, the crack pattern approaches
the first pattern.
2,4.1.5 Goto's investigation:
Goto published(28) photographs of internal cracking which
resulted from pulling both ends of a tension bar embedded in a
concrete prism. By injection of ink into the tensioned specimen,
the internal cracks were dyed. These cracks which formed in large
numbers in the concrete around the deformed bars at about 60° angle
to the bar axis, had a great influence on the bond mechanism between
steel and concrete. Through the formation of such cracks the
concrete around the reinforcing bars presents the appearance of a comb.
The teeth of this comb-like concrete were inclined towards the
54
nearest primary cracks.
2.4.1.6 Investigation of Zong -term cracking:
The early experimental investigation by Iliston and Stevens(48)
was concerned with the formation of internal and surface cracking
and breakdown of bond in reinforced concrete beams and cylinders.
They used Broms technique(41)
of injecting resin into the cracked
specimen. These tests confirmed the finding of Broms(42)
that in
the short term the crack width tends to narrow from the surface of
the concrete towards the steel. Separation on a longitudinal
plane through the steel revealed a resin stain on the steel concrete
interface at each crack indicating the breakdown of adhesion bond.
Their investigation into the crack width on long-term loading in
reinforced concrete beams(49)
revealed that:
(1) The spacing of cracks does not change with time
under sustain loading, but the average crack width does increase
with time.
(2) The increase in width of the cracks which occur at
a decreasing rate with time is caused by reduction in average
tensile strain of concrete between the cracks due to shrinkage and
the time-dependent change of curvature(due to the creep of concrete
in the compression zone). In the tests crack widths doubled in
two years.
(3) There is a breakdown of bond under sustained loading
and the restraint afforded by steel bars to the widths of crack
is reduced.
35
2.5 A BRIEF REVIEW OF NON-LINEAR TECHNIQUES
In the last two decades, the solution of structural
non-linearity has been tackled in many ways by various investigators.
Non-linear behaviour in a structural system is usually in one of
two categories:
(1) Material non-linearity which arises from the stress-
strain relationship which departs from linear behaviour.
(2) Geometric non-linearity which arises from the non-linear
strain-displacement relationship due to the large strains and large
displacements.
In general mathematical techniques that can be used successfully
to treat one type of non-linearity are,with modification,applicable
to the other type. An extensive review of the finite element
,applications to non-linear structural systems is given by Oden(50)
Angyris(51) used initial strain method to solve elasto-plastic
problems by the finite element method. The increase of plastic
strain was computed and treated as an initial strain with no change in
stress. This method fails if ideal plasticity exists or if the
degree of hardening is small. Nilson(52)
and Franklin(53) employed
linear incremental procedure to trace the non-linearity of the
material. Marcal et al.(54) used variable elasticity technique
which would successfully treat any elasto-plastic and perfectly
plastic material. The method suffered the disadvantage that the
stiffness of the structure is changed at each iteration and from
the computational point of view is an expensive operation.
Zienkiewicz et al.(55) developed a method called stress transfer
process to deal with no tension materials using finite element approach.
36
The calculated principal tensile stress was eliminated in the
element and the structure was equilibriated by equivalent applied
nodal forces. A similar approach,"initial stress method", was
later applied to plasticity problems(56)
and this had an advantage
over the initial strain method when dealing with perfectly plastic
materials and an advantage over the variable elasticity technique
as the stiffness of the structure is kept constant during the successive
iterations. A thorough examination of different approaches and their
applicability to different material properties is-given by
Zienkiewicz et al.(57)
2.6 A REVIEW OF THE FINITE ELEMENT METHOD
2.6.1 The Development of the Method
The development of the finite element method is associated
with the growing demand of the aircraft industry for a procedure which
could provide a refined solution for extremely complex aircraft
configurations. The method which is essentially a generalisation of
standard procedures for stress analysis, permits the determination of
stresses and displacements in two and three dimensional bodies by the
same techniques that are commonly applied to framed structures. The
solution for stresses and displacements is, however, greatly
facilitated by matrix formulations and only possible practically with
the use of high speed digital computer.
Hrennikoff(58)
and McHenry(59) in their analysis of plane
stress systems, developed a method of solution called framework
4 0
system or lattice analogy in which a continuum could be represented
by an aggregate of discrete frame bars. Assuming that under a
particular stress system the framework undergoes a compatible
deformation equivalent to that of a continuous material of elastic
37
body, the elastic properties of lattices suitable for that type
of problem were determined. The framework was then analysed by
standard methods for dealing with highly indeterminate systems.
The approximation of the structure by suitable choice of the lattice
elements of restricted properties was, however, very crude and curved
boundaries and orthotropic materials could not be represented.
In fact, it was an attempt to improve Hrennikoff-McHenry
"lattice analogy" for representing plane stress systems which first
led to the development of the finite element concept by Turner et al.(60)
An important part of this pioneering work was an evaluation of the
in-plane stiffness of a two-dimensional body (plate) so as to
establish a relation between the forces and the displacements of the
corner nodes, similar to that of one-dimensional (bar) members. Thanks
to the availability of high speed digital computer the method advanced
very rapidly. Clough(61)
derived the stiffness matrix of triangular
and rectangular elements by applying the principle of virtual
displacement. Wilson suggested that the derivation of an element
stiffness may be based on energy considerations. He extended(62)
the finite element method to non-linear problems, Doherty et al.(63)
and Wilson et al.(64) proposed methods to improve the basic accuracy
of low-order finite elements when subjected to certain stress
gradients. Details of the finite element method and recent advances
can be found in various textbooks (65,66)
2.6.2 Finite Element Study of Cracking in Reinforced Concrete Members
Early work in this area was initiated by Ngo and Scordelis(67)
who studied the behaviour of a cracked beam using triangular
finite elements. The pre-existing cracks were represented by
separating the concrete elements either side of the crack. The
•
38
linkage element was introduced to give steel-concrete interaction.
Nilson(52)
used an incremental approach for a non-linear bond-slip
relationship. He considered crack propagation by allowing a crack
to follow the element bdundaries and progressively twinning the nodal
points as the crack travelled between the elements. This process
required the topology of the structure to change continually due
to crack propagation, hence this presented some undesirable features
as far as programming and predicted behaviour was concerned.
Ngo, Franklin and Scordelis(68), tried to circumvent this problem
by predefining the expected total length of a shear crack (observed
in the experiment) in their study of reinforced concrete beams.
The two sides of the crack were initially held rigidly together by
very stiff linkage elements. By varying the stiffness value of
the linkage elements, the crack propagation was simulated without
renumbering the nodal points.
If pre-existing cracks are not assumed, then criteria are
required for choosing the points where cracks are likely to initiate
and for determining the direction and extent of crack propagation.
Zienkiewicz and Cheung(65) have proposed that the entire element
which had developed tensile stress exceeding the tensile strength
should be assumed to be cracked in the direction normal to the stress.
The load sustained by the stress field associated with the crack
was then redistributed to the rest of the structure and zero stiffness
was assumed for the element perpendicular to the crack. Valliappan
and Nath(69) used the same approach for cracking and applied the
load incrementally. Cervenka(70) used a constant strain finite
element with steel distributed over the element in two orthogonal
directions to study plane stress problems. Zienkiewicz, Valliapan
39
and King(55) presented a "no tension" model in order to study rock
structures. The material was assumed to sustain compression only,
all tensile stresses were therefore eliminated in an iterative
procedure called stress transfer method, and equivalent nodal forces
were applied to the surrounding materials.
Recently the problem of crack propagation has been tackled
using various iterative procedures. The considerable merit of these
procedures lies in the fact that they reduce the computational
effort by making revisions to load vector rather"than to the
stiffness matrix. The initial stress method introduced by
Zienkiewicz et al.(56) for elasto-plastic solutions was further
extended by Valliappan and Doolan(71) for determining the stress
distribution in reinforced concrete structures due to cracking of
concrete and elasto-plastic behaviour of steel. The load was
applied incrementally, if the principal stress in any concrete element
exceeded the tensile strength of concrete in any iteration, the
tensile stress was released and equivalent nodal forces were applied.
The prescribed tension for the cracked- eletent was then assumed to be
zero. The original stiffness matrix of the structure was, however,
kept constant throughout the loading. On a discussion of Ref. (71)
Schnorbich et al.(72) stated that, the pure initial stress approach,
always using the original elastic matrix of the structure as the
basis for solution and interacting with pseudo stresses to correct
the stress state may not be the most economical approach. They
suggested that, periodical up-dating of the stiffness matrix is
desirable. Cracking was considered as the changing of the material
properties of concrete. This manner of introducing cracking allows
40
some shear capacity to be retained in the cracked element. This
allowance for shear corresponds to taking into account the concept
of aggregate interlock across the crack surface. The shear transfer
capacity that remains across the crack changes with crack width.
Nam and Salmon(74)
using isoparametric elements demonstrated that the
constant stiffness method (keeping the same original stiffness matrix)
provides neither an efficient nor correct solution for evaluating non-
linear behaviour of concrete due to cracking in flexural members. The
variable stiffness method was shown to be an effective method for ■
prediction of cracking in concrete flexural members. Hand et al.(75)
in a non-linear layered analysis of reinforced concrete plates and shells
retained some shear stiffness in the cracking plane to account for
aggregate interlock which resulted in a good correlation with the test
, results. Suidan et al.(76) used a three-dimensional isoparametric
20-node rectangular element for analysis of reinforced concrete
structures which included a factor of shear retention in the cracked
plane of concrete. Colville and Abbasi(77)
presented a plane stress
reinforced concrete finite element using models in which the steel
and concrete are considered as one element so as to make the idealisation
of the structure independent of the geometry of the reinforcement.
Lin and Scordelis(78) used triangular layered element idealisation
for the analysis of reinforced concrete shells in which the steel was
presented as smeared layer. While assuming zero modulus for cracked
concrete elements perpendicular to the crack direction, the tensile
stress was released step-wise gradually accordingly to a specified
loading curve. This was to include the tension stiffening effect of
concrete over a relatively long length due to the bond between steel
and concrete.
CHAPTER 3
MATERIAL PROPERTIES AND NUMERICAL TECHNIQUES FOR MATERIAL NON-LINEARITY
3.1 INTRODUCTION
A realistic analytical model of reinforced concrete structures
must reflect the behaviour of the constituent materials (concrete
and reinforcement) and the effect of bond between the concrete and
the steel. In this chapter the behaviour of the materials is
described first, with the appropriate analytical simulation, then
various possible numerical approaches dealing with material non-
linearities are given together with those employed in the analytical
models.
3.2 BEHAVIOUR OF CONCRETE
3.2.1 Microcracking and Structure of Concrete
The physical property of normal concrete as a composite material
depends on the properties of the constituents and the mechanism of
interaction (chemical and mechanical interaction between the aggregates
and the cement) in which stiffer and stronger particles (aggregates)
are embedded in a softer and weaker cement paste. The property of
the hardened cement paste which bonds the aggregate of different
sizes depends primarily on the initial water-cement ratio and the
degree of hydration.
Recently studies on plain concrete have revealed the existence
of microcracks and how they propagate and multiply under the load.
The shape of the stress-strain curve and the failure mechanism is
42
attributed to these irreversible internal microcracks. The
formation of microcracks in concrete is primarily due to strain
and stress concentrations resulting from the incompatible
deformation of aggregates and cement paste caused by shrinkage,
temperature effects, etc. Such cracks can exist even before
loading, but their effect and propagation depends primarily on
the local state of stress which is induced. These cracks have
some effect on the fracture process when the concrete is subjected
to biaxial or triaxial compression but they can initiate failure
at an early stage if the concrete is subjected to a predominantly
local tensile stress state. Concrete both in compression and
tension exhibits an initial linear elastic response. During
this stage the microcracks are initiated at isolated points where
there is a large tensile stress concentration. The cracks relieve
the concentration, redistribution of stress takes place and
equilibrium is rapidly restored. The irrecoverable deformation is
small and the cracks do not propagate. As the load is increased
a second stage is reached where the cracks begin to propagate.
The extent of this crack propagation stage depends on the applied
state of stresses, being very short for brittle fracture under a
tensile stress state and larger forplasticl fracture under a
predominantly compressive state of stress. The third stage occurs
when under the particular system of loading the crack system has
developed such that it becomes unstable and the strain energy
released is sufficient to make the crack self-propagating until
complete disruption and failure occurs. In compression, this stage
takes place at 70 to 90 per cent of the ultimate stress and is marked
by a sharp reversal of volumetric strain distribution. A typical,
43
behaviour of concrete under uniaxial compression is given in Fig. 3.1.
This load stage has been described as the critical load and corresponds
approximately to the long-term strength of concrete. The reason
is probably due to the increase of stress concentration effect under
sustained loads.
The mechanism of crack propagation for uniaxial compression
and tension is shown in Fig. 3.2a and Fig. 3.2b respectively.
The crack propagation path may occur(9):
(1) At the aggregate paste interface.
(2) In the cement paste or mortar matrix.
(3) In the aggregate itself.
The point of crack initiation depends on the relative strength of
the cohesive bond and the local state of stress at those three
localities.
To produce a criterion of failure which could be applicable
to concrete and mortar has proved to be a very difficult task. The
principal reasons are (a) the complex variable structure of concrete,
(b) the complicated mechanism of cracking, and (c) the loading
condition and the specimen size.
Nevertheless the failure of concrete caused by crack
initiation and propagation can be broadly classed into two categories
defined by the mode of failure.
1. Brittle or cleavage type failure: This mode of failure
occurs under predominantly tensile stresses and is characterised
by a short stable crack propagation stage where the microcracks propagate
very rapidly around the aggregate cement paste interface and through
the mortar matrix.
44
2. The ductile or shear type failure: Under predominantly
compressive stresses stable cracks are initiated in the mortar matrix
parallel to the direction of applied compression. The cracks
multiply and extend with the increase of load, the fracture path
divides and travels around the particles in a prolonged and stable
manner. Shear stresses are introduced at the aggregate paste
interface which is sustained by so-called aggregate paste shear
bond strength(79)
until the critical stress is reached.
Under biaxial compression-tension there is a transition from
tensile type brittle fracture (short period of stable cracks) to
shear type ductile failure (long period of stable cracks). The
particular compression-tension ratio at which this transition
occurs depends on the internal structure of the concrete.
3.2.2 Behaviour of Concrete under Biaxial State of Stress
The difficulty in obtaining consistent behaviour of concrete
in a biaxial state of stress is attributed to the numerous parameters
influencing the results of the particular experiment. The main
parameters are:
(1) The characteristics of the concrete such as volume
fraction of concrete and paste, aggregate particle size and distribution,
properties of aggregate and the cement paste, and the bond at the
aggregate-paste interface.
(2) The specimen's size and shape, its moisture condition
and temperature distribution.
(3) The actual distribution of stress or strain and the
effect of the specimen, platens and machine upon these distributions.
(4) Method of loading, i.e. rate of increase of stresses.
45
Recently almost all research on the biaxial behaviour of
concrete has been aimed at expressing the behaviour in terms of
progressive microcracking. Microcrack observation indicates that:
(1) The main cause of derivation from elastic behaviour is
the initiation and propagation of microcracks at the aggregate-
mortar interface. The disintegration and ultimate failure of plain
concrete is associated with the propagation of cracks through the
mortar.
(2) The apparent plasticity of hardened concrete on short
time loading (the curving of the stress-strain relation and
significant permanent deformation on unloading) is not connected
with the flow of cement gel or other components, but is due to the
cumulative effects of progressive microcracking.
(3) The main cause of the increase in both strength and
stiffness of concrete in biaxial compression is due to the confinement
of microcracks.
The results of the strain measurements of the overall behaviour
of a test specimen under biaxial state of stress reveals that:
(1) Significantly higher strength is attainable for a given
concrete in biaxial loading than uniaxial loading(5,10,13). The
strength increase depends on the ratio of the principal stresses.
It. (5,10)
appears to be a maximum (up to 25% higher than the uniaxial
value) at a stress ratio of about 0.5 and diminishes as the ratio
is increased to unity (up to 16% higher than the uniaxial value).
(2) The stress-strain curve exhibits a linear response up
to 30-40 per cent of the ultimate strength.
46
(3) The biaxial tensile strength of concrete is
approximately equal to its uniaxial strength.
(4) In a biaxial compression test, the principal strain
ratio remains practically constant throughout the range of loading(13)
(5) The concrete strain corresponding to maximum stress
varies from 0.002 (uniaxial) to 0.003 (biaxial).
(6) The variation of poisson ratio from biaxial tension (0.18)
to biaxial compression (0.20) is negligible.
The theory of microcracking at its present stage can only
describe the behaviour of concrete in qualitative terms. The
experimentally observed stress-strain relation which reflects the
overall effect of these microcracks is still necessary to describe
the behaviour of concrete in uniaxial, biaxial and triaxial state
of stresses. A typical stress-strain relation in biaxial compression
for various ratios of stresses obtained by Kupfer et al.(5) and Lui
et al.(13) are shown in Fig. 3.3a and 3.3b respectively.
3.2.3 The Idealised Constitutive Relation of Concrete in Biaxial
State of Stress
Experimental data on the behaviour of concrete confirms that
the behaviour is linear elastic in tension. In compression the
behaviour is still linear up to the stress level of 30-40% of the
ultimate strength. Based on these results the following constitutive
relationships were adopted for the analytical members.
1. Tension members (TRC3, T-RC2-3 and TRE3)(38,39,40): As
the state of stress is almost tensile in these members concrete was
assumed to be an elastic linear material. The crack occurs normal
to the direction of the principal stress when this stress exceeds
the tensile strength of concrete as shown in Fig. 3.4.
47
2. The partially prestressed I-Beam (B)(8O'81): The
concrete is in compression around the top face of the member. This
concrete compressive stress, however, does not exceed 35 per cent
of the compressive strength of concrete prior to yielding of the
reinforcement. Hence a linear elastic relationship similar to that
for tension members was assumed.
In principal directions the matrix formulation of the linear
elastic, isotropic behaviour can be written as:
E
1
0
1
0
0
0
1-p
El
C2
E12 1-p
2
2
[ a
where p and E are the elastic constants for concrete and [ D is
the elasticity matrix in principal direction.
3. The prestressed box beam(82): The experimental evidence
of the behaviour of this member indicates regions of high compressive
strains near the support and mid-span. For this member a
constitutive relationship for concrete material was adopted based
on the experimental data in biaxial state of stress and suitable
for use in finite element analysis.
Liu et al.(14) developed an expression concerning the non-
linear behaviour of concrete under biaxial compression;
a1
a2
a12
or
a A + BEE (3.1)
(1-pa) (1+CE-FDE2)
48
in which 6 = stress in the direction considered
C = strain 11 It 11 gi
a = ratio of principal stress in the orthogonal
direction to the principal stress in the
direction considered.
E = initial tangent modulus in uniaxial loading
V = Poisson ratio in uniaxial loading
A,B,C,D = parameters to be found so as to satisfy
the appropriate boundary conditions.
for E = 0 6 = 0
for E = o Da ac
= (1-pa)
for c = c P
a = a p for E = C
P ac Da = 0
in which a and c are the peak stress in biaxial compression and
the strain at peak stress respectively.
Introducing those conditions in equation (3.1) and a
assuming Esee = (Esec
= secant, modulus),
a - CE
E (1-pa) [ 1+(
(3.2) 2) 2-- + ( ) 2 1-pa E sec
equation (3.2) gives a general stress-strain relationship for
concrete in biaxial compression. For uniaxial loading, by letting
a = 0 equation (3.2) is reduced to the equation proposed by Saenz(4)
49
and for E = 2Esec
, the reduced expression gives that suggested
by Desayi(3). The above expression compared satisfactorily with
the experimental results obtained by Kupfer et al.(5)
The use of equation (3.2) in finite element analysis may
give rise to problems(83) when the initial stress method is used to
deal with the material non-linearity. In order to obtain stress in
a principal direction (a) from the strain in that direction (6), the
ratio of the two principal stresses (a) is required. The initially
assumed value (ao) may be far from the correct value (a). The newly
calculated value (a1) based on the initial value (a0) may converge
or diverge from the correct value (a), for which a= F(a). The process
requires a close estimate of the initial ratio (ao) and a reasonable
number of back substitutions. The computational difficulty may be
eased by reducing the required accuracy or introducing some
approximate processes in order to approach the correct value. To
avoid such computational difficulty a formulation based on the ratio
of strains, rather than stresses, is'suggested (with a similar
approach) for the behaviour of concrete in biaxial state of stress.
For an elastic, homogeneous and isotropic material the biaxial
stresses and strains can be written as;
1 2 (61 + pc
2)
I-p
(3.3)
.cs2 2
(pc1 + c2)
1-p
in which al, 61 and a2, c2 are the stress and strain in the two
principal orthogonal directions. E and p are modulus of elasticity
and the poisson ratio. Assuming;
50
Cl 1
= =
2 11
Equation (3.3) can be expressed in terms of ratio of strains.
Ecl
12 a1 _ (1 + py
1)
1-p
Ec2
2
a2 = (1 + py2) 1-p
The equation (3.4) can be expressed as
a = EC ( 2
1-p
£ 2 C l Y2
(3.4)
(3.5)
where a = stress in the direction considered
C = strain
y = ratio of strain in the orthogonal direction to the
strain in the direction considered.
Introducing an expression similar to equation (3.1) in
terms of y the ratio of strains;
a A+ BCE l+py 1-p2 (3.6)
(1+Ce + Dc2
)
and satisfying the similar boundary conditions;
aC
for E = 0 0 = 0 and = E ( 1+ p 2y )
de 1-p
for c = £p = 6p and Da
= 0 de
CE 1+py
1 + 14:1it 2) --- + ( --- ) 2 sec 1-p
cs = • 1-p2 (3.7)
51
The parameters A, B, C and D are found.
A = 0 B = 1
. E ( 1 + lly 2 C ) -
a 1 - p2 c P P
1
in which the notations have the same previous meanings, Substituting
the values of the parameters into equation (3.6),.
D - 2
which is similar to equation (3.2), but expressed in terms of ratio
of strains. Comparison of the above equation with equation (3.5)
shows an additional term which indicates the microcracking effects.
Fig. (3.5) compares the result of the above equation (p = 0.2) with
the experimental results of Kupfer(5) et al. in biaxial compression
The peak stress in biaxial compression (a p) can be obtained
from an idealised(14) failure envelope representing the experimental
results of Kupfer et al. (Fig. 3.6), expressed in terms of the stress
ratio a.
a a < 0.2 - -
P a 1 1.2-a
a 1.3 o. 2
0—P = 1.2
52
an 5 ) a ?. 1 = )B
a
a > 5 -2- . (1 + 1 ) a 1.2a-1
where 13, = concrete strength in uniaxial compression. a is
initially obtained from the initial values of the stresses in the
two principal directions. 6 (strain at peak stress) is assumed
to be 0.0025 for the stress in the major principal direction (larger
numerical value), i.e.
I y I < 1 , cp = 0.0025
and for the stress in minor principal direction (smaller
numerical value), i.e.
I y I > 10
3 6 0.0025 - 0.38 y-0.2
fits the experimental results shown in Fig. 3.3 where poisson
ratio is taken to be p = 0.2.
Differentiating equation (3.7) with respect to c,
E [1 - ( 6 ) 12 P E _
E l+py [ 1 + ( • 2) f-- +( E- )2 12
1-p- Esec 1-p
2 6 P
E P
1.2 a p
53
gives the instantaneous modulus of elasticity of concrete.
Assuming concrete as an orthotropic material in which the direction
of orthotropy coincides with the direction of principal stresses, the
assumed relation (3.7) between incremental stresses and strains in the
principal directions can be written in matrix form as follows.
From equation (3.8) we have
and
then,
1 -
E2
E[ 1 - l )2 6 l+py
1
[ 1 E l+p Yi 2) El ( El )2 J2
1-p2
l+py2
+ ( Esec
EE 1
1-p2
Ep
- ( e!a)2
p
=
[ 1
Au
Aa2
Aa 12J
+ ( E l+p Y2 2 + £2 ( —) 6 p
- AE
Ac2 Ac12
2 ]2
or
1-p2
[Aa] = [D] p [Ac-j
E sec
E1
0
0
2) -- 2 1-p
0 0
E2 0
0 G12
where [D] is the incremental elasticity matrix in principal directions.
54
Since the available experimental data do not give any
information about the shear modulus G12, the special case of
orthotropy in which the additional relation,
11 G12 - E'1
1 2p, +
E"'2 E 1
exists among the principal elastic constants was assumed. As
poisson ratio is already considered in the assessment of shear
modulus, the values E".1
and E'2 are hence obtained by separating
Poisson ratio effect from E1
and E2 respectively, i.e.
1 = and E'
2 = Al
i
A2 2
1+py1 l+py2 where Al 2 - and A2
- 1-p 1-p2
E".1 and E'2 reflect the microcracking effect, whereas G12 takes
account of poisson ratio as well as microcracks. In biaxial compression
E1 and E2 were assumed in the principal compressive stress directions.
In biaxial tension E= EA132 E.--EAwere assumed as the behaviour of
concrete is linear up to failure in both directions (E = initial
modulus in uniaxial compression).
In tension-compression case El = Eptiand E2 = E2 were assumed for
concrete in the principal tensile stress and principal compressive
stress directions.
55
3.3 THE CONSTITUTIVE RELATIONSHIP FOR REINFORCING BARS
The stress-strain relationship of the reinforcing bars and
wires were taken from the corresponding experimental data and
represented by mathematical expressions.
1. Tension members: The stress-strain curve for high
strength reinforcing bar No. 8 (American deformed bar) is given(39)
in Fig. 3.7. The relation is linear up to the stress level of
40,000 p.s.i. with E = 28.4 x 106 p.s.i. The non-linear part was
expressed by an elliptical function.
For Part OA 0 < c < 0.00140845 G = 28.4 x 106 E
For Part AB 0.00140845 < E < 0.0106
(a - 104)2
(E - 0.0106)2
1 (3.9) (96250)
2 (0.009625)2
For Part BC C > 0.0106
o = 106250
where c, in/in.
and a, p.s.i.
The instantaneous modulus of elasticity in non-linear
Part AB is obtained from equation (3.9)
56
De
2. PartiaZZy prestressed I-Beam: The stress-strain curve
for high tensile wire of 7 mm diameter (Fig. 3.8a) was formed from
the experimental data(80)
and load deformation of 7 mm prestressing
wire given in reference (85).
Ew = 1.86 x 105 14/mm2
= 1540 N/mm2 fyield
Mathematical expressions are the following:
For Part OA 0 < e < 0.0078
For Part AB 0.0078 < E < 0.01
a = 1.86 x 105 c
a = - 2.433 x 107e2+4.774 x 105c - 791
For Part BC
e > 0.01 a = 1550
2 in which e, mm/mm and a, N/mm .
The stress-strain relationship for 10 mm deformed bars as
non-tensioned reinforcement and 6 mm compressive reinforcement
and stirrups are shown in Fig. 3.8b. The behaviour was assumed to
be elastic-perfectly plastic.
57
mm/mm N/mm2
For Part OA 0 < C < 0.0023 U = 2 x 105 C
For Part AB E > 0.0023 cS = 460
The matrix formulation in the principal directions can be written as:
al
a2 a12
Es
1
Ps
0
Ps
1
0
0 -
1-ps
El E
2
C.12_
-1.12
2
where Es
and ps are elastic constants of the steel.
3. Prestressed box beam: The stress-strain curves of all
reinforcing bars were assumed to be elastic linear.
For prestressing wires
Ew
212206 N/mm2 and fy
w = 1450 N/mm
2
For mesh reinforcements and additional deformed bars
Es = 2022000 N/mm2 and fy
s = 488 N/mm
2
where Ew
and Es are the modulus of elasticity and fy
w and fy
s
are the yield strength.
58
3.4 BOND BETWEEN REINFORCEMENT AND CONCRETE
3.4.1 Nature of Bond
Bond is a complex system by which stress is transferred
between steel and concrete to make an efficient structural member.
Reinforced concrete can only function as a composite material if
the reinforcement is well bonded to the concrete. The complexity
of the bond is associated with number of variables which constitute
boundary conditions around the concrete layer in the vicinity of
the reinforcement. The most important of these variables is the
type of the reinforcement, and this can be classified under two
types namely, plain bars and deformed bars. The surface condition
of these bars is a dominant factor for defining the cause of bond
transfer along the layer surrounding the bar. Various experiments
have indicated a substantially different mechanism of bond transfer
between these two types of bar. While adhesion and friction between
steel and concrete constitute the bond resistance of plain bars, a
considerable bond strength of deformed bars, especially at higher
steel stresses, is due to the irregular surface deformation and
existence of ribs.
Another variable is the strength of concrete surrounding the
bar. While the bond resistance of plain bars is related to shear
strength of concrete, the concrete surrounding the deformed bars is
in compression against the ribs. The confinement of the bar (cover
thickness of concrete), the condition of casting of the bar (vertical
or horizontal) and the position of the bar cast into the concrete
are also found to affect the bond resistance of the bar.
59
3.4.2 Mechanism of Bond in Plain Bars
The surface of plain bars is relatively smooth. The
deformation consists of small pits on the surface of the bar and
some cross-sectional variation of the bar along its length.
The type of deformation on the surface of plain bars does not
usually vary with the size of the bar, hence the bond resistance
of these bars is proportional to the perimeter of the bar. The
bond of plain bars is composed of:
(a) adhesion between cement particles and,the bar;
(b) keying between concrete and irregularities
(variation of pits) of the bar surface;
(c) friction between the bar and concrete.
Though the shrinkage of concrete can increase the adhesion between
cement gel and the bar, the adhesion is usually small in comparison
with the other two factors. The shear resistance of the mortar within
the small depression is related to the keying between concrete and
the bar as shown in Fig. 3.9. As adhesion is destroyed the key of
concrete within the pits (depressions) constitutes the bond resistance
which depends on the irregularities of the surface. The third
component (friction) acts after the mortar projecting into the bar
depression is sheared. This consists of friction between the bar
and mortar (areas indicated by a in Fig. 3.9) and the friction between
mortar and mortar (areas indicated by b). The failure occurs when
this frictional resistance is overcome and the bar usually pulls
out of the concrete.
60
3.4.3 Mechanism of Bond in Deformed Bars
The bond resistance of deformed bars differs radically from
that of plain bars due to the interlocking of the ribs and surrounding
concrete. Adhesion and frictional resistance also exist but the
great improvement of the bond is related to the bearing pressure of
concrete against the lugs. The bond resistance of deformed bars is
composed of three primary elements:
(1) the shearing resistance due to chemical adhesion
between concrete and the bar;
(2) mechanical interaction or frictional resistance after
the adhesion is destroyed;
(3) bearing of concrete against the lugs.
Before loading, some normal compressive stresses exist around
the interface of the bar due to the shrinkage of concrete. Initially,
chemical adhesion combined with mechanical interaction prevent the
slip. As the steel stress increases the adhesion and frictional
resistance become less important in transferring the stress between
steel and concrete, and the ribs of the bar restrain the movement by
pressing on the concrete. The concrete in front of the rib can
sustain a bearing pressure much greater than the cylinder crushing
strength because of its confined condition. The failure or partial
failure which characterises the high resistance of bond in deformed
bars is the splitting of concrete along the thinest cover of the bar.
This splitting develops even before significant slip occurs at the
end of the bar, whereas splitting of plain bars (depending on the
surface condition of the bar) is often after a substantial slip has
61
occurred. The bond resistance of deformed bars is also associated
with the formation of internal cracks in the concrete around the
reinforcement (Fig. 3.10). These internal cracks will increase at
higher loads so that the concrete adjacent to the steel reinforcement
forms a boundary layer of teeth-like segments (Fig. 3.11) which
resist the load. The layer softens with the extent of cracking.
The property of this layer is, however, extremely complex as the
state of cracking depends on the shape of the deformed bar, thickness
cover of concrete and other factors.
3.4.4 The Main Parameters Affecting the Bond Resistance of Deformed Bars
The complexity of the bond performance of deformed bars in
concrete is associated with several parameters of which the following
are considered to be of prime importance.
1. The profile and surface condition of the bar: The
variation in the angle between the face of the rib and the bar axis
affects the bond and slip of the deformed bars(25)
. If this angle
is small (less than 400) and the surface is smooth, slip can occur
along the face of the rib as the rib can push the concrete away from
the bar. If the angle is greater slip is due to the crushing of the
concrete in front of the rib.
The ratio of the rib height to rib spacing has also been
found(24) to be a characteristic of bond performance of deformed bars.
If this ratio is large, (greater than 0.15) i.e. the ribs are high
and closely spaced, the shear of the concrete plane at the top of
the ribs will govern the behaviour (Fig. 3.12). When this ratio is
smaller than 0.1, partly crushed concrete may form a wedge in front
of the ribs, hence the crushing strength of concrete (which can
sustain a bearing pressure several times its normal crushing strength
due to its confined position) will govern the behaviour (Fig. 3.13).
62
2. The strength of concrete: The integrity of a composite
material such as reinforced concrete, requires a compatible strength
of the reinforcement and concrete. To fully utilise a high strength
reinforcement, high strength concrete is essential to maintain a
composite action (bond). The exact relation between the bond strength
and the strength of concrete is not easy to obtain. A concrete with
high strength certainly increases the bond resistance of the bar
particularly at low steel stress levels. At higher steel stresses
the resistance of bond becomes more complex in deformed bars mainly
due to the formation of internal cracks and the state of stress in
concrete between the ribs. The following relationsbetween concrete
strength and local bond strength are recommended (86,87) :
irf7 T = Kirk— 1
T = bond strength
in lb/in? or N/cm2
K,K1 = a constant
fc
= concrete strength in lb/in? or N/cm2
These relations indicate that the bond strength does not increase
proportionally with the strength of concrete.
3. The position of the bar cast into the surrounding concrete:
The settlement of concrete affects the intimacy of contact between the
bar and concrete. An investigation of this problem revealed that(23)
for a vertical bar the consolidation is better at the top of the lugs
than underneath the lugs. The ultimate bond resistance is, therefore,
more when the bar is pulled against the direction of casting of concrete
than in the opposite direction. For horizontal bar, similarly, the
63
consolidation is better above the bar than below it. This is due
to the accummulation of water and air beneath the lugs which causes
a loss of bond strength.
4. The confinement of the bar: An increase in concrete
cover(21) has been found to produce a higher resistance against splitting
in deformed bars. The improved bond performance is not proportional
to the additional cover thickness. This beneficial effect of
increase in cover is not very significant for large size bars. The
widening of the splitting cracks can be restricted (improved in bond
resistance) if the concrete itself which surrounds the bar is confined
by some kind of transverse compression applied to it due to the
structural actions. The addition of stirrups can also prevent the
opening of the splitting crack along the bar.
3.4.5 A Brief Description of Some Experimental Bond Tests
The bond performance of various reinforcing bars was formerly
determined largely with pull out tests (Fig. 3.14a). In these tests
the force required to pull the embedded bar, the loaded end and the
free end slip were measured. The bond efficiency of the individual
bar was then assessed as the average bond stress and was plotted
against free end or loaded end slip. The bond stress distribution
along the bar is non-uniform. The loaded end may have reached the
ultimate bond strength though the average bond stress may be small.
In this test the adjacent concrete is in compression. This transverse
compression has a beneficial effect on the bond strength and is not
therefore typical of a practical situation. The test arrangement
of Fig. 3.14b is an improvement to the test set up in Fig. 3.14a
in that due to the short length of the bar the bond resistance can
be obtained locally. Various forms of test specimens have been
64
developed to eliminate transverse compression. The bond beam
test (Fig. 3.14c) measures the average bond stresses of the length
t at the ends of the beam. Nevertheless some transverse compression
is applied to the bar from the supports. In the bond test
arrangements of Fig. 3.14d the transverse compression is eliminated.
The measurements of bond and slip in this tensile pull out test at any
location may be affected by the formation of internal cracks even
at small steel stresses. In short specimens Fig. 3.14e formation
of internal cracks may be avoided and the slip at-the loaded end is
close to that of the free end. The bond and slip in this test, however,
may be affected by the reaction of supports.
3.4.6 Idealised Bond-Slip Relation Adopted for the Analysis
The results of the experimental tests on bond performance of
deformed bars have been mostly associated with the relative strength
rather than the real value. The effect of the parameters influencing
the bond strength of a bar has not been determined sufficiently and
independently by tests. It is also true to say that the measurement
of bond and slip is not independent of the test arrangements. The
analyst, however, seeks to model a constitutive relationship for bond
and slip of a point (similar to the stress-strain relationship for
steel and concrete) which could be applicable to his particular
problem. A reliable experimental relationship between the unit
bond stress and the corresponding slip is not available for analytical
purposes. It was, therefore, decided to adopt an idealised bond-slip
relationship for the present analysis based on the following assumptions:
(1) The magnitude of bond stress at any point within the
member is expressed in terms of the slip of that point. The relation
is independent of the location of the point and the surface condition
of a particular deformed bar.
65
(2) The magnitude of bond stress is proportional to the
square root of the concrete strength.
(3) The bond-slip relation along the reinforcement was
assumed to be the form of a multi-linear material with a small
non-linear region.
(4) The bond-slip relation perpendicular to the reinforcement
was assumed to be weak in tension (representing the separation between
the two materials) and strong in compression (transverse pressure
between the two materials).
(5) Certain cracks in concrete adjacent to the bar will affect
the assumed relation (along and perpendicular to reinforcement) in a
destructive manner.
(6) Transverse cracks according to their width and extension
can reverse the bond stress. This is provided for in the relationship
by unstressing and restressing in the opposite direction.
1. The bond-slip relation along the reinforcement: The
initial linear region (line OA in Fig. 3.15) in which a considerable
amount of bond resistance is developed at a very small slip, is
related to the adhesion, friction and the bearing pressure of concrete
against the ribs. The maximum value was taken to be EhrE lb/in2
(f in lb/in2) or 1-1/- c N/mm 2 (f in N/mm2) with the corresponding
12 2.54
slip of 0.0001 in.0( 1 mm). The non-linear region (AB) is assumed
000
to represent the increasing pressure of concrete against the rib.
In this region the resistance of bond is mainly due to the compression
in concrete and adhesions and frictions are insignificant. The
maximum value of this region (bond stress at point B) corresponds to
66
the local bond strength. The magnitude was taken as 11VT lb/in2
(-- c N/mm
2) with the corresponding slip equal to 0.003 in 11
12 76.2
(1000 -----mm). The horizontal region BC is related to the reistance of
the partly crushed concrete against the rib and formation of
compacted powder there. The final slip of the horizontal region is
254 assumed to be 0.01 in (-----1000
mm). The gradual loss of bond resistance
508 after point C is given by CD up to the slip of 0.02 in (1
000 mm).
From point D the very small resistance of bond is due to the inter-
locking of crushed concrete between the two ribs and average of
21/T- lb/in2 12 IT N/mm2) is given to the straight line DE. The
above behaviour is assumed for a location which is not disturbed
by the formation of certain cracks in the adjacent concrete element.
Table 3.1 compares the assumed bond strength with the recommended
maximum bond stress of deformed bars. Table 3.2 shows how the maximum
assumed bond strength varies with the strength of concrete. The
mathematical expression of Fig. 3.15 is given in Table 3.3. As
the relative movement of concrete with respect to steel can be in
positive or negative direction (the slip is positive when the movement
of concrete is algebraically greater than the movement of the steel),
an identical bond-slip relation exists for the negative values of
the slip and bond stress along the bar.
2. The bond-slip relation perpendicular to the reinforcement:
The bond-slip relationship perpendicular to the bar is shown
in Fig. 3.16. The behaviour is linear elastic when there is a
transverse compression between the bar and concrete. In tension
the behaviour is of brittle nature similar to the behaviour of
concrete in tension. The maximum transverse tension was assumed
67
to be 6/T lb/in (0.51/T- N/mm2) which is comparable to the tensile
strength of concrete. Line 0".C. represents the elastic tension
which occurs before the separation of concrete perpendicular to the
reinforcement. Line C'IY. indicates the gradual destruction of
bond perpendicular to the bar. Part D'E' represents a small
constant interlocking of concrete. The above behaviour is assumed
for a location which is not distrubed by the formation of certain
cracks in the adjacent concrete element. The mathematical expression
of Fig. 3.16 is given in Table 3.4.
3. The gradual loss of bond due to the formation of certain
cracks: If a concrete element adjacent to steel has
cracked such that the surrounding bond elements perpendicular to the
reinforcement are in tension (separation of concrete and the bar),
the bond-slip relationship along and perpendicular to the bar will be
affected by that crack. This crack in the analytical model, which
could represent the splitting cracks in the experiment causes a
gradual destruction of bond in the surrounding bond elements. Supposing
that the bond stresses adjacent to the concrete element are
at point M (Fig. 3.15, along the reinforcement) and 11' (Fig. 3.16
perpendicular to the reinforcement). After the formation of that
crack in concrete, they will be assumed to follow the lines MD and
M D respectively.
4. Reversal performance of bond stress: In general the
formation of a crack in concrete adjacent to steel will redistribute
the existing bond and slip. The effect of a crack depends on its
inclination, width and extension. This crack may cause a reversal
behaviour of bond along and perpendicular to the reinforcement. This
68
behaviour is idealised in the given bond-slip relationship as
unstressing and restressing in the opposite direction in the
respective bond element. Bond stresses at N (Fig. 3.17, along
the reinforcement) and N' (Fig. 3.18, perpendicular to the
reinforcement) may then follow the lines NON1 and 14"0"N1 and
travel towards the reverse curve and lie there. If the effect
of a crack is insignificant, the bond stresses may not even reach
point 0 and 0" and remain at arbitrary positions P and P"
respectively. This behaviour can also occur fora bond element
adjacent to a splitting crack as shown by points m, n in Fig. 3.17
and m',n' in Fig. 3.18.
3.5 NUMERICAL TECHNIQUES FOR MATERIAL NON-LINEARITIES
3.5.1 Statement of the Problem
A non-linear material structural problem must obey the
conditions of continuum mechanics, i.e. equilibrium, compatibility
and the constitutive relation of material. The condition of
compatibility is automatically satisfied at the nodes in the finite
element technique, hence the problem is to satisfy the constitutive
relation while preserving the equilibrium of the structure. In the
various methods dealing with the material non-linearity, the non-
linear solution is obtained by solving a series of linear problems
so that the appropriate non-linear condition is satisfied to a
specified degree of accuracy. The solution of non-linear problems
in the finite element method is usually attempted by one of the
three basic techniques.
69
(1) incremental or stepwise procedure
(2) iterative procedure
(3) mixed procedure
For simplification the non-linear constitutive relationship
of an element,
[a] = f ([C]) (3.10)
is considered where the stiffness matrix [ K of the element is a
function of the above non-linearity,
K = K(E,cr) (3.11)
Fig. 3.19a shows a typical stress-strain curve at a point within
the element, say at the element centroid. The nodal load-displacement
relation of this element can be written as;
[ 6 ] = F
(3. 12)
where [ F ] is the nodal forces and [ 6 ] is the nodal displacements
of the element. Fig. 3.19b shows a typical load-displacement
relationship of a node.
3.5.2 Incremental or Stepwise Procedure
The solution of non-linear material behaviour by incremental
procedure is through the sub-division of the load into many small
increments. The load increments are not necessarily equal. The
70
nodal loads are applied one increment at a time and during the
application of each increment the equation (3.12) is assumed to
be linear, i.e. a fixed value of [K ] = EK(c c , ac)' assumed
throughout each increment. The solution of each step of loading
is obtained as an increment of nodal displacements [ AS ] and is
added to the previous accumulated displacement increments to
give the total nodal displacements. [ K ] may take different values
during different load increments. This incremental process is repeated
until the total load is reached. At the application of the ith
increment, the strains and stresses at the centre of the element are
given by;
[o.] = E [ Aa ] j = 1
E[ e .] = Ac j = 1
(3.13)
(3.14)
and the nodal loads and displacements of the element are given by;
[ Fi] = / [AF ]j j = 1
i [Si] E [OS].
1 j = 1
The incremental displacement is found from equation (3.15) in which
E K 1 is evaluated at the end of the previous increment.
E K.1-1 1 [ AS.1 ] = [ AF 1
(3.15)
71
and the stiffness matrix to be used for the next increment is;
[KJ= [Ic2. . (e ., a.) ]
(3.16)
whereEFjand [ E., U. I are the total loads and the total strains
or stresses reached at the end of ith increment., Fig. 3.19c shows
a typical nodal load displacement or centroidal stress-strain
relation in this procedure.
The accuracy of the incremental procedure can be improved by
taking smaller increments of loads at the cost of additional computational
effort. However there are improvements which use some additional
computer cost to better advantage than simply increasing the number of
increments. In the mid-point scheme two cycles of analysis are
performed for each load increment. In the first cycle half of the
incremental load is applied and the stiffness matrix corresponding to
the total stress at the end of half cycle is evaluated and utilised to
compute an approximation to the full increment.
3.5.3 Iterative Procedure-
The iterative procedure is a sequence of calculation in which the
element is fully loaded in each iteration. Due to the assumed initial
constant value of stiffness [ K J = [ K(E , a) ] the constitutive c c
relation (3.10) is not necessarily satisfied. A correction is made
to satisfy this relationship and the self-equilibriating forces due
to this correction are calculated. To maintain equilibrium, a set
of opposite self-equilibriating forces (unbalanced forces) are applied
to the nodes in the next iteration. The process is repeated until
these unbalanced forces become sufficiently small, i.e. the constitutive
relationship is approximated to some acceptable degree. The solution
of such a problem can be accomplished in a large number of ways which
depend on the method of computation of the stiffness matrix EK
and the unbalanced forces.
72
Computation of Unbalanced Forces
In general a constitutive law of a type
Co'J = CD] [E]
(3.17)
which defines a non-linear relationship between stresses and
strains can be written in the form of
[a] = Eao ] [ 5 ]([ 6 1 [co l)
(3.18)
where E D I is a constant linear elastic matrix. Initial stresses
[ao and initial strains [co
] are variables of such a magnitude
as to make up the difference between equation (3.17) and (3.18). One
can concentrate all the corrections in either initial stresses or
initial strains and hence calculate the unbalanced forces. The choice
of method and the speed of convergence obviously depend on the
constitutive law governing the behaviour of the material.
1. Initial strain method: In this method the equation (3.18)
is simplified to
[a] = [B] ([ 6 ] - [ co ])
(3.19)
assuming co = 0 initially, this equation is solved with an
appropriate [ 5 ] matrix to obtain a certain level of stress [a1]
and strain E ela where
or
73
whereas the strain which should have occurred is
[en) ] = [DV. Cal ]
The difference of strains [ = [ lb ] - [ cla ] is now
used as an initial strain in equation (3.19) and the equivalent
unbalanced nodal forces
[Fun =-fxr [ B r [5] CEO ] d vok
are applied. The process is repeated until [Col or [Fun]
become negligible and the final solution is reached.
2. Initial stress method: In this method the equation (3.18)
is simplified to
Ea I = [B] CE:1 C6c,1
(3.20)
with [aj = 0 initially, this equation is solved with an
appropriate matrix [ 5] to reach a certain level of strain 1 ]
with the corresponding stress where
Eala] = [ 5 ] [Cl ]
while the correct stress which should have occurred is
[ alb] = CD] [el l
74-
The difference of stresses cio = E ala -I alb ] is now
used as an initial stress in equation (3.20) and the equivalent
unbalanced nodal forces
Fun f [B] T [aa ] dvoZ
are applied. The process is repeated until no appreciable change
between the two successive iterations exist,i.e. o or
Fun] become negligible.
3.5.3.1. Constant Stiffness Process
In this approach a constant initial stiffness matrix
• [ Ko = , 6c) ] is used throughout the analysis (equation (3.12)).
The problem is solved using initial strain or initial stress method
for the calculation of the unbalanced forces. The successive
corrections are estimated to finally produce the given constitutive
law. Figs 3.19d and 3.19d show the initial strain or initial stress
method performed for a point within the element, say at the centroid.
A typical nodal force-displacement relationship is given in Fig. 3.19d.
3.5.3.2. Variable Stiffness Process
In this approach a linear solution is performed each time with
the ED ] matrix adjusted from the stress or strain level reached
at the previous iteration. The unbalanced forces are similarly
calculated by initial strain or initial stress method. A typical
nodal force-displacement relationship using secant modulus or tangent
modulus to form the stiffness matrix at each iteration are shown in
Figs 3.19e and 3.19f respectively. The corresponding initial strain
or initial stress method in these processes performed at the centroidal
75
point of the elements is given in Figs 3.19e' and e" and 3.19f' and
f".
3.5.4 Mixed Procedure
The mixed procedure utilises a combination of the incremental
and iterative schemes. The load is applied incrementally but
successive iteration is performed after each increment as shown in
Fig. 3.19g.
3.5.5 Comparison of the Basic Procedures
The advantage of incremental procedure is that it provides a
relatively complete description of the load deformation behaviour.
The results are obtained at each of the intermediate states corresponding
to an increment of load which reproduces most correctly the physical
' behaviour. The disadvantage of this procedure is the difficulty to
know in advance what increments of load are necessary to obtain a
good approximation to the exact solution. This method is usually
more time consuming than the iterative procedures.
The iterative procedures are easier to use and faster. The
method is very useful in the case in which the materials have different
properties in tension and compression. The constant stiffness process
(Fig. 3.19d) necessitates a greater number of iterations, however
the saving of computation is considerable as it is not necessary to
invert a new stiffness matrix at each cycle. In variable stiffness
procedure as the stiffness matrix is modified in each iteration
(Figs 3.19e and 3.19f) the rapid convergence to the solution is
obtained at a relatively considerable cost of computational time.
The main disadvantage of iterative procedure is that there is no
76
assurance that it will converge to the exact solution and if
several possible solutions exist, most of the methods will lead
to one possibility only, which may differ according to the method
used. The limitation of the iterative procedure is that the
displacements,stresses and strains are determined for only one
load increment.
The mixed procedure (Fig. 3.19g) combines the advantage of
both the incremental and iterative procedure and minimises the
disadvantage of each method. It can describe the load deformation
behaviour at each load increment with the desired equilibrium accuracy.
The convergence can be more accelerated for subsequent load increments
since the stiffness matrix can be updated with the knowledge that
the equilibrium conditions are fulfilled to a specified degree of
accuracy. The additional computer time is justified by obtaining
higher accuracy in this procedure.
3.5.6 Non-linear Techniques Adopted for the Analysis
Different mixed procedures were used to deal with the material
non-linearities of the members analysed. The unbalanced forces were
calculated using initial stress method. For a strain reached at
the end of each iteration, the difference between the stress obtained
by the analysis and the correct stress from the constitutive law of
the material, was used as the initial stress and converted into
nodal forces. These forces were then applied to the structure in
the next iteration. The stiffness of the structure was updated
according to a pre-defined condition. The sequence of operation
is given later in Chapter 8.
For the reversal behaviour of bond elements a mixed procedure
using secant modulus was proved to be the only successful method.
77
The constitutive behaviours and the procedures are shown in
Fig. 3.20. The bond along the reinforcement (Fig. 3.20a) may
perform a reversal behaviour, the secant modulus approach ensures
that the element is unloaded first (zero stress) before it is
reloaded to the opposite direction. In the direction perpendicular
to reinforcement (Fig. 3.20b) a bi-modular material exists with
different elastic properties in tension and compression. This
behaviour is best simulated via the secant modulus approach.
For monotonic non-linear behaviour of concrete and steel
materials a mixed procedure using tangent modulus approach was found
to be most efficient (Figs 3.21 and 3.22). The process requires
less iterations per load increment.
78
N/cm2 N/mm
2 lb/in2
Recommended Maximum Values
CEB(86) CP110(88)
fb < 4.10 fb
ACI(87),-- lb/in 9.50 Vfc
- 800 I fc
fb = 3.45 /
2 _D ‘
Assumed Maximum Value
fb = 9.16 ii-c- f
b = 0.916 V1T- c
fb = 11 lc
fc Concrete strength in N/cm
2, N/mm2, lb/in
2
fb
= Bond strength in N/cm2, N/mm
2, lb/in
2
D = Bar diameter in in.
Comparison of the assumed and recommended bond stress
Table 3.1
N/mm2 lb/in2 N/mm
2 lb/in2 N/ram2 lb/in2
f 20 3,000 .0. c
40 6,000 70 10,000
fb
4.10 600 5.80 850 7.66 1,000
Variation of the maximum assumed bond stress with
the strength of concrete
Table 3.2
?9
Part x y
OA
AB
BC
CD
DE
0.0001 > x > 0
0.003 > x > 0.0001
0.01 > x > 0.003
0.02 > x > 0.01
x > 0.02
y = 8 x 104x
(x-0,0036)2 (Y-7.5)2
1
+ 2
+ = (0.00355)
2 (3.55)
2
y = 11
y = -900 (x - 0.02)
y = 2
x Slip, in.
fb Bond stress, p.s.i.
Bond-slip relation along the reinforcement (Fig. 3.15)
Table 3.3
Part x'(in) y'
C" 0' C' x' < 0.0004 y' = 1.5 x 104x
C' D' 0.0004 < x' < 0.02 y' = -255(x-0.02) + 1
D' E' x' >- 0.02 ' = 1
x = Separation or contraction, in.
fl = y' _ -c Tension or compression, p.s.i.
Bond-slip relation perpendicular to the reinforcement (Fig.3.16)
Table 3.4
80
stress Uniaxial strain
Volumetric
.8
.6
.4 STRESS-STRAIN RELATIONSHIP
.2 FIG. 3.1
Strain
Crack propagation Crack propagation
Tensile stress concentrations
(a) (b)
T Uniaxial Tension
Uniaxial Compression
€3 .,--- -, ... .... ... -~ f-- t 3 -
'" ~ -£2'£3
- r- +6j
g~~ oii! 1-02 ,Nt-: I
==- ¥.~(2in) (79 In)
+3 tensil£' strain
+1
, ,
\
\~ \ \\ \\\ \\ \\ \\
6j pp = -328 kp/cm2 (4650 psi) (3; f ¥
1.2-A~ .., ........ €,-
L " 1.1-j-
:;-- ......"
/' i~,·t2 1.0 I
09 1 ;// ---, £,--I /1/ O,,J
oj /// I /1/ 6j 1 6'2
I 1/ -- -11 0 --- -11-1 -Iff ---- - 1/-Q52_
LI JI i/ Y €t.~.E3
o -1 -2 -3 mmlm(QOOlinlin) compressIve stram
(a) Kupfer et al (Ref.5) (b)
TYPICAL STRESS-STRAIN RELATIONSHIP OF CONCRETE IN BIAXIAL COIvJPRESSION
11---<1.1.0
500 1000 I~OO 2000 2500
STRAIN (M'CROIKIIN)
Liu et al (Ref. 13)
Br 12
1.
oa /0s.-. 1. .6
K1= ei /es, .4
el, ea te
.2
0 —1 -2 -3
(b)
FAILURE ENVELOP FOR CONCRETE PP UNDER BIAXIAL STRESS
FIG. 3.6
▪ .-no kp/cm2 (2700 Pei)
▪ p°=-315 Sp/cm' (6450 psi)
•—. p,.-590Sp/cm, (8350 psi)
Experimental -02
FIG.3.5
Idealised 2 1. .8
(a)
6'2
Experimental =-328kg./em2(4650p.s.i)
Idealised
COMPARISON OF PLOTS BASED ON RATIO OF STRAIN( =O.2)WITH THE EXPERIMENTAL
TEST(Kupfer,Ref.5) ASSUMED BEHAVIOUR OF CONCRETE IN TENSION MEMBERS AND PRESTRESSED I BEAM
FIG.3.4
B C •■•••••••••
Test Result (Ref.39)
----- Mathematical Simulation
STRESS-STRAIN CURVE FOR REINFORCING BAR (Tension Members)
FIG.3.7
Strain 8 9
110
100
90
80
70
60
50
40
30
20
10
SR
1600
1400
1200
500
1000
cvg 400
• 300 cu
800 4.1
•
200 rn
100
0 1 2 3 4 5
STRESS-STRAIN CURVE FOR NON-TENSIONED STSEI(Ref.80)
FIG.3.8b
200
StrainX103
1 2 3 4 5 6 7 8 9 10 11
STRESS STRAIN CURVE FOR HIGH TENSILE WIRES OF PRESTRESSED I BEAM(Ref.80,85)
FIG.3.8a
600
400
StrainX103
a/e<6.10 PIG.3.13 a/C>0.15 FIG.3.12
Inclined internal cracks t>
V . I - 7.\.
LN : t>
• V • . A . . . Splitting
FIG.3.11
Concrete
FAILURE MECHANISM AT THE RIBS OF DEFORMED BARS
. / Compacted Powder / /
C
C>Orlisiied . V. • . • • . concrete . • . 4 .'
• . • C7 /
83
MECHANISM OF BOND IN PLAIN BARS
FIG.3.9
FIG.3.10
MECHANISM OF BOND IN DEFORMED BARS AND INTERNAL CRACKING
1 84
11-
ONO
I (a) (b)
Pull—out test Pull—out test
4_____4 ,1
(e) Bond beam test
(d)
Tensile pullout test Pull—out test
of short specimen
F1G.3.14
85
Slip 0 I I t I I I I I ,
0 1 2 3 4 5 10 15 20 i n/ woo 254 50,8 127 254 381 508 mm/moo
12 "v/\/}:- N /mm2 tv/Vr Rs!
C
FIG.3.15
2, Bond
Str
ess
Tension
15 381
10 254
0 43
a)
0 1 2 3 4 5 25,4 508 127
D Slip ,
20 in/ woo 508 mm /moo
IDEALISED BOND—SLIP RELATIONSHIP PERPENDICULAR TO THE REINFORCEMENT
FIG.3.16
6
5-
4-
3-
1
0
.2
-3
A
-5
-6
-7
IDEALISED BOND—SLIP RELATIONSHIP ALONG THE REINFORCEMENT
Compression
Bon
d S
tres
s
10
D n/ moo
D
-10
86
-20 -15 -10 -5 0 5 10 15 20
slip
REVERSAL PERFORMANCE OF BOND ALONG THE BAR
FIG.3.17
PsJ
!IV
-A4L4 In/l000
10 15 20 Slip
4
REVERSAL PERFORMANCE OF BOND PERPENDICULAR TO THE BAR
FIG.3.18
Bon
d. S
tres
s
Initial Stress Method Constant Stiffness Procedure Initial Strain Method F
PLOTS OF BASIC PROCEDURES
FIG.3.19
Incremental Procce Load Displacement Relation
(a)
E.
Stress Strain Relation
F
87
(b)
F.Ki K=K(6c4)
Procedure Variable Stiffness F(Secant Modulus Approach)
Ft
Variable Stiffness Procedure Initial Strain Method
(Tangent Modulus Approach) F
Initial Stress Method
Initial Stress Method
Initial
(e')
Strain Method
Mixed Procedure (Tangent Modulus Approach) (g)
Bond Stress
A
Slip
Bond-Slip along the
reinforcement
Bond Stress
Mixed Procedure
(Secant Modulus Approach)
FOR REVERSAL BEHAVIOUR OF BOND
FIG.3.20
4967. ■,
■ 226.
Slip
Bond-Slip perpendicular
to the reinforcement
Compression
61,P2
Mixed Procedure
(Tangent Modulus Approach)
FOR MONOTHONIC BEHAVIOUR OF
STEEL BAR IN TENSION MEMBERS
FIG.3.21
Mixed Procedure
(Tangent Modulus Approach)
• FOR BEHAVIOUR OF CONCRETE IN
BIAXIAL DIRECTION(PRE-STRESSED
BOX BEAM)
FIG.3.22
E,2
88
89
CHAPTER 4
FINITE ELEMENT FORMULATIONS
4.1 INTRODUCTION
The finite element method is essentially a process through
which a continuum with infinite degrees of freedom is substituted
by an assemblage of individual structural components or elements.
The structure must consist of a finite number of such elements,
interconnected at a finite number of joints or nodal points at
which some fictitious forces, representative of the distributed
stresses, actually acting on the element boundaries are supposed
to be introduced. If force-displacement relationships of the
nodes of the individual elements (usually expressed by flexibility
or stiffness matrix) are known, it is possible by using various
techniques of the structural analysis to study the behaviour of the
assembled structure.
The new idea in the finite element method is not the structural
approximation of the continuum, i.e. there is no need for approximation
in the mathematical analysis of this substitute system, but rather
the use of two or three dimensional structural elements This
feature distinguishes the finite element technique from finite
difference or other methods in which the exact equations of the actual
physical system are solved by approximate mathematical procedures.
The important extension of normal structural analysis
procedures which was introduced by the finite element method is the
use of a system of two or three dimensional structural elements to
represent an elastic continuum. Using these elements the structural
90
idealization is obtained merely by dividing the original continuum
into segments of appropriate sizes and shapes, all of the material
properties of the original system being retained in the individual
elements. This capacity for treating arbitrary material properties
is one of the principal attributes of this method.
Either of the two basic approaches (force method or
displacement method) to structural analysis may be applied satisfying
equilibrium, compatibility and the constitutive properties of the
elements. As it has been found that the displacement method generally
provides the simpler formulations and computer programming task,
this method was chosen for the finite element analysis of the members.
4.2 THE FINITE ELEMENT DISPLACEMENT PROCEDURE
The finite element displacement method of analysis of an
elastic continuum may be divided into three basic phases.
1. Structural idealization: The original system is separated
by imaginary lines or surfaces into a number of finite elements.
Though an appropriate subdivision improves the results, in general,
the idealization is not a difficult problem. The elements are
assumed to be interconnected at a discrete number of nodal points on
their boundaries. The displacements of these nodal points are the
basic unknown parameters, just as in the simple structural analysis.
2. Evaluation of the element properties: This evaluation
which is the choice of the deformation characteristic (displacement
function) of the elements is the critical phase of the operation.
For a reasonable representation of the actual continuum each element
must be required to deform similarly to the deformation developed in
the corresponding region of the continuum. The characteristic of
91
the element is represented by the relation between the forces
applied to the nodal points and resulting deformations of the nodes
(the stiffness of the element). A set of functions is chosen to
define uniquely the state of displacement, hence strain and stress
within the element in terms of its nodal displacement. Then a
system of forces concentrated at the nodes and equilibrating the boundary
stresses is determined resulting in a stiffness relationship of the
element.
3. Analysis of the element assemblage: When the element
properties have been defined, the analysis of the stresses and
deformations resulting from any loading condition is a standard
structural problem.
In any structural analysis, the essential problem is to satisfy
equilibrium, compatibility and material property relationship
simultaneously. The above idealization clearly introduces a series
of approximations.
(a) The assumed displacement function will not usually satisfy
the compatibility of deformation of the adjacent element. If a
deformation pattern is specified which provides internal compatibility
within the element and,at the same time,full compatibility of
displacements along the boundaries, then the strain energy in the
idealization will represent a lower bound to the strain energy of
the actual continuum(84)
(b) By concentrating the equivalent forces at the nodes the
equilibrium condition is satisfied at those points only, and not
necessarily within each element and along the element boundaries. These
artificial boundary loads are local and self-equilibrating which have
little influence on the general behaviour of the structure.
92
A sequence of solutions to the problem may be obtained
using successively finer meshes of elements. The sequence may
be expected to converge to the correct result if the assumed
displacement function satisfies certain criteria.
(1) It should not permit straining of an element to
occur when nodal displacements are caused by a rigid body displacement.
The violation of this requirement usually delays but does not prevent
convergence to the true solution.
(2) When the nodal displacements are given values corresponding
to a state of constant strain, the displacement function must produce
the constant strain state throughout the element. The motivation
for this requirement is that a small enough piece of any continuum
may have a simple state, such as constant strain. Violation of this
requirement can result in convergence to an incorrect result.
4.3 FORMULATIONS OF THE ELEMENT STIFFNESS MATRIX
The detailed derivation of an element stiffness matrix is
described in many relevant text books(65-89) The standard procedure
is as follows.
(1) Express the displacement f at a point within the element
in terms of an arbitrary displacement function
f (x,y) = [M(x,y)]x[ a (4.1)
This displacement function should satisfy internal compatibility.
[a ]is the amplitude of the displacement function as undetermined
coefficients. Let the number of these coefficients be equal to
the total number of nodal points displacement components in each
element.
93
(2) Evaluate nodal displacements E e =[ if j ...]
by substituting the coordinates of the nodal points into the
displacement function matrix [M].
[61e = CA]x[al
(4.2)
vector E a ] will then be determined in terms of the nodal
displacements.
Ea] = [A1-4 x Die (4.3)
Equation (4.2) can then be written in terms of nodal displacements
fC x,Y1 = EN(x,Y) 1x C 6 le (4,4)
where C N(x,y) ]= E M(x,y) ]x[A] -1
is the shape function giving the displacement of a point within the
element in terms of the nodal displacements.
(3) With displacement known at all points within the element
the strain at any point can be determined by appropriate differentiation
of equation (4.4).
c (xiar) = [ B (x,y) J x[o
(4.5)
where matrix [BI is called the element strain matrix giving strains
in any point within the element in terms of the nodal displacements.
91i-
(4) Evaluate the element stresses- [
[ c (x,y) ] = [ D ] x [e (x,y) = [ D J x B(c,y) x [ S i e (4.6)
The specific elastic characteristics of the finite element material
are represented by the stress-strain matrix [ D J. These may be
isotropic, orthotropic, elasto-plastic or any other specified
characteristics.
(5) Let [ F ]e = F., Fj ... J
define the nodal forces which
are equivalent statically to the boundary stresses on the element.
Applying the principle of virtual displacement, i.e. to impose an
arbitrary virtual nodal displacement (d[c3 1e) and to equate the
external and internal work done by the various forces and stresses
during that displacement.
(d[ cS ]e) Tx [ F ] = f d[ ]T x [ dv e vol
or
(d[ e) [ F ] e = (dE 6 le) T uvo, B fx Idv)
or
F ]e = (Iva, [ B ]x [ D ]5([ B ]dvoz) [S ] e (4.7)
The stiffness matrix of the element is obtained as:
[1(]e =fvo9, EBrx[D ]x[B]dvot (4.8)
which is seen to be symmetric, i.e. [ K e = [ K ]e . By locating
95
the stiffness matrix of each element into the structural stiffness
matrix [ K J and the element nodal forces in the structural nodal
force [F ], the force-displacement relationship of the structure
can be expressed as:
[F] = [ K x [ 6 ]
where [ S ] is the unknown nodal displacements of the structure. The
symmetric stiffnes matrix E K j is singular because of the inclusion
of rigid body motion. Boundary conditions must be introduced before
the inversion of the matrix becomes possible. The unknown nodal
displacements are then obtained as
-1 [o] = LK] X [F]
The strains and stresses in each element are obtained from
equations (4.5) and (4.6) respectively.
4.4 THE FINITE ELEMENT TYPES SELECTED FOR THE MEMBERS
Various elements have been developed for the analysis of plane
stress problems. For a good representation of the actual state of
stresses in a continuum using simple elements, the number of these
elements must be increased which increases the total number of
unknowns (nodal displacements). On the other hand, as a general
rule , when the order of an element is increased, the total number
of unknowns can be reduced for a given accuracy of representation.
While for these higher order elements the equation solving times may
be reduced, the time required for element formulation is increased.
96
The question may be asked as to whether any advantage is gained
by increasing the complexity of an element. In selecting the
particular type of element, however, the analyst may be aided by
his structural intuition and physical insight to the actual behaviour
of the structure.
According to the experimental evidence a large number of
cracks were seen to have developed in the members selected for the
analysis. The approach for dealing with the problem of cracking
in the finite element method is, to assume a zero stiffness (for any
type of element) perpendicular to the crack direction. It was
therefore decided to use a large number of first order (linear)
rectangular element for the representation of the concrete material.
The elements used for the members are the following:
1. Tension members: Plane stress rectangular finite elements
with four corner nodes (Fig. 4.1a) were used for concrete material.
Bar elements (Fig. 4.1b) represented the reinforcing steel and the bond
between steel and concrete was idealized by linkage or bond elements
(Fig. 4.1c).
2. Bending members: The same plane stress rectangular elements
were employed for concrete in bending members with a modification in
which the shear strain was assumed to be constant over the element
and equal to its centroidal value(63). In the partially prestressed
I-beam the steel reinforcement was idealized by the same modified
shear rectangular elements similar to concrete elements. The linkage
elements were employed for representation of bond between the two
materials. The steel stirrups and compressive reinforcement of
- Aa
Aa y
Au xy
- -1 o 1 o Av 1
O 0 0 0 X
O 0 0 0
Es L Avg
Av3
Av4
97
this member and the entire reinforcing steel of the prestressed
box beam were represented within the corresponding rectangular
concrete element, resulting in a composite element. Hence perfect
bond was assumed for the above reinforcement. The idealized members
selected for the analysis are shown in Appendices 1 to 4.
As the local coordinates of the elements used in the members
coincide with the global x-y direction, the stiffness matrix of
each element is directly formed in the global axis, i.e. no displacement
transformation was required.
4.5 STIFFNESS PROPERTIES OF THE ELEMENTS
4.5.1 Bar Elements
The strain displacement relationships of a typical bar element
with two nodes along its axis (Fig. 4.1a) is given by
Ex
V3 - V1) 1 L
where L is the length of the bar.
In matrix form
Ac x 1 L
-1 0 1 0
O 0 0 0
O 0 0 0
Av
K Avg
Av3
Av4
Acy
Ac xy
The incremental stress-displacement relationship will be:
98
where Es
is the incremental modulus of the bar element (tangent
modulus) given in Chapter 3. The incremental force-displacement
relationship is given by
F1
F2
F3
F 4
AEs
1
0
-1
0
0
0
0
0
-1
0
1
0
0
0
0
0
- AV1 -
AV2
AV3
AV4
L
or
[SF] = Ki' I x [AV I
(4.10)
where A is the cross-section of the bar and [ Ks] is the stiffness
matrix of the bar for the given increment.
4.5.2 Linkage Elements
The linkage element can be conceptually thought of consisting of
two springs with certain defined properties in the two directions(67)
(Fig. 4.1c). The linkage element has no physical dimension and
can be placed anywhere without disturbing the geometry of the structure.
Only the mechanical properties of these elements are of interest.
The incremental relationship between the slip and nodal displacements
along and perpendicular to the bar axis is given by
Ah
AV
-1
0
0
-1
1
0
0
1 X
-AV1
AV2
LV3
-
99
and the incremental bond-slip by
Aa
E
h o
Aa 0 El7
where Eh and E
V are the assumed incremental modulus (Secant modulus)
for the linkage element along and perpendicular to the bar, given in
Chapter 3.
Assuming that the above bond stresses are the average stresses
along the length L (the distance between the centre of the steel
elements on the right and left side of the linkage element) which is
represented by linkage stresses, the incremental nodal force-linkage
stress relationship can be written as
AF1
-1 0 Aah
AF2
= 7TDL X 0 -1 Aa
AF3 1 0
AF4
0 1
Where D is the bar diameter. The incremental force-displacement
relationship can then be established as
where
[ K1 ] =
[AF]
71- DL
=
Eh
0
-Eh
0
[14:]x[AV]
0 -Eh
0
E 0 -E
Eh
-E 0 E y v
(4.11)
100
is the stiffness matrix of the linkage element in that increment.
If linkage elements are placed both at the top and at the
bottom of the steel element (when steel is represented by a
rectangular element) the above stiffness is divided by two to give
the stiffnesses at those positions. When n bars of equal size are
used the above stiffness is multiplied by n. It should be noticed
that the bond or linkage properties in the two directions (along
and perpendicular to the reinforcement) is assumed to be uncoupled.
4.5.3 Rectangular Plate Elements
Let the displacement function which satisfies internal
compatibility and also maintains boundary compatibility between
elements be defined as
Vx = a
lx + a
2xy + a3
y + a4
(4.12) Vy = a
5x + a
6xy + any + a
8
It is seen that the displacement varies linearly in the boundary,
hence the edges of the element displace as straight lines. Adopting
non-dimensional coordinates
x = a and y = y
b
where a and b are the length and the height of the element (Fig. 4.1b),
and substituting the coordinates of the four nodes into the displacement
equation (equation 4.12), the coefficients a1
to a8
can be determined
in terms of the nodal displacements V1 to V8. The equation (4.12)
can then be written as
101
Vx = (1-x) (1-y) V
1 + Tc(1-i)V
3 + x 1 7V5 + Y(1-X)V7
(4.13)
V = (1-x) (1-y) V2 + X(1-Y)V4 + x yV
6 + Y(1-x)V8
Strains are given by
avx av + Y
ay ax 1 aVx 1 aVy b - ay a ax
[B] [v]
The incremental relation between strains and nodal displacements will
be
[ac] = [B] x [AV ]e (4.14)
where [ B is only the function of the position of the point within
the element.
(4.15)
(1-Y) /a 0 (1-Y) /a 0 Y/a
0 -37/a 0
B 0 -(1-X)/b 0 -Tc/b
0 X/13 0 (1-x) /b
- (1-;i ) /b -(1-i)/a -i/b (14)/a i/b i/a (1-x) /b -c/a
[ C
aVx ax av
ay 1 aVy b -
y
a ax
1 avx
and [ AV ]e
QV
102
For a given set of nodal displacements 6x is constant
in x direction but varies linearly in y direction. Similarly
E is constant in y direction but varies linearly in x direction.
The shear strain 6 xy, however, varies linearly in both x and y
directions.
In general the incremental stress-strain relationship in
the global direction (x-y) is given by:(this incremental relationship
is originally written in the principal stress direction)
Aax Aa
Aa
Dl
D2
D3
D2
D4
D5
D3
D5
DO
AE x
Ac
Ac xy
(4.16)
[Aa] = [ DI x [bc]
where ED ] is the elasticity matrix in the global direction for
the given increment.
The incremental nodal force-displacement relationship can then
be calculated as (equation 4.7):
[AF]e = [ivoz[B]7;cED]x[B1dv] x[AV]e (4.17)
where the stiffness matrix corresponding to the given increment
will be
f vo B 1TX [ D ]3cE B :1CIV (4.18)
or
3.03
for a constant thickness t of the element
L rlr iTr 11- 1 LK.] = "1"t J L B J X L D Jx LB] dxdy o o (4.19)
The results of performing the multiplication [ B ]7;< [ D ] xI B ]
and integrating the product over the element area are given in
Appendix 5.
4.5.4 Rectangular Plate Elements with Constant Shear
The major disadvantage of a rectangular plate element with
linear variation of displacement is that it behaves badly under
pure bending. Supposing that this linear element is used to model
a beam in pure bending stresses as shown in Fig. 4.2a, the exact
displacement for this type of loading is illustrated in Fig. 4.2b.
But the element cannot displace so, its sides must remain straight
as it deforms (Fig. 4.2c). The linear variation of bending stresses
(Fig. 4.2a) can be activated by the element (Fig. 4.2c) according to
the assumed displacement function (equation 4.12). Figs 4.2d and
4.2e show the induced shear stresses and y stresses in the element
when it is subjected to pure bending. These stresses which are the
cause of error are undesirable and should be reduced. The shear
strain should clearly be zero throughout the element in pure bending.
The finite element shear strain is given by:
1 avx 1 avy — b a —
Dy 3x
Differentiating the assumed displacement function (equation 4.13) and
inserting the values of nodal displacements associated with the pure
bending shown in Figs 4.2b or 4.2c
104
V2 V
4 V6
V8
= 0
and
V3
V 7
= -V1 = -V5
= (1 - 2x) V xy 1
It is seen that shear strain exists for all points within the
element except for the centre where x =1/2 . This observation led to
the suggestion(63) that when forming element matrices, terms in
matrix EB a associated with shear strain should always be evaluated at
the element centroid (X = y = 1/2. while other terms in E B]
remain unaltered. Performing this procedure in the present analysis
greatly improved the behaviour of the element when subjected to bending.
The trial example given in Fig. 4.2f shows this improvement. Results
of the analysis and comparison with the beam theory are given in
Table 4.1.
4.5.5 Composite Elements
When perfect bond is assumed to exist between the steel
reinforcement and the surrounding concrete, the composite element(77)
can be employed. The advantage of this model is that the structural
idealization will be independent of the geometry of the reinforcement
(in contrast to the models in which the reinforcement is represented
by bar elements, hence the location of the steel indicates the location
of the nodal points). Instead the reinforcement is included directly
within the element. The stiffness matrix [ Kc, s
of the composite
element must be computed as the sum of the stiffness of the concrete
and the steel components. The importance of the location of
105
the reinforcement is, however, considered in the element stiffness.
Since the rectangular elements with modified shear were used to model
the bending members, the stiffness of the reinforcement was included
in the stiffness of these elements.
Fig. 4.3a shows an arbitrary reinforcing bar located inside
a concrete rectangular element. Assuming an elastic, isotropic
and homogeneous material for concrete the elasticity matrix of the
concrete element in any direction is given by;
1 Ecl Ec2 0
Ec 3 Ec2 Ecl
0
0 0 Ec6_
The elasticity matrix of the reinforcement along its direction is
Esl Est
0
Es J Es2 Es4 0
0 0 Es6_
The elasticity matrix of the contained reinforcement along the
bar direction will then be;
El E2 0
[ = LES J E E2 E4 0
0 0 E6
The stiffness matrix of a contained bar is given by;
[Ks] = vs [ B ]Tx s
B jdv
106
in which [B.] is the strain matrix of the rectangular element with
constant shear (see Appendix 5). Since [ B] is a function of the
coordinate of the points within the element,[K]must be integrated
as a line integral over volume of the steel contained in the element.
In general the bar may not be even straight. In the members analysed,
however, the reinforcing bars were straight and parallel to the sides
of the concrete element (in x and y directions). The derivation of
the stiffness matrix of a single bar is therefore given in the global
directions.
1. Reinforcing bar in y direction: Fig. 4.3b shows a a1
vertical bar located at a distance a from the y axis. Let m = 1 a
be the non-dimensional coordinate of the bar. The stiffness matrix
of the bar is then obtained as;
1 = A
s .b I[Brx[ Es K 131 dy (4 . 20)
for which the position of the reinfOrcement at x = m (0 < m < 1) is
inserted into the corresponding [ B] matrix (constant shear element).
Performing the multiplication and integration, the upper diagonal
terms of the 8 x 8 symmetric matrix [ Ks v
] of the vertical reinforcement
are given in Appendix 5.
In order to show how the effective stiffness of the vertical
contained reinforcement is produced at the nodal points of the concrete
rectangular element, assume the simplified elasticity matrix of the
vertical reinforcement to be;
- 0 0 0
O E 0
0 0 0
107
Which considers the stiffness of the bar only along its length.
(a) for m = 0, i.e. the bar is located along the y axis
(Fig. 4.30,
K(2,2) = K(8,8) =
EA K(2,8) = K(8,2) = bs
and
K(4,4) = K(6,6) = K(4,6) = 0
(b) for m = 1- 2 i.e. the vertical bar is located in the '
middle of the element (Fig. 4.3d),
1 EAs K(2,2) = K(8,8) = — • 4 b
1 EAs K(4,4) = K(6,6) =
1 EAs K(2,4) = K(6,8) = • —b- 4
1 EAs K(2,6) = K(4,8) = 4 b
1 EAs K(2,8) = K(4,6) = -
(c) for m = 1, i.e. the vertical bar is located at a distance
a (length of the element) from y axis (Fig. 4.3e),
K(2,2) = K(8,8) = K(2,8) = 0 EA
K(4,4) = K(6,6)
K(4,6) = K(6,4)
EAs b
b
b
b
EA
b
108
2. Reinforcing bar in x direction: Fig.4.3f shows a b1
horizontal bar located at a distance b1 from the x axis. Let n = —
b
to be the non-dimensional coordinate of the bar. The stiffness
matrix is given by
1 E K
s 1h
= As.a oEB 1Tx[Es] x B dX (4.21)
for which the position of the reinforcement at y = n is inserted
into the corresponding EB 1 matrix. The results of the integration
are given in Appendix 5. The distribution of the horizontal bar
stiffness at the nodal points can be similarly demonstrated as for
the vertical bar.
It should be noted that while concrete material of the composite
element was assumed to behave according to the specified law, the
calculation of the stiffness matrix of the contained reinforcement is
based on the homogeneous, isotropic and linear concrete behaviour. In
other words the elasticity matrix of the contained reinforcement
[ Es ],
[ E =
is obtained assuming
[Es]
E Eel
E
Ec
1
pc
0
11
1
0
0
0 1-11_ -TE
1-v2
(EC and pc arethe elastic constants of the concrete)
which is independent of the non-linear behaviour of concrete. It
can however vary according to the prescribed steel behaviour. In
the present analysis as the elastic properties of the reinforcing
109
bars in horizontal and vertical directions were equal,
isotropic linear elastic behaviour was also assumed for the
reinforcement, i.e.
1 p s o
Ps 1 0
1-P s 2
(Es
and vs
are the elastic constants of the steel).
It was further assumed that if a crack appeared within the
composite element in any direction;
[ Es] [ Bs]
In general the stiffness matrix of the contained reinforcement
in a given increment in the global axis is given by equations (4.20)
and (4.21),
1 r r -
for vertical bar [ = V f [Bix t. EjxL. B -1 dy
s v v o s 1
r - r -- for horizontal bar [Ks jh = Vh t
o [ B j x r E
sj x B] ax
I where L Es ] is the incremental elasticity matrix of the contained
reinforcement and Vv and V
h are the volume of the horizontal and
vertical bar. The stiffness matrix of the rectangular concrete
element with modified shear in the same increment is given by;
1 1 _T x r, rBI dxdy - - K
c] = a.b.t IippL [ o I 0 [B
Es
Es
1-Ps2
0 0
110
Then, the stiffness matrix of the composite element for the same
increment can be written as;
p
cs = Ki + E Ki I
1 (4.22)
where p is the number of the reinforcing bar within the concrete
element.
4.6 TRANSFORMATION MATRICES
The incremental elasticity matrix of the bar elements, linkage
elements and the contained reinforcement of the composite elements were
originally written in the global axis. The stress-strain relationship
of the reinforcement presented by rectangular elements and the
constitutive relationship of concrete is, however, given in the
principal stress direction which generally differs with the global
direction for which the element stiffness matrices are assmbled and
the solutions(displacements, strains and stresses) are found. Hence
it is necessary to relate the principal directions and the global
(x-y) direction. The derivation of the required transformation
matrices are given in Appendix 5.
4.7 CRACKING AND CRUSHING OF CONCRETE ELEMENTS
1. Formation of the first crack in an element: If the
principal stress in any direction exceeds the assumed tensile strength
of concrete a crack will appear within the element normal to that
direction. The cracking is considered as a change of the material
property of concrete from previously assumed behaviour (isotropic or
orthotropic) to a new prescribed orthotropic behaviour. The stiffness
111
of the element perpendicular to the crack is assumed to be zero,
but some shear capacity which takes account of the aggregate
interlock across the crack surface is assumed to remain. The
shear transfer capacity across the crack can be a function of the
crack width. The elasticity matrix of the cracked element
perpendicular to the crack direction is written as
[Di l e
0
0
0
0
(E)c
0
0
0
eG
The excessive stresses which cannot be sustained under the
constitutive law of the cracked element are converted into the
nodal forces (unbalanced forces) and are applied to the surrounding
elements. When a crack has formed in an isotropic material,
E
where E is the initial tangent modulus of concrete in uniaxial
direction. When it has formed in the assumed orthotropic material
(E) c = p= 0
y2 = 0
The expression for E; is given in Chapter 3.
G E
2(1+p) is assumed as the shear modulus of the cracked
element.
112
The shear retention factor a has originally a unit value (0 = J.)
for the uncracked element. After cracking this value must be
reduced. The effect of varying values of from 0.2 to 0.5 as a
function of the crack width was insignificant, hence a constant
value of (3 = 0.4 was assumed for the shear retention factor across
the crack.
2. Formation of the second crack in the element: The new
constitutive behaviour assumed for the cracked element in the crack
direction will give a principal stress direction which generally
differs slightly with the previous direction causing the formation
of the first crack. This is due to the introduction of shear
retention factor (for f3 = 0 the principal stress direction remains
unchanged after the formation of the first crack). If the tensile
stress of concrete in the new principal direction exceeds the
concrete tensile strength, the second crack is assumed to form in the
element perpendicular to that direction. The element is assumed to
have zero stiffness, i.e.
[ Di] cc = C ol
All the stresses in the element are therefore converted into nodal
forces and released.
3. Crushing of a concrete element: For an originally assumed
isotropic concrete material, if the principal stress in an uncracked
or cracked element exceeds the compressive strength of concrete in
uniaxial direction, the element is assumed to have crushed.
113
For an originally assumed orthotropic uncracked concrete
element if the compressive stresses in the two principal directions
satisfy the failure criterion of concrete (see Chapter 3), the
element is assumed to have crushed. When a tensile stress exists
in one direction, then the element is crushed if the compressive
stress in the other direction exceeds the compressive strength of
concrete in uniaxial compression. For a cracked element, crushing
occurs when the compressive stress in the new principal direction
exceeds the compressive strength of concrete in uniaxial compression.
When the element crushes, its stiffness is assumed to be
zero similar to the formation of two cracks. All the stresses
(tensile or compressive) are therefore converted into nodal forces
and released
= [ 0 ]
4.8 CALCULATION OF STRESSES AND UNBALANCED FORCES DUE TO THE
MATERIAL NON-LINEARITIES
The term for material non-linearity in the present context
includes:
(a) The non-linear constitutive relationship of a
given material (concrete, bond and steel).
(b) Yielding of the linear elastic steel, cracking
and crushing of concrete.
(c) Change in the stiffness of the contained reinforcement
due to the formation of a crack in the concrete
material of a composite element.
114
The term increment does not necessarily mean the load increment
but a stage for which the stiffness matrix of all the elements is
updated and kept constant. The true behaviour of the material is,
however, a continuous non-linear relationship. Hence it is necessary
to maintain the equilibrium within the element (constitutive relationship)
by successive iterations in each increment during which the undesired
stresses are treated as initial stresses in the element and are converted
into unbalanced forces to be applied to the structure.
r Assuming LD j is the stiffness matrix of an element in the global
direction at the beginning of the increment at the nth iteration within
this increment
&in x = [ D1 x 6cn x
where [ Acsn ]x and [ Acnx are the resulting stresses and strains of the
element in this iteration. The total stresses in the global direction at
the end of this iteration are given by
E 6n ]xu L a n-1 lx E Ix
where [ 6n-1 ]x is the total balanced stresses at the previous (n-1)
iteration. Let the principal stresses corresponding to the total stresses
l u in global direction [ an jx
, to be [ an jp , i.e.
r a [ r u L n
i p L n ]
From the constitutive relation of the material in the principal stress
direction
[a n ]p f( [ En ]p)
where [ Enp
is the total strain in the direction of the principal
stresses. The unbalanced stresses in the principal direction are
then calculated as
115
[ Aa p = [ an P - [an ] nP
and the unbalanced stresses in the global direction are found
as;
[ Aa ]u n x
R 1T [ Au ]p11
These unbalanced stresses in the global direction are treated as
initial stresses and are subtracted from the total stress [ a n x
]u to
obtain the balanced stresses,
E ] x [ an - Aan]:
To maintain the equilibrium, the equivalent unbalanced nodal forces
are calculated as;
r- F Bi, L n v
r L j
T Fixo Jx
dv
and applied to the surrounding structure.
Since the stresses and the strains of an element are assumed
to be represented by its centroidal value, the above integration is
reduced to a simple matrix multiplication, i.e.
EFn x n lu = EB iT [Aa x .v
where V is the volume of the element and the constant matrix [ Bc]
is formed by inserting the coordinates of the element centre
- (x = 1 1
y = into the strain matrix [ B ] given earlier. 2
116
For materials whose principal direction coincides with the global
direction (bar and linkage elements) the unbalanced stresses and
forces are calculated directly in the global direction.
117
TYPE OF ELEMENT Y displacement at n X stress at m
LOAD A LOAD B LOAD A LOAD B
Plate rectangular 70.60 72.30 -2117.64 -2854.34
Plate rectangular with constant shear
98.40 99.70 -2950.80 -3983.68
Beam theory 100. 103 -3000 -4050
Test problems for plane stress rectangular elements and comparison
with the beam theory (Fig. 4.2f)
Table 4.1
(d) Shear Stresses
A
Simple Bending Stresses
x
(a)
(b)
X
(c)
Y -Stresses (e)
56.251
187.50 1
Load BL
t
1001 1
m
Test problem for the Plane Stress Rectangular
Element
5x2=10
(f) Load B
1000 156,25
1187,50
156,25 A
4
118
V
Bar Element
Rectangular Plate Element
Linkage(Bond) Element
FIG.4.l
RECTANGULAR ELEMENT ERROR DUE TO PURE BENDING STRESSES
FIG.4.2
E =1500
VI= 0,25 t =1
119
(a) (b)
R.xia 57.y/b
- - - -',. R
(0) (d)
(e)
(f)
COMPOSITE ELEMENT
FIG.4.3
120
CHAPTER 5
BEHAVIOUR OF CONCENTRICALLY LOADED TENSION MEMBERS
5.1 OBJECTIVE OF THE ANALYSIS
The study of the behaviour of concentrically loaded tension
members is fundamental to prediction of crack width and spacing of
reinforced concrete structures. Concrete is very weak in tension,
hence reinforcement must be provided in the tension region of a
structure. The formation of a tensile crack due to excessive tensile
strain is caused by the low extensibility of concrete which is not
able to follow the same elongation as that of the reinforcement. As
the concrete in a concentrically loaded tension member is loaded through
the periphery of the reinforcement (Fig. 5.1), the bond between the two
materials is a dominant factor in the behaviour of the member. The
analysis of these members, therefore, is expected to give an insight
into the effect of bond on the formation, width and spacing of tensile
cracks, particularly around the reinforcement.
In order to study analytically the behaviour of these members, two
rectangular tension members of different lengths were selected from a
series of tests conducted by Broms(38,39) The purpose of the analysis
of the short member was to follow the formation and the shape of internal
cracks. The long member was chosen to study mainly the spacing and
the width of the primary cracks. The precracking, cracking and
post-cracking behaviour of the members as obtained by the analysis is
described in detail and checked with experimental data. The members
selected were those for which well documented experimental results
121
were available(38,39,40)
for comparative purposes.
5.2 AN ASSESSMENT OF THE BEHAVIOUR OF CONCENTRICALLY LOADED RECTANGULAR
TENSION MEMBERS
When a tensile load is applied to a free reinforcing bar the
displacement of the bar is determined by its properties and the load.
If the load is applied to the same bar embedded in a concrete block
the movement of the bar is restrained by the property of the surface
contact of the two materials. This restraining surface causes the
concrete around the bar to move, hence tensile stress will develop in
concrete. Fig. 5.2 shows the gradual transfer of stress from bar to
concrete along the member. Section A close to the end is loaded
only from the concrete surface around the bar, the resultant force at
that section is at the bar level and is equal to bond force developed
near the end. The force in Section B is the result of the two following
forces:
(1) The force in the adjacent concrete section (Section 18)
which is caused by the bond force between this section
and the end.
(2) The bond force between the two adjacent sections B and
B.
As a result of diminishing bond stresses towards mid-span, the
resultant longitudinal force in the upper half of the concrete tends
to rise as shown at Sections B, C and D, until it approaches the mid-
cover height, Section E. The longitudinal concrete stresses in any
section due to the resultant force at that section are also shown in
122
Fig. 5.2. At Section E the bond force is fully developed, bond stress
is very small and the whole section is in uniform tension. The
resultant stresses in each section can be considered as the result of
a force placed at mid-cover height plus a bending moment on that
section. Thus vertical concrete sections of a concentrically loaded
tension member are subjected to bending as well as axial force when
the external load is applied to the protruding ends of reinforcement.
The amount of axial force is small at the loaded end (smaller bond
force) and increases towards the mid-span. The bending moment, on
the other hand, is larger around the loaded end (the tensile force is
at greater eccentricity with respect to mid-cover height) and decreases
towards the mid-span. This bending moment which causes the concrete
to deflect sideways away from the bar (Fig. 5.3) is responsible for the
presence of high lateral tensile stress near the load point and lateral
compressive stress in the concrete adjacent to the steel towards the
mid-span. The combination of longitudinal stresses due to the axial
tension force and due to the bending can thus give rise to both tensile
and compressive stresses in the vertical sections and also to a
variation of compression to tension stresses as one passes along the
member. Transverse stresses in the concrete are also introduced and
these vary from tension to compression as one travels from the end
to the mid-span. If the lateral tensile stress exceeds the tensile
strength of concrete near the end, a longitudinal crack will initiate
from there. A transverse crack will also form in concrete adjacent to
steel if the longitudinal stress exceeds the strength of concrete in
any section. From Fig. 5.4 it can be seen that any transverse crack
initiated in concrete adjacent to steel near the end will have a
limited extension due to compressive stresses at the top face. The
formation of these cracks is also seen to be connected to the formation
123
of longitudinal crack near the end as both are affected by the
amount of bending moment. At a section where more uniform
longitudinal tensile stresses are present (Section D or E in Fig. 5.4),
the initiation of a transverse crack in concrete at the level of
reinforcement will propagate and travel the whole depth of the section,
and hence form a primary crack.
For the same applied load if the member is assumed to have a
larger depth the bending moment is greater at any section (greater
eccentricity of the force in concrete sections with respect to the
mid-cover height). This bending moment will affect the distribution
of longitudinal stress in the concrete along the member so that the
section with near uniform tensile stress where formation of primary
cracks are expected is removed further from the end. It is seen,
therefore, that the position of a primary crack (primary crack spacing)
is related to the depth of the member.
5.3 DESCRIPTION OF THE MEMBERS SELECTED FOR THE ANALYSIS
Two concentrically loaded tension members TRC2-3 and TRC3 as
tested by Broms(38,39) were selected for the analysis. The length
of the long member TRC2-3 was chosen to be 32 in. (the experimental
length of this member was 6 ft.). This member was considered to be
long enough to study the formation of primary cracks. The length of
the short member TRC3 was the same as the experiment. The dimensions
and material properties of the members are given in Table 5.1.
The modulus of rupture was used for the tensile strength of concrete
as well as for the combined bending and axial tension strength. The
modulus of elasticity of concrete was calculated as(4):
105 Vi-i; Ec
in lb/in 2 1+0.00617c
124
where fc
is the compressive strength of concrete in p.s.i. The assumptions
adopted for behaviour of materials (steel, concrete and bond) and
description of numerical techniques to deal with non-linearity of
materials are given in Chapter 3.
Due to symmetry (Fig. 5.5) only a quarter of the member was
analysed. The finite element model of the members with appropriate
boundary conditions are given in Appendix 1.
5.4 LOADING CONDITIONS OF THE MEMBERS
The members were loaded up to 70 kips (steel stress of about
88,700 p.s.i. at the end) equal to the final load of the experiment
when the steel is assumed to yield. The increment of load varied
throughout the application of loading in the analysis. Smaller
increments were applied when cracks were initiating in the member and
larger increments were chosen at the later stages of loading. The
average increment was about 6 kips.
5.5 BEHAVIOUR OF SHORT TENSION MEMBER TRC3
5.5.1 Behaviour of the Member before Cracking
Load was applied to the protruding ends of the reinforcing bar
to the total of 10 kips before cracks appeared in the member. The
distribution of longitudinal stresses in concrete adjacent to steel are
shown in Fig. 5.6a. At small load (3 kips) the stress is uniform
in the concrete except near the end. At higher loads the position
of maximum stress shifts towards the mid-span and non-uniform
distribution results. The distribution of longitudinal stress in
concrete sections parallel to the reinforcement and at 1.15, 2.15 and
3.37 in. from it is given in Fig. 5.6b at 9 kips load. Very small
125
compressive and tensile stresses exist near the top face of the
member. Fig. 5.6c shows high transverse tensile stresses in
concrete adjacent to steel and near the end even at small loads.
Transverse compressive stresses are present around mid-span. Steel
stress distributions are given in Fig. 5.6d. Bond stress distributions
in Fig.5.6e show a sharp fall near the loaded end at small loads. There
is, however, in this region an increase of bond stress with load.
Corresponding slips are shown in Fig. 5.6f.
Fig. 5.7 (a,b) illustrates the relative magnitudes of longitudinal
and transverse stresses within the member just before cracking. The
stress contours (longitudinal and transverse) as shown in Fig. 5.7 (c,d)
indicate two regions of tensile stress concentrations namely, longitudinal
stress at mid-span and transverse stress at load point.
5.5.2 Behaviour of the Member after Cracking and Comparison with
Experimental Data
As the load reached 10 kips a longitudinal crack initiated
from the loaded end and spread 1.5 in. towards the centre. At the
same load transverse cracks also appeared with very small width and
extensions near the centre. (The formation of secondary cracks at
this load stage in the member was recorded by strain gauges in Broms
experiment and the presence was confirmed when this member was cut
open and examined(39)
.) The relative magnitudes and inclinations
of principal stresses just before cracking is shown in Fig. 5.8a. The
width and extension of the first cracks formed in the member are
illustrated in Fig. 5.8b. The widening and extension of these cracks
and formation of a new transverse crack at further increase of load
( to 20.00 kips is given in Fig. 5.8c. Experimental observations
38)
also indicate that a longitudinal crack with visible width appeared
in this member at this load stage. Fig. 5.8d shows the widths and
126
propagations of cracks at 30 kips. Two different shapes of
secondary cracks exist in Fig. 5.8d. The maximum width of one
of the cracks is at 0.70 in. from the reinforcement. The shape
of this crack is affected by the formation of surrounding secondary
cracks. The maximum width of a secondary crack of small extension
is at the level of the reinforcement. The typical shapes of these
two secondary cracks were observed in rectangular tension members tested
by Broms(40)
The width and extension of cracks for the applied loads
of 40, 50 and 70 kips are given in Fig.5:8e,f, and g respectively. All
the cracks widened under increasing loads. The width of the horizontal
crack is considerable at the load point but it decreases rapidly towards
the centre and its extension is limited by the formation of a transverse
crack at its tip. The experimental crack pattern of this member is
given in Fig. 5.9 for comparison(38). The numbers indicate the load
at which the crack was observed. From Fig.5.8e, f and g it is reasonable
to state that the shape of secondary cracks of the first order (i.e.
of relatively large extension) are affected by the formation of
adjacent secondary cracks of second order. Their maximum width
does not occur at the level of reinforcement. The extension of
secondary cracks of second order is limited to the vicinity of
reinforcement (due to the presence of secondary cracks of the first
order) and their maximum width is at the level of reinforcement. The
crack pattern in the member at ultimate load is shown in Fig. 5.8h.
The distribution of steel stress in Fig. 5.10 indicates that the
variation of stress remains almost unchanged after the secondary cracks
are fully developed within the member (20 to 30 kips). Secondary
cracks cause a local increase in steel stress depending upon their
size. The unit elongation of the member as compared with the unit
elongation of free reinforcement is given in Fig. 5.11. After the
127
formation of transverse cracks the unit elongation of the member approaches
that of the free reinforcement. This indicates that the steel stress
can be calculated very closely on the basis of a fully cracked section
and is confirmed by the distribution of steel stress in Fig.5.10 and
the results of Broms experiments on rectangular tension members(40)
The bond and slip distribution are shown in Fig. 5.12 for applied load
of 10, 20, 30, 50, 70 and 80 kips. As the load increases the bond
stress near the end drops. Some transverse cracks of relatively large
extension produce reversal bond stress in the member. The slip
distributions show that the amount of slip is significant near the
end at higher loads, but it is negligible within the member where
secondary cracks are present. It is also interesting to note the
distribution of bond and slip at 80 kips (steel stress of over
100,000 p.s.i.) in Fig. 5.12f. As the steel is yielding at this load,
the amount of bond stress is very small indicating that the reinforcement
is coming out of concrete block. No reversal bond stress is seen at
this load stage.
The transverse distribution of the primary crack width is a function
of the deformation of the concrete between the primary cracks (which
may be given by the short tensile specimen) and the slip of the steel
at the crack, Fig. 5.13a.
Fig. 5.14a shows the displacement of the steel bar and the
distribution of the displacement across the end concrete section of
the short member as obtained by the analysis and the corresponding
experimental measurements. At low loads the displacement of the
concrete is linear across the section and relatively small
compared with the displacement of the reinfOrcement. For higher loads
the variation of concrete displacement is almost linear being maximum at
128
the level of reinforcement. The deformation of the concrete at
the top face, i.e. maximum distance to reinforcement, is seen to
be negative at all stages of loading indicating that this face will
undergo increasing compression of relatively low magnitudes as
load is increased. The results of the analysis also shoa that,
for a point at 3.4 in. from the bar, the displacement is -almost zero
at all stages of loading as if the concrete is rotating about that
point. A close agreement is obtained between the analytical results
and the experimental measurements. The result of the analysis up to
a load of 50 kips lies between the measurements obtained for members
TRC3-1 and TRC3-2. The dotted lines corresponding to experimental
measurements of concrete deformation at 70 kips are probably unreliable
as mentioned in the report(40) The deviation of steel deformation
obtained by the analysis with experimental one at higher loads requires
some comments. The stress-strain relationship of the steel bar as
shown in Fig. 5.13b is non-linear after the stress exceeds 40,000 p.s.i.
The minimum possible stress in the steel can be calculated on
the assumption that the whole concrete in the section is at the
ultimate tensile stress. The steel stress in the section will then
be:
fs
P - Pc
As
where P = total applied load
Pc
= load carried by concrete
As
= area of steel bar
fs
70000 - 16674 0.7895
129
evaluating Pc
Pc = (3.50 x 8.10 - 0.7895) 605 = 16674 lbs.
and calculating fs
67544 p.s.i.
The elongation of steel measured in the experiment at 70 kips
is about 0.0119 in. (Fig. 5.14a) which corresponds to average steel
strain of 0.004 and average steel stress of 70 000 p.s.i. Thus the
reported steel deformation would appear to be that which corresponds
to the concrete being at its ultimate tensile strength. Nevertheless
the steel stress at the end is 88700 p.s.i. and the experimental
crack pattern of Fig. 5.9 clearly indicates the formation of secondary
cracks and horizontal cracks within the member. The experimental
measurements of Fig. 5.14a also indicate that concrete near the top
and bottom face are in compression or very small tension. All of
this suggests that the average steel stress in the member is far more
than the above calculated value. Measurements of the total elongation
between the load points of tension members in Broms experiments as
mentioned previously confirm that the average steel stress can be
very closely calculated on the basis of fully cracked section.
According to this calculation the end displacement of steel at 50 kips
(steel stress of 63300 p.s.i.) and 70 kips (steel stress of 88700 p.s.i.)
are 0.096 in. and 0.194 in. respectively. These are the upper bounds
to elongation of steel at the end. The corresponding values obtained
by the analysis are 0.095 and 0.185 in. as shown in Fig. 5.14a.
130
Fig. 5.14b compares the results of the analysis with the
assumption of linear stress-strain relation for steel with the same
experimental results. It is seen that the two results now are very
close even at higher loads.
The magnified deformation of the member at the applied load of
30, 50 and 70 kips is illustrated in Fig. 5.15 where the relative
deformation of the side of concrete and the steel can represent the
shape of the primary crack of the corresponding long member.
5.6 BEHAVIOUR OF LONG TENSION MEMBER T-RC2-3
5.6.1 Behaviour of the Member Prior to Cracking
The behaviour of the long member TRC2-3 in pre-cracking phase
is similar to the short member TRC3 . The distribution of concrete
longitudinal stress adjacent to steel, 1.15 in. and 3.40 in. from
steel are shown in Fig. 5.16 for three stages of loading. All concrete
longitudinal stresses become uniform at a distance from loaded end.
Very high transverse tensile stresses exist near the load for long
members as shown in Fig. 5.17. The distribution of steel stress, bond
stress and slip are also shown in Figs.5.18, 5.19 and 5.20 which are
very similar to those of short members. The distribution of transverse
and longitudinal stresses in the form of isobar charts are given
in Figs.5.21 and 5.22 at 9 kips load just before cracking. Transverse
tensile stresses at the end and compressive stresses towards the middle
of the member are seen in these figures. Transverse stress is
almost zero at the centre of the long member where uniform longitudinal
stress exists. The maximum tensile stresses in this member at cracking
load are transverse stresses close to the load.
131
5.6.2 Formation of Cracks and Post-cracking Behaviour of the Member
At an applied load of 9 kips, the tensile stress in the
concrete adjacent to steel at the loaded end exceeded the concrete
strength in tension and the first crack initiated from the load point
and extended about 2 in. along the reinforcement. As a result of
redistribution of stresses in the concrete due to the longitudinal
crack a few transverse cracks appeared in front of this crack and
propagated at an inclination of 60 to 70° with the reinforcement.
The load was raised to 11 kips causing the extension of previous
cracks (longitudinal and transverse) and formation of new transverse
cracks in the vicinity of reinforcement towards the centre of the
member as shown in Fig. 5.244. The limited extension of these transverse
cracks and the appearance of compressive stress at the top of the
member (Fig. 5.23) imply that the bending action (as mentioned in
section 5.2) dominates the behaviour of the member near the end. The
propagation and inclination of the cracks at 11 kips are given in
Fig. 5.24a. The width of transverse cracks is very small at the
time of formation (less than 0.0003 in.), they are not visible and
further loads of greater magnitude are required to widen them.
According to Broms experimental observations the secondary cracks
were seen after the formation of primary cracks. The present
analysis, however, indicates that there are two distinct stages for
secondary cracks:
(a) Initiation of these cracks: At this stage they are still
invisible.
(b) The widening of the cracks: At this stage they become visible.
132
The formation of a longitudinal crack will affect the bond between
concrete and reinforcement particularly near the end. This crack,
as a splitting crack,causes a considerable reduction of bond resistance
close to the loaded end.
5.6.2.1 Formation of the First Primary Crack
As the load was increased to 13 kips the extension of the
longitudinal crack was followed by the initiation of another transverse
crack more perpendicular to reinforcement at a distance 7.60 in. from
the end. This crack which formed similarly to previous transverse
cracks propagated across the whole section of concrete and hence
became a primary crack in the member. The spacing of this primary
crack (7.6 in.) is close to the experimental spacing (8.0 in.). The
position of this section from the end is a crucial parameter, here the
section had developed almost a uniform longitudinal tensile stress
which allowed the crack after initiation near the reinforcement to
travel the whole section. The width of the primary crack is such
that it is visible at the time of formation. The longitudinal crack
at this load had extended about 5.0 in. from the end but only the
first half of the crack has an appreciable width. The width of
the cracks and their extensions after the formation of first primary
crack are shown in Fig. 5.24b.
5.6.2.2 Formation of the Second Primary Crack
After the first primary crack was fully developed in the member,
a longitudinal crack (similar to the longitudinal crack at the end)
initiated on the right hand side of this crack and extended inwards.
The formation of a number of transverse cracks at the level of
133
reinforcement and their propagation and the further extension of
the second longitudinal crack in the second half of the member was
similar to that which had occurred during the formation of the first
primary crack. The second primary crack formed at the same load
as the first primary crack and at 8 in. distance from it. Figs. 5.25a
and 5.25b show the crack pattern and the crack widths after the
formation of the second primary crack. The width of the second primary
crack is almost constant throughout the depth and is visible at the
time of formation. The width of the first primary crack is, however,
more affected by the extension of surrounding secondary cracks.
5.6.2.3 Width of the Cracks at Later Stages of Loading
The member was further loaded up to 25 kips by increments of 6 kips
during which the cracks widened and extended. Fig. 5.26 shows the
crack width at this stage (steel stress of about 32 ksi). Secondary
cracks of similar shape to those in the short member also developed
here. Small secondary cracks with maximum width at the level of
reinforcement will affect the shape of the other secondary cracks
with greater extensions, and all of which will define the shape of
the primary cracks as demonstrated in Fig. 5.27. The width of the
secondary cracks in Fig. 5.26 indicates that some of these cracks are
visible at this stage. The wedge shape of primary cracks becomes
more noticeable as the secondary cracks widened. The crack width
at the top face is two to three times larger than at the level of the
reinforcement. The same ratios have been measured by Broms for
primary cracks(38,40) for steel stresses exceeding 30,000 p.s.i. The
widening of secondary cracks in the present analysis coincides with
the appearance of secondary cracks in the experiment for only at this
134-
load stage is their width sufficient to make some of them visible.
The width of the cracks at 37 kips (steel stress of about 47 ksi)
and 49 kips (steel stress of about 62 ksi) are given in Figs.5.28
and 5.29 respectively. Transverse cracks appeared within the end
longitudinal crack length at a load of 49 kips. The maximum width
of about 0.002 in. for the secondary cracks, and 0.014 in. at the top
face for the primary cracks are seen in Fig. 5.29. The extension
of the longitudinal crack near the end has almost reached the other
longitudinal crack initiated on the left hand side of the first primary
crack. The crack widths of the member at 61 kips (steel stress of
about 77.50 ksi) in Fig. 5.30 indicate that the secondary cracks with
small spacing have all widened. The longitudinal cracks are still
approaching each other along the reinforcement and are very close to
one another. The wedge shape of the primary cracks still exists at
the steel stress of 77.50 ksi. The ratio of the width at the top face
to the width at the level of reinforcement is about 2.5 and 2 for
the first and the second primary cracks. It is seen that these ratios
are slightly reduced at this load stage probably due to the loss of
bond around the primary cracks. The crack pattern of the member at
67 kips is given in Fig. 5.31.
5.6.3 Analytical Results and Comparison with the Experimental Data
The distribution of steel stress just before cracking and for
the whole range of post-cracking loads is shown in Fig. 5.32. The
variation of steel stress after the formation of primary cracks is
very small. The difference between the stress in steel at the two
primary cracks is due to the orientation of cracks in those sections.
The second primary crack is almost perpendicular to the reinforcement
and stress in concrete is almost zero in the direction of reinforcement.
135
The formation and extension of secondary cracks within the member
are also responsible for the distribution of steel stress.
Fig. 5.33a shows the average unit elongation of the member
against steel stress at the end. Part A corresponds to the unit
elongation before cracking. In Part B due to the formation of a
longitudinal crack near the end, bond is partly destroyed and the
steel stress rises locally. Part C traces the formation of few
transverse cracks in the member near the end and Part D corresponds
to the formation of primary cracks. Part E is the unit elongation
of the member after the formation of the primary cracks up to the
final load. The unit elongation of reinforcement without concrete
is also drawn in Fig. 5.33a. After the formation of primary cracks
it is seen that the two elongations are very close to each other.
The experimental curves corresponding to unit elongation of cylindrical
and rectangular reinforced concrete tension members with cover
thickness of 3 to 5 in. are given in Fig. 5.33b for comparison.
The concrete longitudinal stresses near the top face (3.55 in.
from reinforcement level) are given in Fig. 5.34 at post-cracking
loads. After the formation of primary cracks, small tensile or
compressive stresses exist at the top face due to the widening and
extension of secondary cracks. At higher load, however, more tensile
stresses are built up.
Highly irregular bond stress distributions as seen for short
members_also appeared in the long member as shown in Fig. 5.35. The
variation of slip (Fig. 5.36) indicates that a considerable slip
exists very close to primary cracks. Between the two primary cracks
the slip is almost negligible due to the formation of secondary
cracks. The slip at the end of the member (Fig. 5.36f) was however
affected due to the formation of secondary cracks near the end at 61 kips.
136
The magnified deflected shape of the member at applied loads
of 25 and 61 kips are given in Fig. 5.37. The shape of the
longitudinal and primary cracks shows that the member is not deflected
symmetrically. This is attributed to the crack propagation which
initiated from the loaded end and progressed inwards. The extension
of longitudinal cracks are also greater on the right hand side of the
primary cracks.
The experimental crack widths of this member are compared with
the analytical crack widths in Figs. 5.38 and 5.39. The.measured
total crack width (the sum of the width of all cracks along the axis
of the member) is shown in Fig. 5.38 as a function of applied total
load. It is seen that very close agreement exists between the
analytical and experimental total crack width after the applied load
of 37 kips. The elongation of the reinforcement in air is also
drawn in that figure. The total crack width obtained analytically
approaches the curve corresponding to the elongation of reinforcement.
This suggests that the average concrete stress adjacent to reinforcement
is almost zero at higher loads.
The width of the primary cracks obtained by the analysis is
plotted against the experimental measurements of a primary crack
width for this member (Fig. 5.39) at 40, 55 and 70 kips. Considering
the complex nature of cracks in concrete the two results are
remarkably close.
Finally, it is interesting to compare Goto's experimental findings
of crack distribution in concrete tension members(28) with the results
of the present analysis. Fig. 5.40a shows the internal cracks of a
long member reinforced with 3/4 in. deformed bar and Fig. 5.40b shows
137
the internal cracks of a second member reinforced with 1.26 in. deformed
bar. The crack pattern of the two experimental results is comparable
to Fig. 5.31, the analytical crack pattern of the member TRC2-3. The
splitting of concrete at the primary crack sections and at the end as
seen in the experimental figures (the positions of the injected resin
along the reinforcement) is also comparable to the horizontal cracks
and their considerable width along the primary cracks in Fig. 5.30 for
the member analysed.
5.7 CONCLUDING REMARKS
(1) In reinforced concrete tension members the shape of the
primary cracks are determined by the formation of secondary cracks.
The calculation of primary crack width is far from reality if secondary
cracks are ignored.
(2) The variation in concrete stress in cross-sections of
concentrically loaded tension members can be considered as due to a
tension force plus a bending moment.
(3) The analysis shows that most of the secondary cracks form
before the formation of primary cracks. The width of the secondary
cracks unlike that of primary cracks is such that they are not visible
at the time of formation. The appearance of secondary cracks in
experiments can be said to correspond to the widening of these cracks
which are formed earlier.
(4) The deformation of concentrically loaded tension members
is not symmetrical between primary cracks.,
(5) A large number of secondary cracks exist in the vicinity
of reinforcement at higher loads.
138
(6) The formation of secondary cracks of the second order
(small cracks around the reinforcement) will affect the shape of
secondary cracks of the first order (extending towards the top and
bottom faces) all of which will make the wedge shape of the primary
cracks.
(7) Primary cracks and some secondary cracks (depending on
the width and extension) when formed, will cause a reverse in bond
stress and slip around them.
(8) The formation and the extension of longitudinal cracks and
secondary cracks near a primary crack are interrelated.
(9) Once primary cracks are formed the variation in the
distribution of steel stress in the member remains relatively constant.
(10) The close agreement obtained between the results of the
analysis and the experiment suggeststhat a two-dimensional analysis
is capable of predicting satisfactorily the behaviour of the reinforced
concrete rectangular tension members.
139
Member Cross section in.xin.
Length in.
Concrete strength
p.s.i.
Modulus rupture p.s.i.
Modulus of elasticity p.s.i.
TRC2-3 3.50 x 8.10 32.00 3450 469 4343080
TRC3-1 3.50 x 8.10 8.00 5140 605 5012980
Dimensions and material properties of concentric members
Table 5.1
Lower Mid-Cover
LOAD — -bearehIrie-OT the member===-7--
Concrete Longitudinal Stresses
— Mid-Cover
I I "
FIG. 5.1
1313
Upper Mid-Cover
A RECTANGULAR TENSION MEMBER
Reinforcement
A
•
Longitudinal Stress Distribution
Tension
r Compression 'FIG. 5.4
FIG.5.2
Line of application of resultant forces in concrete
FIG. 5.3
140 Top Face
Bond Stress magnitude between the two adjacent sections
Transvers Stress Distributioi
32.00
MEMBER T-RC3 Analysed Part
t inf25 Reinforcing
Bar
MEMBER T —RO2
14-1
0 oi
Half of the area ofthe steel bar =0.3927 in2
elements Thickness of element El' 0.2 2. 50 in. E2 El 2r . Area of the bar assumed for element E -0.192in
Thickness of element E 0.192 2,=3.50 0.50 - 3.11in.
MEMBERS IDEALISATION
FIG.5.5
Areaof the bar assumed for element E 0.2
Concerte Transverse Stresses to steel
Stress p.s.i
500
400
300
200
100
p.s.i (a)
MEMBER T-R03
FIG. 5.6 11000
10000
8000 g kips
3 4000
6000
Distance from end face,in.
2 3 4
9 kips
2000 Concrete Longitudinal Stresses
adjacent to steel
Stress p.s.i
(b) 500
40
300
200
100 3,37
2 3 4 Distance from end face,in.
Concrete Longitudinal Stresses AT 9 kips
Stress p.s.i
Distance from bar(in.)
.1
2,15
1,15
0
3 N.:,
Distance from end face,in. 2 3 4
Distribution of Steel Stresses
Stress P.s.i
Slip(103in.)
Slip Distribution
14-2
600
500
400
300
200
100
(0)
Distance from end face,in.
4 3 kips 6 • 9 :•
adjacent.
g kips 600
5 (e) 00
400
300
200
100
2 3 4
Distance from end face,in.
Distribution of Bond Stresses
(r)
2 3 4
Distance from end face,in,
(a)
10 Kips +
MEMBER T-RC3
Stress Scale
I I
.-k
100 ps i op-u-ysoo ..
14~
+Tension
-Compression
(b)
+ -t-
+ .-€-
Concrete Longitudinal stresses I Concrete Transverse stresses
10kipS ~.
(c)
Compression o,~ __ ----:
Tension
_...J..- • __ .~ ~ ~ 10klDS ~
Cd)
I I
Compression I
~100
~ . . . I .~ Longitudinal Stress Contour I Transverse Stresses Contuor
STRESSES IN CONCRETE JUST BEFORE CRACKING (AT 1.0 KIPS)
. FIG. 5.7
kips 10 4__
c
MEMBER T-R03
1144
Crack Width Scale 04410/1000 In.
Relative Magnitude and Inclination of
CRACK WIDTH AFTER THE FORMATIU OF Principal Stresses just before cracking
FIRST CRACKS (10 KIPS)
CRACK WIDTH AT 20 AND 30 KIPS
FIG. 5.8
(h)
70 kips • 4-- 70kips 4—
MEMBER T —R03
Crack Width Scale 0?—po/l000 In*
CRACK WIDTH AT 40 AND 50 KIPS
14.5
CRACK WIDTH AT ULTIMATE LOAD (70 KIPS) FINAL CRACK PATTERN
FIG. 5.8
T-RC3-2
Numbers indicate observation of cracks at given kips
EXPERIMENTAL CRACK PATTERN MEMBER T-i.R03(Ref.38)
146
FIG. 5.9
MEr-lEER T-RC3
90
.,-1 I ~ --IOklps til ~ ... ~ 80~ 80 (I)
H ~
j7+ .r! til
;4 ... 70 til C!l (I)
H
~I ~ ill
60~ .-I 60 (I) (I)
~ CI:l
50~ 50
401-30 klPSI
40 !Reinforcement with Concrete Shell
3°L 301- \ 20
20
10
10klps
- Distance from end face ,in. in. o 2 3 4 .001 .002 .003 .004
DISTRIBUTION OF STEEL STRESSES . AVERAGE UNIT DISPLACEMENT OF STEEL ~ -..J
FIG. 5.10 FIG. 5.11
3.48
50 kips
2 3 4 In. Distance from end face
(c)
1 2 3 4 in , Distance from end face
3
600
cn •
Pi
02 400
ti
200 0
-200
1
30 kips
Distance from end face
0
-200
-300
600
• P. rn 400
Co 4-3
pi
•
200 0
pa
1o
2
1 2 4 in. Distance from end face
MEMBER T-RC3 BOND STRESS AND SLIP DISTRIBUTIONS
AT 10, 20, 30 AND 50 KIPS
FIG. 5.12
200 -400
-600
p▪ i to
1 2 3 4 n.
•
0 Distance from end face
Co
1 (a)
10 kips
600
400 -200
Bond Stress,p.s.i
1 2 3 4 im Distance from end face
Bon d Stress, p.s. i
600
400
200
20 kips
1 2 4 'In- Distance fro:: end face
1
-
3 4 I n. Distance from end face
600
400
200
1 2 3 41n Distance from end face,in.
-200
-400
70 kips
2 4In
Distance from end face,in.
BOND STRESS AND SLIP DISTRIBUTIONS AT 70 AND 80 KIPS
FIG. 5.12
Bon d Stress, p.s.i
0
.0 1 2 3 4 m Distance from end face,in.
800 a)
q:1 g 400 0 ro
200
2 3 4rn Distance from end face,in.
80 ki p s
(f)
15
10
149
MEMBER T—RC3
FIG. 5.13a Final Position of Steel
.001 .007 .003 .006 .002 .004 .005
Ji/2 Crack I width
Final Position <c of Concrete
:Original Position of (Steel and Concrete
Strain in./in.
150
Steel Stress-Strain Relation
5.13b
.0165 70
EXPERIMENTAL \ (T-RC3-11T-RC3-2)
----Steel \ ----Concrete
.014 \ 1 ■ 1
ANALYSIS ■
----Steel
.012 \ \
-70 \ ----Concrete \\ ■I
.010 I1
I \
50 \ \
\\\
--50 Numbers indicate •008 loads in kips
\\ I\ \\
006
\70 ■ \
■ .004
.002
.75 2 2,75
Distance from Reinforcement lini
a) 4)
0 0
End Deformation of Steel
151
MEMBER T-RC
PRIMARY CRACK WIDTH
FIG. 5.14a
(1) 0)
.014 O 0
rd
End Deformation of Steel
1
1
\ 012- 70 \
70 \ \1 \1 \
.010 \ 1 \1
Ak 50 50
\\ loads in kips
.006 0 \V 30 30 \ \70
004 qo
31).".5p -■
.002 _JO -
10
.008
70
.75
EXPERIMENTAL
(T—R03-1,T—RC3-2)
----Steel
----Concrete
ANALYSIS
----Steel
----Concrete
Numbers indicate
I0
2.75
MEMBER T—RC3
Steel Stress—Strain Relation is assumed linear
152
Distance from Reinforcement ,in.
PRIMARY CRACK WIDTH
FIG. 5.14b
153 (a)
30 KIPS Steel Stress=37900p.s.i
MEMBER T—R03
crack Width Scale 10./woo in. 1 5
(b) 50 KIPS
Steel Stress=63200p.s'4
MAGNIFIED DEFORMATIONS FIG. 5.15
(c) 70 KIPS
SteelStress=88500p.s.i
ADJACENT TO STEEL
9 kips
6 " 3 //
Distance from end face,in.
Stress ,p.s.i
400
300
200
100
(a)
u; 600
• 500 ra S O
•
400 4.,
300
CONCRETE TRANSVERSE STRESSES ADJACENT TO STEEL
200
100
K P5
FIG.5.17
0 12 14 16
Distance from end face,in.
154-
MEMBER T-RC2-3
10 12 14 16
(b)
1.15 in. DISTANCE FROM STYRT,
• 300
g kips (1) V) 200 Ft FJ co 100
6
3
Distance from end face,in.
4 6 8 10 12 14 16
Stress , p.s.a
. (0)
3.40 in. DISTANCE FROM STEEL
3 //
end face,in. 10 12 14 16
CONCRETE LONGITUDINAL STRESSES
FIG. 5.16
0 2 4 6 8 1.0 12 14r
16
MEMBER T —RC2 —3
DISTRIBUTION OF STEEL STRESSES
FIG. 5.18 600
400
9 kips 6 3
Distance from end face,in.
DISTRIBUTION OF BOND STRESSES
FIG. 5.19
10 12 14 16
Distance from end face,in.
kips
6
Rips DISTRIBUTION OF SLIPS
FIG. 5.20
10 12 14 16 Distance from end face,in.
Stre
ss ,p. s.i
500
400
300
200
100
200
155
MEMBER T —RC2 -3 FIG: 5.21
CONCRETE LONGITUDINAL STRESS CONTOUR BEFORE CRACKING (9 kips)
FIG. 5.22
CONCRETE TRANSVERSE STRESS CONTOUR BEFORE CRACKING (9 kips)
p.s.i
-25
-50
-100
.5
6 .4 13kips
.4 2 1.5 1 5
MEMBER T—RC2-3
FIG. 5.23
Transverse Cracks
CRACK WIDTH AFTER THE FORMATION OF FIRST PRIMARY CRACK
FIG. 5.24
11 kips
(a)
CRACK PATTERN BEFORE THE FORMATION OF FIRST PRIMARY CRACK
Crack Width Scale oX4o/looin.
(b)
1
.31 1,31
.7 .31 .21 .21 1.
1 1 1
.21
.5
(b)
.31 .21 zu,..___IL
.7 I I
.4 .21 .31
.4 1 AI Al
\ (a)
\ \ \ \
\ \ H \ \ \ _i_3___ \ \ /
\ \ _.,\,.- / L_
13k9s 411-■.-
CRACK PATTERN AFTER THE FORMATION OP SECOND PRIMARY CRACK
Crack Width Scale oiLegi000
13kiPS
I 1 I
5 1
1 1 i
I I
6 1 A li .3 11
.4 1 A : al
1 1 i I 1 . Al 1.._
1 .2: ,......_
16
t 3 A6
1' S I
3 2 i .4 A A
CRACK WIDTH AFTER THE FORMATION OF SECOND PRIMARY CRACK
FIG. 5.25
Secondary Crack of greater extension d2
C;A C3
1b 4 1b 1P VPIQ V
111110111•11■■•■■•1111111111111
Secondary Cracks of smaller extension
FIG. 5.27
37 kiln
7 5 3 1.3 .6
MIBER T-RC2-3
CRACK WIDTH AT 25 KIPS
FIG.5.26
2 Crack Width Scale 0 lop000in.
g4
1.1 7.6
I I 16 6.4 i .5
-.....41 ••
.3 I 16. 2.
.5 1 68
1. .4 , 3.8
8,7
13 8.6 4,8 3 2 1 .2 1.5 .8 A
CRACK WIDTH AT 37 KIPS
FIG. 5.28
A V 2.
9 10. 10.7
25 kIPS
Primary Crack
19,6
17,
14.6 25
13.2
MEMBER T —RC2 -3
FIG. 5.29 Crack Width Scale
16
22,4
1 1.6 33 7
CRACK WIDTH AT 61 KIPS
FIG. 5.30
CRACK WIDTH AT 49 KIPS
r--
kips 67 ' 4—
MEMBER T-RC2-3
c
CRACK PATTERN AT 67 KIPS
FIG, 5.31
61 kips
43
80
CD CD O
•
70 A
C.2
ri 0 • 60 Co
50
MEMBER T -RC2 -3
Primary Crack Primary Crack
162
40 31
30
19
Second Primary Crack 13 /i
I0
First Primary Crack 13" 9 iiBefore Crackinm
0 2
4 6
8 10 12 14 16
Distance from end face,in.
STEEL STRESS DISTRIBUTION
FIG. 5.32
50 A C D
30
20
10
60
40 Reinforcement
Reinforced only Concrete
60
T -C2-2 T -C2-4
- T -C2-5 - T-RCI-I
T -RCI-2 — REINFORCEMENT
(WITHOUT CONCRETE SHELL
__L_ 0 0.001 0.002 0.003 0.004 in.
50
17 0 40
0 30
a a
0 20
10
70,000
60,000
50,000
140,000
30,000
20,000
10,000
STE
EL S
TRES
S,
lb/s
q in
AVERAGE UNIT ELONGATION
FROM REF. 40
MEMBER T-RC2-3
M 80
M 70 r-i 0 0
.001 .002 .003 .0 4. in.
AVERAGE UNIT ELONGATION OF STEEL
TIG. 5.33a FIG. 5.33b
g kips
Before Cracking
10 12 14 16 Distance from end face,in.
9 kips
After the First Primary Crack
12 14 16 Distance from end face,in.
200-
a 300 -
(b)
•ra
9 300
a 200
H 100 4) cc
4
a a
100 4-3 to
0
-100
After the Second Primary Crack
9 Kips
Distance from end face,in.
(d.)
14 16
37 kips
10
Distance from end face,in.
200
100
Stress ,p.s.i
0
-100
61 kips
Distance from end face,in.
9 300 sa,
200 a
p,(1) 100 4-) co
0
-100
164
MEMBER T -RO2 —3
CONCRETE LONGITUDINAL STRESS AT 3.55 in. FROM STEEL
FIG. 5.34
600 MEMBER T—RC2-3
11
! (a)
/ 11\\ Distance
from end
\ ,facelin.
;:t617-1 9 Before Cracking
9 After the first Primary Crack
12 14
----9After the second\ 1 Primary crack 1i
DISTRIBUTION OF BOND STRESS
FIG. 5.35
kips 19
37 61
Distance from end face,in.
200
100 v/
0 2 4 1
100
200
300
400
500
600
600 •
500
400 Q) ;-1
300 Cf/
165 • • 500
to 400 1)
300 CQ
200
100
0
100
200
300
400
500
Slip(10-3in.) • 1
MEMBER T-RC2-3
Distance from end face,in. 12 14 16 6 8 10
(a) BEFORE CRACKING
(d) AT 19 KIPS Slip(10-3in.)
Distance from end face,in.
(b) AFTER THE FIRST PRIMARY CRACK
2 (c) AFTER THE SECOND PRIMARY CRACK
Distance from end face,in.
Distance from end face,in.
Distance from end face,in.
• 16
Slip(10-3in.) (e) AT 37 KIPS
Distance from end face,in.
4
VARIATION OF SLIP
FIG. 5.36
16
166
1
O
-1
3
2
r
T—RC 2-3 Crack Width Scale 013-10/i000 Ifl.
( a)
DEFLECTED SHAPE AT 25 KIPS
FIG. 5.37
DEFLECTED SHAPE AT 61 KIP
168
MEMBER T-RC2-3
EXPERIMENTAL
ANALYSIS
Reinforcement Without Conncrete Shell
Steel Stress,ksi 20 40 60
80 100
10 20 30 40 50 60 70
Applied Load,kips
TOTAL CRACK WIDTH (Ref.40) (L = 72in.)
FIG. 5.38
MEMBER T—RC2-3
EXPERIMENTAL
3.50 in.
I First Primary Crack ANALYSIS —1* -- Second Primary Crack
\\ in.\ I
.02 .03 .04
Steel Stress = 50700psi
Steel Stress = 69700 psi Steel Stress = 88700 psi
Applied Load = 40 kips Applied Load = 55 kips Applied Load = 70 kips
FIG. 5.39 PRIMARY CRACK WIDTH (Ref. 40)' rn 0
secondary crack t 3,000
secondary crack pCmary crack (3,000 "hd) (1900, Kgiem2 )
L..
splitting face
It*
inj e
c tin
g ho
le
(a)
I 1. primary crack
( 570 Kg/.,,,2 ) primary crack
( 570 "hr.2) primary crack
( 570 K9/cm2 )
splitting face
—1
primary crack (1,050 1(gic )
primary crack (1,050 Kg/crg)
primary crack ( 1, 050 Kg/cmz )
secondary crack ( 3,000 Kgfcm2 )
primary crack (1,850 Kg/c4
primary crack (1,500 "/.?)
injecting hole
170
(b)
injecting hole
INTERNAL CRACKS FORMED BY TENSION PULL—OUT ON BOTH ENDS OF THE BAR (Ref. 28 )
FIG. 5.40
173.
CHAPTER 6
BEHAVIOUR OF ECCENTRICALLY LOADED TENSION MEMBERS
6.1 OBJECTIVE OF THE ANALYSIS
Eccentrically loaded rectangular tension members resemble
the part of a flexural member located between the two• major transverse
cracks (Fig. 6.1). They do not however have compressive forces
applied to their boundary. Unlike the concentric members the
behaviour of these members, i.e. crack initiation and propagation is
governed by the member length and the eccentricity of the reinforcement
and thus the position of the external load. The importance of these
factors and their influence upon the pattern and the width of cracks
are studied analytically in this chapter. Two long members and three
short ones with different reinforcement eccentricities were chosen
for this purpose. The concrete cross-section of these members was
similar to that of the concentric members.
6.2 DESCRIPTION OF THE MEMBERS
The eccentric members selected for the analysis are the
following:
Short member Si: This member which is identical to Broms(38)
short member TRE3 was analysed for comparative purposes with
experimental data.
Short members S2 and S3: These members are similar to the
short member S1 except for the eccentricity of the reinforcing bar
(measured from the concrete centre line).
1'72
Long member Li: Similar to member Si except that the length of
this member is twice that of member Si.
Long member L3: Similar to member S3 and twice the length of
that member.
All members are reinforced with a single reinforcing bar similar
to concentric members.
The dimensions and material properties are given in Table 6.1. Due
to the symmetry, only half of the members were analysed (Fig. 6.2). The
finite element models with the appropriate boundary conditions are given
in Appendix 2.
6.3 CONDITION OF LOADING
All members were loaded up to 52 kips (steel stress of about
66000 p.s.i.). The application of the incremental loads was similar to
that of the concentric members described earlier.
6.4 BEHAVIOUR OF SHORT TENSION MEMBERS
6.4.1 Short Member S1 (Member TRE3(38)
)
This member with eccentricity of 2.30 in. was loaded up to 11.50 kips
(steel stress of about 14600 p.s.i.) by three increments in the pre-
cracking stage. The longitudinal and transverse concrete stresses
adjacent to the steel and the steel stresses are shown in Figs.6.3a, b
and c respectively. Fig. 6.3d illustrates the variation of the concrete
longitudinal stress near the top face, bottom face and side face
(at the level of steel) just before cracking. Due to the
eccentricity of the steel, the top face undergoes increasing compression
of small magnitude. The maximum stress in concrete is adjacent
to the bar in mid-span. Fig. 6.4a shows the position and the width
of the first crack. This crack initiated adjacent to the steel and
propagated towards the top face and also reached the bottom face.
As the load was increased to 20.50 kips another transverse crack
formed between the first crack and the end. This was, however,
173
limited and did not reach the bottom face and is to be compared to
internal cracks of concentric members. The width of the previous
crack at mid-span widened and a horizontal crack initiated from it
at the level of the steel. Fig. 6.4b and 6.4c show the widths
and inclinations of cracks at 20.50 kips. Fig. 6.5a shows the
width of the cracks and formation of new cracks up to 29.50 kips.
Note that at this stage a horizontal crack formed close to the
load at the end of the member. The formation of a horizontal
crack at the end in the corresponding concentric member of identical
cross-section (member TRC3) occurred at a much earlier stage (10 kips).
Thus the large eccentricity of the steel allowed higher loads to be
applied before a horizontal crack is formed at the loaded end. In
this member unlike the concentric member, the internal transverse
crack near the end precedes the horizontal crack and the redistribution
of stresses caused by the internal crack accelerates the formation
of the longitudinal crack. Hence, an interrelation between internal
transverse cracks and the horizontal crack exists in eccentric members.
The extension of the horizontal crack in the eccentric member is less
than the concentric one. Fig. 6.5b and c illustrate the width,
extension and inclination of all cracks at 52 kips. The transverse
crack in the centre has widened and lengthened. The width of this
crack near the bottom face is considerably larger than the width
above the reinforcement. This is due to the formation of internal
cracks which extend above the reinforcement and which control the
width of the longer cracks. The horizontal cracks have widened
but not lengthened.
Fig. 6.6 shows the experimental crack pattern obtained by
Exams(38). The numbers on this figure refer to applied load in
kips. It can be seen that the analytical central and longitudinal
174
cracks compare favourably with this figure. The smaller
theoretical cracks are,however, only of the order of 2 x 103 in,
maximum width which is of the order of the sensitivity of his
measuring device. The extension of the horizontal crack at the
end was limited in the experimental model. The formation of the
horizontal crack at the root of the transverse crack in'the centre
is not seen in the experiment. However the appearance of horizontal
cracks which start from primary or secondary cracks and their spread
along the reinforcement has been observed by Bromst tests on
rectangular tension members.
6.4.2 Crack Formation in Members S2 and S3 and Comparison with other
Short Members
The short eccentric tension members S2 and S3 with reinforcement
eccentricity of. 1.75 and 1.15 in.were analysed similarly to the member
Si. The precracking behaviour of these members which is between the
concentric member TRC3 and the eccentric member S1 (with the largest
eccentricity) is not given here.
Members S2 and S3 were loaded up to 12 kips when a transverse
crack which initiated at the steel level formed in the centre of the
members. At the same load a horizontal crack formed close to the
load point and spread inwards. In member S2 with a larger
eccentricity than S3, the transverse crack reached the bottom face
of the member and the horizontal crack spread only 1.5 in. from the
end. In member S3 (smaller eccentricity) the extension of the
transverse crack in the centre did not penetrate to the bottom face,
but it remained an internal crack. The horizontal crack in this
member, however, propagated further (2 in.) from the end. Figs 6.7
and 6.8 show the crack widths and inclinations for members S2 and S3
175
at 20.50 kips and 21.00 kips respectively. Comparison of the
behaviour of these members with member Si (largest eccentricity)
and the corresponding concentric member (TRC3 in previous chapter)
of identical properties reveals that as the eccentricity of the
members is increased horizontal cracks form at later stages and
spread less along the reinforcement. On the other hand, as the
eccentricity is increased, transverse cracks are more likely to
reach the bottom face and hence become a primary crack. Thus
the formation and extension of these cracks are strongly affected
by the eccentricity of the steel.
Figs 6.9 and 6.10 illustrate the crack width and pattern of
members S2 and S3 at 52 kips (steel stress of about 66000 p.s.i.).
The internal cracks of shorter length have, their maximum width
above the reinforcement. The total width of the cracks is, however,,
greatest at the level of the reinforcement due to the existence
of small internal cracks. It is also seen that, like concentric
members, small internal cracks control the shape of the other cracks.
The internal crack in the mid-span of member S3 remained internal even
at higher loads probably due to the extension of the horizontal
crack in that member. The behaviour of this member was similar to
concentric members.
Fig. 6.11 (a and b) demonstrates the magnified deflected
shapes of member Si (largest eccentricity) and member S3 (smallest
eccentricity) at final load of 52 kips. Only the external cracks
are shown in these figures.
176
6.5 BEHAVIOUR OF LONG ECCENTRIC TENSION MEMBERS
Members Ll and L3 whose lengths are twice the short
members S1 and S3 represent the behaviour of long eccentric members
TRE2 and TRE3(38) in Broms' tests. However in the analysis,
the tensile strength of concrete was made equal to that of short
members for comparative purposes.
6.5.1 Behaviour of the Members before Cracking
6.5.1.1 Long Member Ll: This member with 2.30 in. eccentricity
sustained 10 kips before cracks appeared. Concrete longitudinal
and transverse stresses, and the bond and slip distribution at 10 kips
just before cracking are shown in Fig. 6.12. The longitudinal
tensile stresses at the bottom face of the'member increases from zero
to a maximum value at the centre (Fig. 6.12a), whereas the longitudinal
stresses in concrete at the level of steel decreases at a distance from
the end and becomes uniform near mid-span. The distribution is
similar to the short member S1 except that the position of maximum
stress due to the length of this member now shifts to the bottom face.
The concrete transverse stress also approaches zero as the length
of the member increases. The maximum concrete stress is the
longitudinal stress at the centre where a crack similar to flexural
cracks initiates.
6.5.1.2 Long Member L3: The distribution of concrete stresses, bond
stress and slip for this member with 1.15 in. eccentricity at
9.50 kips is given in Fig. 6.13. The distribution of concrete
longitudinal stress in Fig. 6.13a shows that for this member (where
eccentricity is smaller) the maximum longitudinal stress is somewhere
between the end and the middle of the member at the steel level. The
177
maximum concrete tensile stress is, however, like concentric
members, the transverse stress near the load point. (Fig. 6.13b).
The first crack in this member is then a horizontal splitting crack
at the end. The existence of high transverse stresses near the
end is attributed to the larger cover thickness of this member
(smaller eccentricity) as mentioned earlier.
6.5.2 Post-Cracking Behaviour of the Members
6.5.2.1 Formation of Cracks in Member Ll: The first crack in
this member occurred at a load of 10 kips and initiated from the
bottom face at mid-span and propagated above the reinforcement by an
amount equal to that below it. At 12 kips another crack appeared
adjacent to steel (secondary crack) at a distance of 3 in. from the
previous crack,spreading towards either face and reaching the bottom
of the member. At 14 kips some secondary cracks appeared in the
member and as the load reached 16 kips one of the internal cracks
extended upwards and also reached the bottom face. The inclination
and the width of the cracks at 20 kips are given in Fig. 6.14a and b
respectively. The width of primary cracks are maximum at the
bottom face. The internal cracks are also seen to close the width
of these cracks near the level of reinforcement in a similar manner
to the way in which primary cracks are closed in concentric members.
The shape of this kind of flexural crack was seen in Broms experiments(40)
An increase of load beyond 20 kips resulted in widening and extension
of the previous cracks as well as the formation of additional internal
cracks. Horizontal cracks also initiated from the root of the
primary cracks at the level of the reinforcement spreading along the
member and approaching each other at higher loads. The formation of
178
the horizontal crack at the end was, however, after the appearance
of some secondary cracks near there. Fig. 6.15a and b give the
crack pattern and crack widths at 52 kips. The extension of
horizontal cracks are less than those of concentric members.
6.5.2.2 Formation of Cracks in Member L3: The first crack to
form in this member (smaller eccentricity) was a horizontal crack at
the end when the load reached 9.50 kips. The propagation of this
crack to about 2 in. resulted in the formation of two internal cracks
at or near the tip of the crack in the member similar to that of the
long concentric member. At 12 kips two more internal cracks formed
within the member. One extended upwards while the other spread
below the reinforcement. At 16 kips another crack initiated at
the level of reinforcement and reached the bottom face of the member.
The distance of this primary crack was about 6.12 in. from the end,
almost twice the lower cover thickness of concrete which agrees well
with the measured average crack spacing of concrete tension member
in Broms' tests (38'39). More internal cracks formed at higher loads
together with widening of other cracks. Fig. 6.16a and b correspond
to the behaviour of this member at 20 kips. The loads at which the
cracks were formed are also given in Fig. 6.16a. Comparison of
Figs 6.14 and 6.16 indicates that:
(a) for the member with smaller eccentricity (Fig. 6.16)
a horizontal crack has formed near the end, whereas
the member with larger eccentricity (Fig. 6.14) has
no horizontal crack at this stage;
(b) in member L3 (Fig. 6.16) only one primary crack
has developed within the member, but in member Ll
(greater eccentricity) there are three transverse
179
cracks which have reached the lower side of the
member (Fig. 6.14)-
(c) the sequence of cracking in the member with smaller
eccentricity is from the end towards the centre,
whereas in the member with larger eccentricity the
sequence of cracking is from the centre (where the
first primary crack is formed) towards the end;
(d) in member L3 (smaller eccentricity) the formation
of primary cracks is affected by the formation
and extension of the horizontal crack which forms
first. In member Ll (larger eccentricity) the
formation and extension of horizontal cracks are
affected by the primary cracks which form first.
Further applied loads for member L3 resulted in the formation
of a few additional internal cracks and the extension and widening of
previous cracks. The crack pattern and the crack width of this
member at 52 kips (steel stress of 66000 p.s.i.) is given in Fig. 6.17.
No more primary cracks are seen to have developed in the member but
horizontal cracks have extended all along the reinforcement.
6.5.2.3 Distribution of Steel Stress and Elongation of the Members:
The steel stress distributions before and after cracking for long
members Ll and L3 are given in Fig. 6.18. The steel stress is nearly
uniform over a length near the centre before cracking. The first
transverse crack at mid-span in member Ll has caused an almost
symmetric distribution of steel stress between the crack and the end
as shown in Fig. 6.18a at 10 kips. The horizontal crack in member L3
has increased the stress in steel near the end as given in Fig. 6.18b
at 9.50 kips. The distribution tends to become uniform at higher
180
loads due to the formation of more internal cracks in both members.
The average unit elongation of the members are plotted in
Fig. 6.19 which is similar to unit elongation of concentric members.
The discontinuity in the elongation of these members is attributed
to the formation of numbers of transverse cracks which occurred at
the steel stress of about 20 to 25 kips. The elongation is
very close to the elongation of free reinforcement (given for
comparative purposes).
6.6 CONCLUDING REMARKS
(1) In eccentric members the formation and extension of
cracks was affected by the member length and the eccentricity of
reinforcement.
(2) In short members as the eccentricity of.the reinforcement
increases the transverse cracks forming at the level of reinforcement
tend to penetrate to the bottom face. Horizontal cracks form
later and their extension is limited. As the eccentricity is reduced
the crack formation approaches to that of concentric member in that
the horizontal cracks form earlier and extend at higher loads, the
transverse cracks tend to remain as internal cracks.
(3) The first crack in the long eccentric member with larger
eccentricity (2.30 in.) was a transverse primary crack initiated
from the bottom face (similar to flexural cracks). The sequence
of cracking was from the mid-span (initiation of the first crack)
towards the end.
(4) The first crack in the long eccentric member with smaller
eccentricity (1.15 in.) was a horizontal crack at the load point
(similar to concentric members), the sequence of cracking was from
the end towards the mid-span.
181
(5) If the eccentricity of reinforcement is increased,
more transverse cracks reach the bottom face of the member. If
the eccentricity is reduced, the horizontal cracks in the member
at the level of reinforcement extend further.
(6) The formation and extension of horizontal cracks and
transverse cracks (primary or secondary) are interrelated so that
the formation of one before the other will control the formation
and propagation of the other.
(7) The spacing and the shapes of the primary cracks are
in agreement with the experimental results.
(8) The formation and extent of horizontal cracks at the
position of primary and some secondary cracks are confirmed by
experimental results.
6.7 EXPERIMENTAL AND ANALYTICAL WIDTHS OF THE CRACKS
The experimental crack widths measured at the side of the
members are compared with the analytical crack widths obtained
by idealised two-dimensional models. Fig. 6.20a shows the
experimental crack width at the side (width b) and directly over
the reinforcement (width c). Fig. 6.20b shows the analytical
crack width of the same member (width a). The width of a crack
at the side face in the experimental model narrows from the side
towards the reinforcement. This is related to the formation of
internal cracks around the reinforcement and across the width of the
member. The width of a crack in the idealised model is uniform
across the width of the member. If no internal crack forms, the
difference between the width b and the width c should reduce to
a negligible value and they can be represented by the analytical
crack width a. To compare the width a with the width b and c,
182
the following parameters must be considered.
(1) Cover thickness of concrete at the side face:
According to Broms'experimental results(39,40)
the average
spacing of all cracks which appear on the surface of concrete at
any given level is twice the distance of the reinforcement from that
level. Thus, only internal tracks between this spacing can affect
the crack profile. If the concrete side cover is small, i.e. the
member is thin, the crack width b and c become close, and if the
analytical and experimental crack spacing are equal, then b or c
can be represented by the analytical crack width a.
(2) Bond resistance:
With a relatively efficient bond resistance more internal cracks
are formed around the steel. If bond resistance is small (formation
of very few internal cracks), the width b and c will become close,
hence can be more accurately predicted by the analytical width a.
(3) Distance from the reinforcement:
The width of the crack across the member width becomes more
uniform as its distance increases from the reinforcement (width b'
and c#, Fig. 6.20a) hence can be represented by the analytical
width a' (in Fig. 6.20b).
From the consideration of internal cracks it can be stated that
for the same analytical and experimental crack spacing, the analytical
crack width (a or a'') gives an upper bound to the experimental crack
width directly over the reinforcement (c or c') and a lower bound
to the experimental crack width at the side face of the member (b or b'').
183
Member Cross section in. x in.
Length in.
Eccentricity in.
Concrete. compressive strength p.s.i.
Modulus of rupture p.s.i.
Modulus of elasticity p.s.i.
S1 3.5 x 8.10 8 2.30 4280 629 4698045
S2 3.5 x 8.10 8 1.75 4280 629 4698045
S3 3.5 x 8.10 8 1.15 4280 629 4698045
Ll 3.5 x 8.10 16 2.3 4280 629 4698045
L3 3.5 x 8.10 16 1.15 4280 629 4698045
Dimensions and material properties of eccentric members
Table 6.1
Top Face
(e)
CD
184
FIG. 6.1
Bottom Reinforcing Deformed Bar Face (1 in. Dia.)
elements E2 Area of the bar assumed for
element E1(at the level of rein- Ei
members forcement) 0.2in2. The rest of the steel area was
Ll t == 2.85 divided equally betweenplements
S2 t =3.01 E2andE3(above and below ,the
S3, L3t=-237 steel level)each 0.2927
IDEALISED MEMBERSFOR THE ANALYSIS
FIG. 6.2
MEMBER S1 (T-RE3) 800
11,50 kips
kips 400
; U) a)
tn. 600 Pa
tif Stress, p. s.i
(b)
Distance from end face,in.
2 3 4
Distance from end face,in.
3 lops 6 a
200 200 kips
Transverse Stresses 11.50 u -200
185
(a) 600
400
Longitudinal Stresses
Stress, p.s.i
600 10000
400 8000
At Bottom Face
(d)
14000
ui 12000 4) Fd
Adjacent to Steel
200 6000
Distance from end face,in.
4 4000
At Top Face
(c)
11.50 kips
//
It
CONCRETE STRESS ADJACENT TO STEEL
2000 CONCERTE LONGITUDINAL STRESS BEFORE CRACKING
(11.50 KIPS)
FIG. 6.3
Distance from end face,in.
1 2 3 4
DISTRIBUTION OF STEEL STRESSES
(a)
0
20.50kips KOs
1150
1.
4i11.
CRACK WIDTH AT 11.50 KIPS
Steel Stress =14600p.s.i
CRACK PATTERN AT 20.50 KIPS
Steel Stress =26100p.s.i 00 01
CRACK WIDTH AT 20.50 KIPS
Steel Stress =26100p.s.i
MEMBER S1 (T—RE3) Crack Width Scale oLitv"" In.
Fig. 6.4
Crack Width Scale oXpoZi000 10. FIG. 6.5
CRACK WIDTH AT 29.50 KIPS
Steel Stress =37500p.s.i
C
(b)
1.4 .5
2,2 27
15 2.2 2. 12
1,3 1.5 1,2 1.5 tg 52kips
1. 1.25 1,1 .2 34 .8
.7 .7 6,2
3.9
13,
CRACK WIDTH AT 52 KIPS
Steel Stress a66200p.s.i
CRACK PATTERN AT 52 KIPS
Steel Stress =66200p.s:i
52kips
2,5 1.2
MEMBER S1 (T—RE3)
MEMBER S1 (T-RE3)
Numbers indicate observation of cracks at given kips
EXPERIMENTAL CRACK PATTERN (From Ref. 38)
188
FIG. 6.6
MEMBER S2(Ecc. =1.75in.)
FIG. 6.7
CRACK PATTERN AT 20.50 KIPS
Steel Stress =26100p.s.i
Crack Width Scale 0410 o
CRACK WIDTH AT 20.50 KIPS
Steel Stress =26100p.s.i
MEMBER S3(Ecc. =1.15in.)
FIG. 6.8 Crack Width Scale oiLliohooa
41n.
CRACK PATTERN AT 21 KIPS
Steel Stress =26700p.s.i CRACK WIDTH AT 21 KIPS Steel Stress =26700p.s.i
Crack Width Scale olLito/l000irt MEMBER S2(Ecc. =1.75in.)
FIG. 6.9
(b)
11
.15 Is
1.4 28 12
1.3 1,3 • 11 3. 2.2 1,2 1.3 48
CRACK WIDTH AT 52 KIPS
I4ps 52
Steel Stress =66200p.s.i Steel Stress =66200p:s.i
(a)
C
kips 52
4--1—
4 in
nACK 'PATTERN AT 52 KIPS
.9 1.4
22 1,7 .7
MEMBER S3(Ecc. =1.15in.)
FIG. 6.10 Crack Width Scale o?„„lio/woo ln.
c
(b)
1,8
52Rips 1.
28 .7 X22
26
3. 1.4 3.1
2,5
.3
CRACK WIDTH AT 52 KIPS
Steel Stress =66200p.s.i
kip 52
s
(a)
< I
I
ain.
CRACK PATTERN AT 52 KIPS Steel Stress =662001y.s.i
•••
(a) (b)
MEMBER S1 (T—RE3) MEMBER S
3
Deflection Scale 011--1 1o/woo irl. 1
MAGNIFIED DEFLECTED SHAPE AT 52 KIPS
FIG. 6.11
MEMBER L1(Ecc. .2.30in.)
FIG. 6.12 A
700
• 600 U) B
U) 0 400
cc
(a)
200
Distance 6om end face,in. 5 6 7 8
C
194
Distance from end face,in.
8
200
6 7 8 Distance from end face,in.
CONCRETE LONGITUDINAL STRESSES AT 10 KIPS
• 600
,17 400 4) co
(b) 200
CONCRETE TRANSVERSE STRESS,ADJACENT TO STEEL AT 10 KIPS
•rl 7 BOND STRESS DISTRIBUTION AT 10 KIPS
0 6 5 4
m 3 (d) 2 1 0 1 3 4 5 6 7 8
Distance from end face,in.
SLIP DISTRIBUTION AT 10 KIPS
Distance from end face,in.
=1.15in.) MEMBER L3(Ecc.
FIG. 6.13
• 500 V2
P:1. 400
• 200
B
C 6 7 8
0 600
co co co 400
CD
200
(b)
Distance from end face,in. 8
600 U)
u)co 400 a)
to
200
6 7 8 Distance from end face,in.
BOND STRESS DISTRIBUTION AT 9.50 KIPS
(d) Distance from end face,in.
3 4 5 6 7
7 tO, 1 6 0 I-1 5
0
195
CONCRETE LONGITUDINAL STRESES AT 9.50 RIPS
CONCRETE TRANSVERSE STRESS ADJACENT TO STEEL AT 9.50 KIPS
SLIP DISTRIBUTION AT 9.50 KIPS
CRACK WIDTH AT 20.KIPS
Steel Stress =25450p.s.i
196 MEMBER Ll(Ecc. =2.30in.)
CRACK PATTERN AT 20 KIPS
in. rack Width Scale q-4""l C
FIG. 6.14
A.
(b)
.6
1,2 ,45 1,4
16
0 Cl oi
.6 13 .6 14
Kips 20 .6 1, .5 16
.7 1,1 12 .5 2,
.6 16 .3 2,4
.3 6
Steel Stress =25450p.s.i
Direction of Propagation
• • — ki r" 20
ps
kips 101 12kiPs
(a)
0
12
15 25 15 .5
1Z I 14 1,4 1.2 34 121 4 42
.4 66
sp
102
24
MEMBER Ll(Ecc. =2.50in.)
FIG. 6.15
197
c
(a)
0
52kips 1 4K
ps
—
CRACK PATTERN AT 52 KIPS Steel Stress =66200p.s.i Crack Width Scale °}2 41011000
(b)
1,1
22
.7
52kips
L
.5 18 1 1.3 12
2. A 13
.8
4.2
22
7,5
8In
3
46 7
32 .2 12 18 .7 12
Al
,31
CRACK WIDTH AT 52 KIPS Steel Stress =66200p.s.i
Crack Width Scale °1o/main'
(b)
12 17
2 .7 1,5
.4 .7 .65 .7_ 141
.6 .5 1. :4 —.2 .5- 1:1
.75 .7 12
A 2,6 1.
3,5
3 21 1.3
198
MEMBER 1,3(Ecc. =1.15in.)
FIG. 6.16 (a)
CRACK PATTERN AT 20 KIPS Steel Stress =25450i).s.i
20kips 9.5 kips 16kip 161 16ki
12
kips
_ g.skIPs
I
20kips •-
CRACK WIDTH AT 20 KIPS Steel Stress =25450p.s.i
1 CRACK PATTERN AT 52 KIPS Steel Stress =66200p.s.i t n.
Crack Width Scale 010/"
199
MEMBER L3(Ecc. =1.15in.)
FIG. 6.17
(a)
(b)
22
311
3,5
1,6 1.2 .7 2, 15 1,2 3,8 14 1.3 1.7 1. 1,3
26 37.6 61 18
1,2
, 1,6 1. 1.2
L__
1.2 .8 12 7.7
12.
3.2
kips r 52 _
1-1-1
CRACK WIDTH AT 52 KIPS Steel Stress =66200p.s.i
52 kips
co 70
U] 0] a
uo 4.3
60 a
4-> to
50
(a)
0 g 70
U]
0 k
to 4-Z
rcil, 60 a
cc 4-)
50
1 3 5 6 7 8 Distance from end face,in.
30
20
20
10
MEMBER L1(Ecc. =2.30in.) MEMBER L3(Ece. =1.15in.)
52 kips
(b)
34 40
30 -
20
10
20
10
10 4 kips
1 i 3 4 5 6 7 8 Distance from end face,in.
STEEL STRESS DISTRIBUTION STEEL STRESS DISTRIBUTION
FIG. 6.18
201
----- MEMBER Ll(Ecc. =2.3Oin.)
L . 16 in. --MEMBER L3(Eec. =1.13in„)
40
20
-- ." .
,. v,' !-'
,/. ' /,.
../. ,, ;' -'
/ /
/
/;/
4///
Free Reinforcement
/
. / // / ,
/'' /1
//./ IiI
.4/ 1: ..
//,' //
/,-- /If'
- f §
in. .001 .002 .003 .004
AVERAGE UNIT ELONGATION
FIG. 6.19
(‘'
202 (a) EXPERIMENTAL MODEL
-e) z 1
/.. 1 /*-
/ / 1 / 1 i
/ I / I / I /
Z i I O
,1 - -- • - - I
/ ■ 1 I / 1 s.)
z
1 ' '
I / I / , , Op) V I z
/ - -4/ IDEALISED TWO DIMENSIONAL MODEL
/ FIG. 6,20
203
CHAPTER 7
ANALYSIS OF FLEXURAL MEMBERS
7.1 ANALYSIS OF A PARTIALLY PRESTRESSED BEAM
7.1.1 Objective of the Analysis
With the use of high strength steel in reinforced concrete
beams, working stresses have increased in magnitude in these members.
The concrete therefore cracks at earlier stages of loading and this
may result in excessive cracking and deflection at service loads.
Thus the structure is required to be checked against various
serviceability limit states of which cracking is an important factor.
For compliance with the limit state of cracking, a reasonably good
estimate of the widths of the cracks is necessary.
The purpose of the present work is to predict analytically
the behaviour of cracking, bond and deflection in a partially pre-
stressed beam. The beam was selected from a series of tests on
flexural cracking of partially prestressed concrete beams of I-
sections conducted by Desayi(80)
The analytical reinforced concrete
model used for tension members was partially modified to simulate
the present flexural member. The results of the analysis are
compared with the well documented experimental data available for
this member
204.
7.1.2 Modilialtical.modelofeBetham
The finite element model adopted for this member differs
with the one developed for tension members (full details of the
models are given in Chapter 4) in the following respects:
(1) Bar elements representing the main tensile
reinforcement were replaced by rectangular elements.
(2) Stirrups and compressive reinforcement was incorporated
into the corresponding concrete elements.
(3) The simple rectangular plane stress element was
improved in the shear term (Chapter 4) to obtain higher accuracy in
bending.
(4) The effect of initial prestressing force was represented
as a point load applied to either end of the member: the increase
of prestressing force at each stage of loading was simulated by an
incremental applied force F = S w .hw in which
Sw
= stiffness of the prestressing wire
and
h = incremental horizontal displacement of the
anchorage points at the ends.
7.1.3 Description of the Experimental Member and Idealisation
The experimental program undertaken by Desayi consisted of
testing nine post-tensioned prestressed concrete I-beams. The
number of prestressing wires and deformed bars used as non-
tensioned reinforcements were the variables of the test. The
detailed properties of the beams tested are available in Refs. 80
and 81. The member selected for the analysis, Beam B3, contained
205
both prestressing wires and non-tensioned reinforcement. Fig. 7.1
shows the overall length and the cross-section of the beam. The
1-beam was 150 mm x 300 mm x 6300 mm overall under two symmetrical
point loads placed 1800 mm apart. Stirrups were provided to avoid
shear failure. Before the casting of the beam a 30 mm diameter
inflated rubber tube was introduced and subsequently withdrawn leaving
a hole for high tensile wires. The behaviour of three beams with
only prestressing wires as the reinforcement (beams Al, Bl and C1)
indicated that very insignificant bond existed between the wires
and the surrounding concrete(80,81)
The details of prestressing wire are as follows:
Type of prestressing wire
7mm diameter
Number of prestressing wire
2
Modulus of elasticity 1.86 x 105N/mm
2
Stress at 2% offset strain. 1200 N/mm2
Ultimate strength of wire 1540 N/mm
2
The properties of the deformed bar used as non-tensioned
reinforcement were as follows:
Type of deformed bar 10 mm diameter
Number of bars 4
Modulus of elasticity
2 x 105 N/mm
2
Yield strength
460 N/mm2
Assumed Poisson ratio 0.3
206
The stirrups were 6 mm diameter bars and their material
properties were assumed to be similar to the 10 mm bars. The
arrangement of stirrups is given in Fig. 7.1a.
Table 7.1 shows the concrete properties and the amount of
prestressing force in the member.
The concrete poission ratio was assumed to be 0.2. The
modulus of rupture was assumed as the tensile strength of concrete
in the analysis.
The stress-strain relationship of prestresSing wire was not
available in the experimental data. This was extracted from
Ref. (85) as shown in Chapter 3.
Due to symmetry only half of the member was analysed. The
finite element model with corresponding boundary conditions and
loading are given in Appendix 3. Fig. 7.2 shows the idealised
member for the analysis. The beam was divided arbitrarily into a
coarse mesh for the shear span and a progressively finer mesh in
the flexural span where the flexural cracking was of interest.
7.1.4 Condition of Loading
After the application of the prestressing force the transverse
load was applied to the member in increments up to 28000 N where some
of the non-tensioned reinforcements yielded. The analysis terminated
at 30000 N when the yielding of a few steel elements resulted in
very large slip in some bond elements. The selected load increments
were such that certain stages of loading in the analysis would
coincide with the corresponding load of the experiment for comparative
purposes of the results. The average load increment in the analysis
was about 3500 N.
207
7.1.5 Analytical Results of the Beam and Comparison with Experimental
Data
7.1.5.1 Application of Prestressing Force:
The model was first loaded by a horizontal compressive force
equal to the prestressing force of the experiment (given in Table 7.1).
The compressive stress in the concrete and at the soffit in the mid-
span (5.768 N/mm2) was very close to the experimental measurement
(5.79 N/mm2). All concrete sections along the beam were in
compression except near the top of the beam.
7.2.5.2 Formation, Width and Spacing of Flexural Cracks:
The transverse load was then applied to the member up to
11700 N before any crack appeared in the member. At this load
all the elements at the bottom of the beam within the flexural span
were at or above the assumed tensile strength of concrete, which
resulted in the formation of four flexural cracks in that region.
The cracks initiated from the lower side and propagated almost
perpendicularly to the reinforcement. Fig. 7.3 illustrates the
sequence of crack propagation after the first crack appeared in
the member. The propagation of the first crack (Fig. 7.3) to a
certain height (45 mm) causes the formation of the second crack
whose propagation depends on further extension of the first crack.
The third crack forms similarly as the extension of second crack
reaches the same height. The extension of this crack is controlled
by the extension of the second crack whose extension depends on the
first crack. The formation and propagation of new cracks are
therefore the result of the extension of preceding cracks. A series
of flexural cracks is thus formed in the member at cracking load.
208
The crack pattern and crack width at 11700 N are given in Figs 7.4a
and 7.4b respectively. The cracks are vertical and their width
varies almost linearly from a maximum at the lower side and there
appears to be no local closure effect at this stage. At 14000 N
(see Fig. 7.5a) the previous cracks extended and new cracks formed
between the existing cracks so that their density was greater near
the point load. Also at this stage two cracks appeared in the
shear span. Three of these cracks (denoted by I) initiated at
the level of reinforcement (internal) and propagated both upwards
and to the soffit. The extension of these cracks was, however,
limited. The increase of load to 17000 N resulted only in the
propagation of a few cracks. The crack pattern and crack width
at this load are given in Fig. 7.5. The crack widths again vary
approximately linearly from the soffit with the exception of one
which has two small adjacent cracks on either side and here there
is local closure. Figs 7.4a and 7.5a can be compared with the
experimental crack pattern of this member given in Fig. 7.6.
Stages 7, 10 and 13 in the experiment correspond aprpoximately
to transverse load of 11700, 14000 and 17000 N (the load stages
in the experiment are based on a uniform increase of the central
deflection as indicated by a dial gauge). At 20000 N more flexural
cracks appeared between the previous cracks. The cracks were
more inclined as they propagated upwards, as shown in Fig. 7.7.
Two more cracks initiated from the reinforcement and reached the
lower side. Load Stage 15 in Fig. 7.6 corresponds to the experimental
crack pattern of this member at working load (20000 N). The width
of the cracks at the level of reinforcement obtained by the analysis
is given in Table 7.2. It can be seen that the widths of the
209
cracks are, in general, no longer varying linearly from the soffit
but because of the number of intermediate cracks local closure
takes place. The maximum value given in Table 7.2 (0.135 mm) is
reasonably close to the experimental maximum crack width of 0.1163 mm
given in Ref. (81).
At 24000 N high tensile stresses were built up•at the top
of the flexural member. The inclination of the maximum principal
stresses (tensile) varied between +30° to -30° with the reinforcement
resulting in the formation of some inclined cracks. Almost all
the large flexural cracks reached underneath the top flange at
28000 and 29000 N. The inclination of these cracks is shown in
Fig. 7.8a. Some internal cracks also formed at the level of
reinforcement in finer mesh of the central region. Formation of
horizontal cracks at the level of the reinforcement in two regions
indicated the presence of high lateral tensile stresses as seen
previously for tension members. The cover thickness of this member
was, however, relatively small, hence higher loads were required
to produce cracks in those sections. Comparison between Fig. 7.8a
and the experimental crack pattern at load Stage 21 (Fig. 7.6),
especially the inclination of cracks formed later below the top
flange indicates good agreement between the two results at 29000 N.
This agreement even extends to the first appearance of bifurcated
cracks adjacent to the top flange, 5 in number in the experiment and
4 in number in the analysis. It is also interesting to see the
various profiles of the flexural cracks at this stage in Fig. 7.8b.
Their width has increased significantly since no more cracks appeared
in the beam at higher loads.
210
Some reinforcing steel elements (non-tensioned reinforcement)
yielded at 28000 N. At 30000 N more steel elements yielded and imposed
very large slips in some bond elements for which the equilibrium
equations were not satisfied at the prescribed number of iterations.
The computation was, therefore, terminated at this load. The yielding
of reinforcing steel was according to the assumption of elastic-
perfectly plastic stress-strain relation. The stress in the prestressing
wire at this load was about 1235 N/mm2 (ultimate stress in wire is
1540 N/mm2). The ultimate load stage in the experiment (Stage 23)
corresponds to transverse load of 35000 N.
7.1.5.3 Distribution of Steel Stress, Bond Stress and Slip:
The distribution of the reinforcing steel stress along the
flexural span at all stages of loading is given in Fig. 7.9. Under
prestress, the stress in the reinforcing steel was 24.50 N/mm2
compression and this was transformed to a stress of 19 N/mm2 tension
when the transverse load of 11700 N (just before cracking) was
applied. The distribution was altered drastically due to the formation
of first flexural cracks at 11700 N. The maximum stresses then
corresponded to cracked sections and minimum stresses to between the
cracks. At 14000 N new cracks appeared in the member and the steel
stress was redistributed. The distribution of steel stresses was
similar at 14000 N and 17000 N as there was no increase in the number
of cracks. However at the higher load, the stress variation is larger
due to the increased bond stresses. More cracks appeared at 20000 N
and the steel distribution changed accordingly. After this load no
significant change occurred in the stress distribution (similar to
tension members at higher loads). Figs 7.10 and 7.11 show the
variation of maximum steel stress (non-tensioned reinforcement) and
211
stress in prestressing wire against the applied load. The maximum
stress in non-tensioned reinforcement increases linearly with the
load before and after cracking. The tensile stress in the wire
(constant along the beam) although itself in the, linear stress-
strain range, has a non-linear variation with the load but the assumption
of no bond for the prestressing wire must be borne in mind.
Distribution of bond stresss and slip (along and perpendicular
to reinforcement) at 11700 N (cracking load) and 11700 N are given
in Figs 7.12 to 7.15. The bond stress along the reinforcement
(horizontal) reverses its direCtion between two cracks as seen for
tension members.
The positive bond stress perpendicular to the reinforcement
(vertical) indicates tension between steel and concrete and negative
values are compression between the two materials (Figs 7.12b and
7.13b). Concrete above the reinforcement tends to separate from
the steel at any cracked section and exerts a compressive force
on the reinforcement between the two cracks. The opposite behaviour
is seen for the concrete below the reinforcement as demonstrated.
This behaviour is more clearly demonstrated in Fig. 7.13a. The
corresponding variation of slip (horizontal and vertical) are shown
in Figs 7.14 and 7.15. The horizontal slip increases at the cracked
section with higher loads. The amount of separation between
concrete and top of reinforcement at cracked sections (Figs 7.14b
and 7.15b) indicates that the splitting between concrete and steel
can occur there. This separation was responsible for the formation
of horizontal cracks in concrete at the level of reinforcement
at sections A-A and B-B (Fig. 7.15b) later in the loading (see
crack patterns at 20000 N and 29000 N). The distribution of
212
bond stress and slip at higher loads is given only for the central
region of the beam where the member is represented by a finer mesh
as shown in Figs 7.16 to 7.19. Due to the formation of flexural
cracks bond stress and slip were redistributed. The slip along
the reinforcement is seen to have increased significantly at
29000 N.
7.1.5.4 Deflection of the Beam:
The moment-deflection plot of the beam is compared with the
experimental measurements in Fig. 7.20. The analytical values of
mid-span deflection are slightly less than the experimental values.
dM The stiffness a — of the analytical structure is, however, almost
exact in the post-cracking phase.
7.1.5.5 Comparison of Crack Width and Spacing:
The plot of crack width at the level of reinforcement versus
steel stress for all flexural cracks obtained by the analysis is given
in Fig. 7.21. The experimental values and the experimental average (81)
crack width are also shown there. It can be seen that there is a
variation in steel stress at each load stage which is not reflected
in the experimental results. The experimental and analytical
spacing of flexural cracks at the level of reinforcement and at
the lower side of the beam (see Fig. 7.16) are plotted against values
of m in Fig. 7.22, where
M - M Cr m = M
ult - M
cr
and
213
M = Bending moment
Mor = Moment at cracking
Mult
= Moment at ultimate load
The calculation of m for the analytical plot was also based
on the experimental values of Mcr
and Mult
for direct comparison
with the experiment.
7.1.5.6 Analytical Results of the Crack Width and Spacing of the
Member with Elements of Equal Size along the Flexural Span:
The crack width scatter obtained analytically and compared
with the experimental values in Fig. 7.21 may suggest that the
scatter is not independent of the element size which varies along
the flexural span. It was, therefore, decided to divide the span
(900 mm) into 21 elements of 42.85 mm length (the minimum experimental
average crack spacing of the member at ultimate load was about 40 mm).
On loading the behaviour of this model was similar to the previous
model. The initiation of the first flexural cracks, the propagation
and formation of new cracks within the span were not affected
significantly when the size of the elements was changed. Fig. 7.23
shows the crack pattern and the crack width of this model at 20000 N
(working load). The shape and the spacing of flexural cracks in
Fig. 7.23 can be compared to that of the previous model (Fig. 7.7). •
The average crack spacing of this model is plotted in Fig. 7.24.
Fig. 7.25 shows the width of the cracks against the steel stress
similar to Fig. 7.21 for the first model. It is seen that a large
scatter of crack widths still exist for this mode. The agreement
214-
between the analytical results and experimental values of the
crack width in Fig. 7.25 suggests that the large scatter of the
crack width is due to the formation of new cracks which affect
the width of the surrounding cracks as explained earlier. It
should be mentioned, however, that the analytical scatter is
independent of the heterogeneous nature of the concrete as concrete
was assumed homogeneous and was equally divided along the flexural
span. Table 7.3 gives the mean crack width and the standard
deviation for the first and the second model and the corresponding
average steel stress in the cracked sections.
7.1.6 Concluding Remarks
(1) Once the first crack formes in the member, formation
and extension of each new crack initiating from the lower sides
is governed by the extension of preceding cracks. Hence'a series of
flexural cracks form in the member at cracking load.
(2) After the formation of the first set of cracks, the
initiation of a new crack between the two previous cracks depends on
the spacing of the cracks. If they initiate from the lower side, they
are likely to propagate.to the same extent as previous cracks,
especially if they form early. As the spacing is reduced most of
the cracks initiate at the level of reinforcement. The extension
of these cracks downwards will reach the soffit if they form early
but their propagation upwards is usually limited due to the small
spacing. If these cracks form at later stages, their extension
will be limited only to the vicinity of reinforcement and they
will remain as internal cracks.
215
(3) The short flexural cracks which form later and reach
the soffit will reduce the width of the larger cracks at the soffit
and at the level of reinforcement. The cracks which remain
internal will reduce the width of the surrounding flexural cracks
only at the level of reinforcement.
(4) Very insignificant change in longitudinal steel stress
distribution was seen in the member at higher loads (similar to
tension members).
(5) At any cracked section the concrete tends to separate
from the top of the reinforcement. High tensile stress can therefore
develop in concrete resulting in the formation of horizontal cracks
at the level of reinforcement in those sections.
(6) Large scatters of the analytical crack widths which
agree with the experimental scatters were obtained through the
assumption of homogeneous concrete material.
(7) The variation in the analytical crack widths at each
load stage is independent of the material properties, since these
have determinate values, and is due entirely to the sequential
propagation of cracks. The analytical sequential propagation of
cracks is carried out at predetermined load increments, whereas
the experimental propagation is continuous under continuously
increasing load. On the assumption that these two methods of
propagation give similar results, it can be suggested that the
variation in the experimental crack width is as much due to
continuously propagating cracks as to the random nature of concrete
strength and bond. Furthermore if the mean and standard deviation
of the size of the analytical cracks is calculated at each load stage,
216
then once the crack pattern is established, the coefficient of
variation of the crack widths remains sensibly constant.
7.2 TWO-DIMENSIONAL ANALYSIS OF A PRESTRESSED BOX BEAM
7.2.1 Objective
(1) To check that the representation of the reinforcement as
contained reinforcing bars inside the corresponding concrete
elements is satisfactory for heavily reinforced concrete members
failing in web compression.
(2) To incorporate the assumed non-linear behaviour of
concrete under a biaxial state of stress in the program.
(3) To confirm that the two-dimensional model of reinforced
concrete developed for a prestressed box beam subject to in-plane
loads and failing by web compression predicts the behaviour of the
member satisfactorily.
(4) To compare crack pattern, strain distribution and
deflection with experimental results
7.2.2 A Brief Descriation of the Experimental Investigation
An experimental investigation was carried out by Edwards
at Imperial College (University of London) concerning the structural
behaviour of a prestressed box beam with thin webs under combined
shear and bending. The work investigated the shear strength of
prestressed beams with thin webs which fail in inclined compression.
The detailed description of the experiment (test arrangement,
prestressing and loading regime, etc.) is given elsewhere(82) The
webs were precast in panels- set into the bottom flange concrete
and made continuous with the top flange by means of continuity
217
reinforcement. A central diaphragm of 200 mm thickness was
placed at mid-span. The box beam was prestressed by a total
of 22 prestressing cables at three sections. The force in each
cable was 105 kN except for the four outer cables which received
half that value. The dimensions and the reinforcement details
are shown in Fig. 7.26. An upward vertical load of 150 kN
was applied to the under side of the diaphragm at the time of
prestressing and maintained there. This was to avoid any
significant tension in bottom flange in mid-span.- Demec points
were attached at selected points initially and, after cracking,
along some concrete struts. Dial gauges were located to measure
the central deflection. The beam was then subjected to a loading
regime. The vertical load was increased from 150 to 750 kN. After a
full set of readings the load was reduced to 150 kN. The structure
was then reloaded up to 1050 kN during which period inclined cracks
appeared in the webs at 840 kN. The beam was unloaded to 150 kN
and then reloaded to 1050 kN and this was followed by 10 cycles of
loading between those loads. The load was later increased by various
increments from 1050 to 2000 kN. At this stage it was considered
necessary to strengthen the load frame and increase the jacking
capacity. The load was, therefore, reduced to 150 kN. The beam
was finally loaded from 150 to 2440 kN at which failure took place
in the webs.
7.2.3 Analytical Model of the Box Beam
The box beam was simulated by plane stress reinforced concrete
elements. Rectangular elements with modified shear terms (Chapter 3)
represented concrete material. Reinforcement consisted of mesh bars
of close spacing in vertical and horizontal directions in webs and
218
flanges and additional vertical bars in webs and prestressing
cables in top flange. As the reinforcement was considered
to be well distributed within any concrete element,perfect bond
was assumed to exist. The stiffness of each individual bar in
any concrete element was, therefore, calculated and incorporated
into the element stiffness. The stiffness of reinforcement
could, however, be updated according to its constitutive relation
and the cracking of the corresponding concrete element as described
in detail in Chapter 4. The concrete was assumed to follow a non-
linear behaviour in a biaxial state of stress. The prestressing
forces of the experiment were applied at the same positions to the
analytical model. The diaphragm was represented by rectangular
elements of the top flange width at the mid-span section. The
idealisation of the box beam and reinforcement arrangement for the
analysis are given in Fig. 7.27. The presence of the cables was
ignored when the compressive forces (prestressing) were applied to
the beam.
7.2.4 Material Properties of the Box Beam
The concrete strength in the analysis was chosen from various
tests on the strength of the webs. These tests consisted of the
28 day cube strength, the strength at the time of testing and the
load-strain curve of cracked and uncracked sections cut from a
cracked web panel. The initial modulus of elasticity for concrete
was taken from previous tests. The concrete tensile strength
was calculated as(86)
:
ft
= 0.78 + 0.6 fc
where
219
ft
= tensile strength of concrete (N/mm2)
fc
compressive strength of concrete (N/mm2)
The stress-strain relation of 15.2 mm diameter strand
(prestressing cable), 3.18 mm diameter reinforcement mesh and
additional reinforcement together with constitutive relation
for concrete material in biaxial state are given in Chapter 3.
Table 7.4 summarises the material properties assumed in
the analysis.
Due to symmetry only half of the member was analysed. The
finite element arrangement boundary conditions, support and load
positions are given in Appendix 4.
7.2.5 Condition of Loading
Prestressing forces and vertical load of 75 kN were first
applied to the half model of the beam in the analysis. Increments
of 225 kN and 200 kN were then applied in the precracking phase.
In the post-cracking phase smaller increments (100 kN) were applied.
The final load in the analysis was 1250 kN (2500 kN for the full
model) at which the compressive stress in the web near the support
was 45.34 N/mm2 (compressive strength of concrete was assumed to be
47.70 N/mm2). The experimental failure load was 2440 kN at which
crushing of concrete and buckling of steel occured in the webs near
the support.
7.2.6 Analysis of the Beam and Comparison with Experimental Results
The prestressing forces were replaced by three horizontal
forces of 1260, 315 and 525 kN at three nodal points in the middle
of the top flange. These points were at 0, 1550 and 2325 mm
220
distance from the end as shown in Fig. 7.27. The beam was
first subjected to these forces and a vertical force of 75 kN
in the mid-span. The vertical deflection in mid-span was 1.73 mm
downwards. Maximum compressive stresses of about 12 N/mm 2
developed in the top flange near the third prestressing force
(Fig. 7.27) and in the mid-span. The inclination of very small
tensile stresses in the web varied between 70° and 90 with the
horizontal axis. At 300 kN the central deflection was almost
zero and the inclination of web tensile stresses along the beam
were between 55° to 70° approximately. The magnitude of compressive
stresses near the third prestressing force was reduced to 10 N/mm2.
As the load was increased to 500 kN the tensile stresses in the web
exceeded the assumed tensile strength of concrete. The inclination
of these stresses in the web varied between 57o and 65
o only. The
central deflection was 1.39 mm upwards. The position of the
maximum compression near the third prestressing force shifted from
the top flange to the top part of the web towards the second
prestressing force. The maximum compressive stress of the top
flange in mid-span also travelled vertically to the bottom part
of the web near the vertical load. The magnitude of these
stresses was- 13 and 11 N/mm2 respectively. At this stage
inclined cracks initiated from the upper side of the web. The
propagation of these cracks was in the same direction as their
inclination (towards the mid-span as well as the lower side of the
beam) as shown in Fig. 7.28. The direction of these cracks varied
between 25° and 33° with the horizontal axis (principal stress
directions were approximately between 57° and 65°). Most of
221
the cracks, however, formed at 30. These angles of cracking
are very close to experimental crack inclinations which varied
between 28° and 32° in the webs. The analytical crack pattern
of the member at this load (1000 kN for full model) is given
in Fig. 7.29. Crack initiation in the experiment was at 840 kN„ The
experimental crack pattern at 1050 kN is given in Fig. 7.30.
More inclined cracks appeared in the web particularly near
mid-span as the load was increased. At 750 kN flexural cracks
formed almost vertically in the top flange near mid-span. Some
later cracks initiated from the lower side of the top flange as
an extension of inclined cracks in the web, and crossed the flange.
Flexural cracks first developed at mid-span and then developed
towards the support. Crack patterns at 950 kN from the analysis
1900 kN for full beam) and 2000 kN from the experiment are
shown in Figs 7.31 and 7.32 respectively. The analysis indicates
that the maximum compressive stress is still in the web adjacent
to the top flange but moves towards the support at this stage.
The magnitude of this compressive stress was about 35 N/mm2. The
mid-span deflection was 13.60 mm. The vertical load was then
increased from 950 kN to 1250 kN at 100 kN increments. At
1250 kN flexural cracks in the top flange spread along the beam
towards the support. These cracks were more inclined at the
lower side of the flange near the support. The analytical crack
pattern at 1250 kN (2500 kN for the full model) given in Fig.7.33
can be compared with the experimental one at failure load (2440 kN)
shown in Fig. 7.34. A maximum compressive stress of 45.35 N/mm2
was found in the web near the support at 2500 kN. Failure load
222
(2440 kN) in the experiment was caused by spalling and crushing
of concrete in the web adjacent to the support region.
Fig. 7.35 shows the position of demec gauges in the
experimental model which recorded the maximum compressive strain
readings during the test. The positions at which maximum
analytical stresses occur and which were compared with the
experimental data are also shown in this figure. The plot of
the analytical compressive strains corresponding to the maximum
compressive stresses against the vertical load is•given in
Figs 7.36 to 7.38. The corresponding four relevant experimental
readings (four webs) are also shown in these figures. The
comparison is reasonably good. The variation of the analytical
maximum compressive stresses against the vertical load is shown
in Fig. 7.39. An approximate linear relationship exists for all
the points before and after the cracking load.
The plot of load-deflection of the beam is also compared
with a series of experimental plots in Fig. 7.40. Considering the
residual deflection due to loading and unloading (Fig. 7.40), the
analytical result is close to the experimental data. The analytical
model indicates a less stiffer structure at higher loads. This is
probably due to earlier and possibly more extensive cracking in
the model.
223
CONCLUDING REMARKS
(1) Within 1000 to 2000 kN of the central load in
which the box beam was not subjected to loading and unloading, the
analytical deflection at mid-span agrees closely with the
experimental deflection (Fig. 7.40). Considering the residual
deflection of the experimental model below 1000 kN, the analytical
deflection compares favourably with that of the experim'ent. The
deflection obtained by the analysis at higher loads (above 2000 kN)
is also in agreement with the experimental deflection of the final
loading regime when the initial residual deflection is taken into
account.
(2) The analytical compressive strains corresponding to
the maximum compressive stresses are in agreement with the maximum
strain readings, especially at post-cracking stages (Figs 7.36 and
7.38).
(3) The orientation of the analytical shear cracks in
the web is similar to those observed in the experiment.
224
STRENGTH OF CONCRETE N/mm2
NET PRESTRESS IN WIRES
Compressive Strength (f c)
Split Cylinder Tensile Strength
Modulus of Rupture
Force N Stress N/mm2
74.4 3.94 5.57 73007 950
Concrete properties and prestress in wires
Table 7.1
0.109 0.110 0.038 0.129 0.052 0.135 0.131 0.035 0.102 0.108 0.089 0.096 0.130
Crack widths at the level of reinforcement (mm)
(At 20,000 N, Fig. 7.7b)
Table 7.2
225
FIRST MODEL Elements of Different Size
SECOND MODEL Elements of Equal Size
Load Ave. Mean Standard Mean Standard Steel Crack Deviation Crack Deviation Stress Widths Widths
N N/mm2 mm mm mm mm
11700 90 0.0544 0.0086 0.0566 0.0065
14000 140 0.066 0.020 0.082- 0.018
17000 205 0.103 0.0266 0.083 0.0255
20000 270 0.099 0.0357 0.0992 0.030
24000 360 0.131 0.0429 - -
28000 450 0.157 0.0416 0.1604 0.0545 •
Mean and standard deviation of crack widths
Table 7.3
CONCRETE STRENGTH N/mm2
INITIAL MODULUS OF ELASTICITY N/mm2 I
Compressive Tensile Concrete Strand Mesh and Additional Reinforcement
47.70 3.60 32360 212206 202000
Assumed material properties of the box beam
Table 7.4
1
150 1, I 150 L 15e150mn IDEALISED FINITE ELEMENT DIVISIONS
FIG. 7.2b FOR THE BEAM
1
150
Pre—stressing Force
-1.
four 10mm.dia.
1mrr
-159, gat 35 Thm
6minks
ls at 150 MM b,150MM.
1 1
two7 wires 111111.
3000
150 mm.
=118.58MM.
1 1
1 1 III I 1 I 1 r 1
1 1 1 1 r
CROSS SECTION OF THE BEAM
FIG. 7.1b
Reinforcement
LEVEL 6- 7-
6 mm. Cha.
Wires
1
1
1
mm. 900 mm. did.
ones
DETAILS OF THE BEAM B3
FIG. 7.1a
Finite Element Nodes
Reinforcement
M 900 M.
LEVEL
mm. 3150
—
IDEALISED CROSS SECTION FOR THE ANALYSIS
FIG. 7.2a
---- Level of Reinforcement
SEQUENCE OF CRACKING OF THE FIRST FLEXURAL CRACKS AT 11.70kN (CRACKING LOAD)
FIG. 7.3
4
i
6
2
227
FIG. 7.4a CRACK PATTERN AT CRACKING LOAD,11.70kN (Stage7 in the Experiment) 0.1 0•5 mm. Crcak Width Scale 1-."4"4
Top Flange
CRACK WIDTH AT CRACKING LOAD,11.70kN
FIG. 7.4b
Bottom Flange Steel Level
Top flange
.035
05
.03
04 .045
06
.05
. __2(E ____________ _ ______ X06 .07
Bottom Flange Steel Level 1- - - ____
-023
.02 AU
035 A 2 .044 .026 -...
-1.0$ • 0 3
17 kN. C.
Numbers indicate extension of
cracks at given kN (I) Crack initiated from the level of the steel
14 14
14 14 14
I 1 1 14 Steel Level
17
Top Flange
14
14 14
Bttom Flange
FIG. 7.5a CRACK PATTERN AT 17kN (Stagel3 in the Experiment) ASMM Crack Width Scale
C 17kN.
17
Top Flange
CRACK WIDTH AT 17kN
FIG. 7.5b .067
.10
.05
.083
.11$
.12
413--
Bottom Flange
Steel Level\A65
.125
.10
I ATF .08
.086 .113
t .125
,045 .04
.07 .075
.095 .10
.108 .11 •
*12 .12 .12 .14 .13 1
.035 .025
.07 ,043
.09 -058
.10 .06 5
01i. .115 .0 75 .12 .08 0
230
Stage -_7
t=======~O~.=====I~~'~==1:L~======i~I~O===~~9===-~9==~~~'O~=p~~~~8~~_=8==~==W='~~z===~9E=~~=9===I~I=O=====i---l0
EXPERlfJIENTAL CRACK PATTERN OF BEAfvl B3 (Ref .80)
FIG. 7.6
20 kN FIG. 7.7a 0.1 to mm.
Crack Width Scale -I CRACK PATTERN AT 20kN (Stagel5-16 in the Experiment)
Top Flange
Bottom Flange
r
teel Levet
45 .03
07 .04
488 .05
.10
11 .057 .066
,07 .09
.12 .11
.05
.07
.11
.15
.16
.15
.115
.10
.09 .08
.07 .11
.13
.15
.12
.075
FIG. 7.7b
20 k N.
Top Flange
Numbers indicate extension of (1)Crack initiated from the level of the steel
2
at given kN
20
20\
20 cracks
20
Bottom Flange _ateel_Level
CRACK WIDTH AT 20kN
29 kN
Top Flange (I) Crack initiated from the level of the steel Numbers indicate extension of
cracks at given kN
24 29 29
24
Bottom Flange
\24
FIG. 7.8a 29 kN
29
29 29 X9
24
I
CRACK PATTERN AT 29kN (Stage2l in the Experiment)
29
29
28 9
29128 29/
0.1 0.5 mm I Crack Width Scale 1-"-"I
,0 16 .20 1
Top Flange
.022 .036
FIG. 7.8b 061 .108
.09 .185
08 .25 Bottom
Flange .09 .27
Steel 14 .24 Level 16
.23 .19 .19
„035
.12
.18
.23
.025
.10
.16
21
.025
.06
.085
.11 026
.035
.11
,15
.14
.016
.066
.10
.14
.015
.04
.05
0
.03
.09 .04
.127 .07
,14 .085
24
18 ,15 12
12
.22
12
.12
41.8 _4232838
.19
.095
.325
.10
04
.16 \
.17 =78-7
.215
.08
.10 067 .097
.16 106 .17 , .07 20 .09
.10
.04 .09
_L._
\.195 f 1818-,112
2C5 .117
CRACK WIDTH AT 29kN
Reinforcement yields at this level
c
24
300-
CNJ
"-- 460 — — — Z -,------ 9K
I Load Point
200
100
0 2175 2 50 2300
2500
2700
2900 3100 3150
Under Pre-stressingLoad Distance from end face, mm.
-50- DISTRIBUTION OF STEEL STRESSES
FIG. 7.9
950 1050 100 200 300 400 500 1150 1250 1350
FIG. 7.11 FIG. 7.10
MAXIMUM STRESS IN REINFORCEMENT STRESS IN PRE-STRESSING WIRE
Load kN 29
25
20
15
10
0
—Yielding of Reinforcement--
Cracking Load
Stress, N/mm2
__ Yielding of Reinforcement
Cracking Load
Stress, N/mm2
~ Load Point
(\J
~ :z:t .. 7 m m ~6
.j..)
tIJ rd 5 ~
24 r-I ctl
.j..) 3 s:1 (\)
.~ 2 j:..j o ::r:
(\J S
~ ~ .. m
olm
-1
-2J'\ -3+ -401-
-5'-
-6
tI2 ~ 2
+=> tIJ
rd s:1 0 0 P=l
r-I
2 -1 'M .j..)
~ -2 l>
\ \ \ \
\ \ \
\
Crack
\
~ Steel moves to the left of concrete Steel moves to the right of concrete
\ \
\ \ , \
\ \ ~ \ \ \ \ \ '\
\ \ ~
+ Tensile
Compressive
~ ~ ~ ~ ~ ~ ,\ .\ \\
\ , \ \
-- Below Reinforcement
-----Above Reinforcement
(b)
Crack
", I " I "-, ___ J "-
.......
HORIZENTAL AND VERTICAL BOND STRESS DISTRIBUTION AT 11.70kN
FIG. 7.12
r-- ..... ................ ,
\ \
Crack
\ , , I \ I
"I I I , \ Distance from I end face, mm.
3150
Crack
Distance from end
1\) \>I \J"I
5
•
7
a
•
6• \
-p 5- (4
O
▪
4 •
H
•
3 0
g 2
Fai 1
0 2250
1
1
1
2400 1
1
/ 2550 27,00
Distance from m.
2850 3000 end face,
3m140
(a)
/ -2 .
-3 -
-4
-5
-6- Below Reinforcement
-7 - ---- Above Reinforcement (b) Hi- Tensile
-- Compressive
Crack Crack Crack
28 0\
Crack
I \ I \
•
Crack
Distance from end face, mm.
", -\ 315%.
Crack
HORIZENTAL AND VERTICAL BOND STRESS DISTRIBUTION AT 17kN FIG. 7.13
Load Point -17 Steel moves to the left of concrete
Steel moves to the right of concrete
Concrete Above Reinforcement
Tension (Separation)
Tension (Separation)
Crack Crack Ow■— Compression
411111111111111P.
Compression Compression Tension
Concrete Below Reinforcement
FIG. 7.13c
237
25 2550
Distance from end face, mm.
31§A. (a)
28 0
•••■■
----Below Reinforcement
Tensile ----Above Reinforcement
- Compressive
„ .05 E .04- - 03- P, ' .02 CQ .01 H 0 - 0 .01 •
(1) -.02 ,i
x-.04- -.05
N5
(a)
Distance from \\end face, mm.
3152
2550
2p50
5'
N
A (b) \ / \
2:50.
/?' B /. / /Distance from
/ end face, mm. 31
/ • • .•••..••• ••••••
• .03 9 ,02 4 .01
0 H °
A FIG. 7.15 HORIZENTAL AND VERTICAL SLIP DISTRIBUTION AT 17kN
Distance from end face, mm.
Vt 31 0
0 g
•
.03 4 .02 H .01 w 0
01 0-.02 -1-'-
•
.03 a>
/". •••., (b) I ,... .... . ., / / S.-
FIG. 7.14 HORIZENTAL AND VERTICAL SLIP DISTRIBUTION AT 11.70kN
Load Point
'Steel moves to the left of concrete -- Steel moves to the right of concrete
d .05 .04
pi •,-1 .03 co .02 H .01 O 0 c.9 -.01 -.02 -.03
43 -.04 -.05
Distance from end face, mm. 31$q
Below Reinforcement
/ ----- Above Reinforcement
FIG. 7.17 (b)
Crack Crack Crack Crack
31 Distance from end face, mm.
R) \A ■4010
Distance from end face, mm.
"17 8 N
■, 7
cn
•
6
463
•
5
g•
4 ro
P 3
4-) ii 2 0
O
•
i
0
-1
-2
-3
- 4
-5
-6
-7
Steel moves to the left of concrete Steel moves to the right of concrete
Ng 7-
\ 6-
m
•
5-a ti3 4 ,c3 O 3- P
Hor izental
0 2R25
2-
Distance from end face, mm. 3:110
HORIZENTAL AND VERTICAL BOND STRESS DISTRIBUTION AT 20kN -8 HORIZENTAL AND VERTICAL BOND STRESS DISTRIBUTION AT 29RN
FIG. 7.16 (b)
T1 2 0c■I o cla s Crack Crack Crack Crack
H Z
•
1 O ^ • 1 , N
i -. 31 "-I N0 4-> 0 g-1 O -P • GO. —1
-1- Tensile -- Compressive
• + Steel moves to the left of concrete 00 .12
H JO cc
H 08 a • .• 0
0
A O pa
Distance from end face,mm. 31 0
HORIZENTAL AND VERTICAL SLIP DISTRIBUTION AT 20kN
-- Steel moves to the right of concrete
Horizental Slip,mm.
.08
.06
.04
.02
0
-.02
-.04
-.06
Distance from end face,mm. 3
---- Below Reinforcement ---- Above Reinforcement
-17 Tensile
__ Compressive
Distance/from end face,mm. 50
• a a
H .04
1 .02 0
a
-.0
(b) (b) • .0 P4 •
▪ .02 /Distance from end face,mm.
3150 0 0
-.02
-.04
O
FIG. 7.18
-.04
HORIZENTAL AND VERTICAL SLIP DISTRIBUTION AT 29k11
FIG. 7.19
•08
1
.07r-----~----r-----r-----r-----r-----~--~0-----L-----~----~----J o Experimental Values
06~fc=Concrete OompressiveStrength ~ Analytical Values • I I I
M= B~nding ~1?ment I b= Breadth of Flange
I I
h= Overal Depth •05 1 I
P4r-'~~~-r-----t~~-t-----r-----r----~----~----~----~----J-----J Ct;
C\J ,.q
~ 031-.=:...----+~~
02 t---+""",, --+---
01 III
j[ L/1?OO~/,500 0 10
1~/300 IL!200 20 30 40
MO~illNT-DEFLECTION PLOTS OF BEAM E? m~
7 2 L=6300
FIG. 0 0
Central Deflection, mm. (~)
L/100 I I I
~r- --.~.- -- - -~------- --
50 60 70 80 90 100mm.
~ .....
(\J 500
~ ~
• II'
tQ 400 co ())
~ .p tJ)
M ~ 300 ()) .p U2
200
100
o
~
l( 7fX}'l..
1'.
)( I(jC.
?l d!:A .
)( It X l( X)(IiC' X ~ IIJ I\,I(~~
1- )( IU( If. X
I( " lI. It
V X J( .( ~ 1< Jt l( • I( x >< I'.
I(
/ '" 'v. x xJ(
,c. " J( x x J( I(
IC
X /~ X~"lIlG( I(JC. - Experimental Average: l( ! ~ tzZ2I Experimental Values
)(xxx ~nalytical Values V'~.,' i(X X
1<.
XXl(~
Crack \-Jid th, mm •
.1 .2 .3 .4
CRACK WIDTH AT STEEL LEVEL OF BEAM B3
FIG. 7.21 ~ I\)
I
--- --- Experimental
Experimental
I
and
-- Analytical Analytical
I at Level at Level
at Level at Level
7 6
.\\
-
--- Stevens(Ref.49)
7 6,
----Illston
N ■ ■
.... \ ■ ■\
, ra
ge C
r- ■
-------- ■
---7.:-- -,-
.
___ Values of m
N
0 .1 .2 .3 .4 .5 .6
7 .8
240
200
160
120
80
40
CRACK SPACING VERSUS m PLOT OF BEAM B3 FIG. 7.22'
k 2 0 N ' SECOND MODLE(Elements of equal length)
4-- Bottom Flange .ateel Level
o K N. In 0 0.5 in •
Crack Width Scalef-Lu-LI FIG. 7.23a CRACK PATTERN AT 20kN
Top Flange
(I) Crack initiated from the level of the steel
Top Flange
.033
.077
.12
126
,06
.078
.08
.05
192
.04
.08 12
BottomFlange
Steel Level
n .08
11 xri Al 06 7.17 09
•08 .0 5 .23
.04 .05 .04 .05 .05 .03 .035 .03
.06 .09 .05 .075 .055 .029 .00 ,045
43 .11 .06 .10 Ass .04 .11 .0 6
.08 .125 .065 .12 .07 .072 .102 .864
.085 125 . 7 - 412- - -al 0 a2.99- -...9.§.8-
.08 .14 .06 .1 4 .06 .12 .09 2 .071 .10 ; .16 .06 .07 .134 .106 .074
FIG. 7.23b CRACK WIDTH AT 20kN
• .1 .3 .2 .5
SECOND MODLE(Elements of equal length)
I
---- ---- Experimental
Experimental
I
Analytical
and
at Level
at Level
at Level 7 6
Stevens(Ref.49) 6 and 7
----Illston
ge C
rack
aa10•114 ValMON. a, ammo 0
Values of m 1 - _
CRACK SPACING VERSUS m PLOT OF BEAM B3
FIG. 7.24
240
200
160
120
4
80
SECOND MODLE(Elements of equal length)
500 E
0 400 0 P CQ
0 300 CR
200
100
X
X
X. K A X X.AX
A X xXXX X
x
X X, X
A X
X xx.
Ai XX
X XXX
armlumummiliwwwviminimmirmwm
)900c
X X
„K 1.0
Experimental Average
Values
Values
rmm Experimental x.gx Analytical
0,0( ummardimmimmumwm■wm
we >oc soft.
----
............ X X
K x<zX
y
----- MK* X X
Crack Widthlmm.
0 .1 .2 .3
CRACK WIDTH AT STEEL LEVEL OF BEAM B3
FIG. 7.25 CTt
25,4 50.8 min. 3,175
3.175
tTP°
25,4 @50,8, rifil.
3,175 11/ fu
25 0
3:175
100b t I
er 4+4 , 9.525
750 25
2 0
59
o rn
40
800
a rn 0
40
Web-Top Flange Reinforcement
EXPERIMENTAL MODLE OF THE BOX BEAM FIG. 7.26
Reaction 4750 MM.
1090 3660
1400 mm.
0 8+8, 9.525
0 MM. 4+4,9,525
50.8 50. 0 / 38 .175 116/
Span and Reinforcement Arrangements
775
L nun.0 115.2 strand
MM. 775 775 775 775 775
0 I0
Diaphragm and Cross 'Section of the Beam
L
Plan-Top Flange and Cables
50,8 C , 25,4 cc
Web-Bottom Flange Reinforcement
111Q( 366( /
i I
Finite Element Divisions i tlf
152 >1. --l- _
12;< . +- __ . Sections r- ----. Strand ~n mm. E 775
E - -- - - Pre-stress~ng I
I 1550 1 kN 5x105 kN. d 3 x105. __ ~~_~
12x10skN. __ ~=~ 1
-! . Forces = - __ /0" -l ... t esslng ______ = __ _
Pre-s r ~~-_- -=-___ 28x3.175 _
= h Reinforcem _~ = = _. -i " . 1 Mes ___ -_ _ _ .... ~I Lono-itudl.na -_-_ = _ -= _ __ _ III I Eo y _~ _ .
It) -..=.-=-=--== _ _ :11 I" -=- II
= = =-- I" 5x3,175 pi "~I II
IU==. 11/ III 'II 1,1 h' II I!I
o 5x3,175
I" ~I Vertical M I R' esh I elnforcement
I mm ~.4 .
'
II II
III III
11rIp III II: III
III I'
II I; II II I,
-----w II III
2M} II' III II'
1
19~
lao
I I. 13 8
Diaphragm and Cross Section
IDEALISATION-OF THE BOX BEAM FIG. 7.27
1550 715
I ~I
Ij
IJ
co ~
"'" 'It N
"'" 'It N
l It)"",
'It N
1.0"", 'It N
,N ~
1.0 'It
~
.F-l 8)(25.4 mm >c 4>(25.4~ 4X25.4%Jrl-
~Ill III: I!I' III
11111111 1111 o 11'1 II~ 2><9.525
0 __ JIll 2X9.525~
J::.
1:1 9: ,II, 11111:" Additional Vertical 11,\ ,III
III, I III 111,1 1 Reinforcement 1,1/ ,II II IIII ',II I,ll It!li,tI 1III 1"1
~ ())
# 2 3
Directions of Crackl CRACK PROPAGATION IN THE WEB
Propagation FIG. 7.28
\\NN\NNNN \\\\\\\ \\,\\\
r4r
CRACK PATTERN AT 1000kN (Approx. Load Stage 14 in theExperiment)
tk N. 1000
FIG. 7.29
SOUTH-WEST WEB. LOAD STAGE 14 SOUTH-EAST WEB. LOAD STAGE 14
NORTH-EAST WEB. LOAD STAGE 14 FIG. 7.30 NORTH-WEST WEB. LOAD STAGE 14
M11( C I IM11 )
\\\
NNNN \\\\\\\\ \\ \\\■\\ \N
N \NN \\ \\\
\\NNN NNNN\\ \\\\\\
k N. 1900
CRACK PATTERN AT 1900kN (Approx. Load Stage 23 in the Experiment)
PIG. 7.31
252
i
SOUTH-EAST WEB. LOAD STAGE 23 SOUTH-WEST WEB. LOAD STAGE 23
NORTH-WEST WEB. LOAD STAGE 23 NORTH-EAST WEB. LOAD STAGE 23
4
\ \ : c ( l < c l 1 I HI( t li ■ H ) ( H I ) ."----.:\ \ ,\, ...----...\\ .''---.. .1.\\. ,̀, -̀. ."--..\:".." '''s.,. "---....\ "--:-...>-...\ ---.... \ . ■,"--->-,>-,...\ : \\>,...\ `,, '-...\\N, \ \ \ \ \ \
N\N \\ N\ \\ .\\ .-.,;., \ \ \ ,\\ \ \'\\ \* \ \ .\\\\ \'`-,\\\\
N \ \\\\\ \\ \ \ \ \\\\ \ \, \ \\\.
N \\NNNN \ \ \\ \\\\ \ \ \\\ \--•-■
CRACK PATTERN AT 2300kN (Approx. Load Stage 30 in the Experiment)
i 2500k N.
FIG. 7:33
SOUTH-EAST WEB. LOAD STAGE 30
NORTH-WEST WEB. LOAD STAGE 30
..-11446,0122"7- essmsi_moiszonitima*Nifiesuffe...;;—INA 254
SOUTH-WEST WEB. LOAD STAGE 30
NORTH-EAST WEB. LOAD STAGE 30
255
.A 3b
C 44 x •
SOUTH-WEST WEB
Load Point
74 ,,C
61 Bx
.A 60
Load Point POSITION OF DEMEC POINTS E 30,31,44 etc. [ A(Element 54)
POSITION OF ANALYSIS POINTS ,E (Element 93) C(Element 362)
SOUTH-EAST WEB
FIG. 7.35
A. 75
7.6 T3
. C
89
NORTH-EAST WEB Load Point
1
1
1
I
C 59X
46 b..
.A 45
t Load Point NORTH-WEST WEB
0 500 1000 1500 0 500 1000 1500
2300
2000
1500
---- Analysis (Elemefit No. 54)
Gauges 30,60,45 and 75
PIG. 7.36
/
1000
500 /
/
Strain X106 150 150
2300
2000
1500
Average Value of the Strain
Analysis (Element No. 93)
Average Value.of the Strain Gauges 31,61,46 and 76
FIG. 7.37
Strain x106
POSITION A POSITION B
tV
!J I
/I I I Compressive Stress,N/mm2
1 10 20 30 40 50
POSITION C
ELEMENT NO.
Position A ---- 54 Position B =--- 93 Position C ---- 362
VARITION OF MAXIMUM COMPRESSIVE STRESSES
FIG. 7.39
---- Analysis (Element No. 362)
Average Value of the Strain Gauges 44,74,59 and 89
Strainx106
1500
2300 a
1-1
•rl
2000 _ a)
1500
1000
500
150 0
0.---
x -----.
x Experimental o a] Analysis
Numbers indicate Load Stages
of the experiment
Mid-Span Deflection,mm. 1 1 I 1 1
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
2
4
6
8
10
12
14
16
18
20
22
LOAD-DEFLECTION PLOT OF THE BOX BEAM
FIG. 7.40
259
CHAPTER 8
COMPUTER PROGRAM
8.1 INTRODUCTION
A finite element computer program has been developed on
CDC 6400 and 6600 machines at Imperial College. The program which
is coded in Fortran IV language predicts the behaviour of the plane
stress reinforced concrete members. The finite element formulations
are based on the displacement (stiffness) approach. The program
deals with the following non-linearities expressed as mathematical
functions;
(a) The assumed reversal bond-slip relationship along
and perpendicular to the reinforcement.
(b) Constitutive relationship of the reinforcement.
(c) The assumed behaviour of concrete in biaxial state
of stress.
(d) Cracking of concrete elements in one or two directions
and crushing of concrete.
Steel and concrete elements can be connected by linkage
(bond) elements, otherwise perfect bond can be assumed, in which
case the stiffness of the reinforcement (according to its position)
can be calculated within the respective concrete element.
The stiffness of the structure is determined as an assemblage
of the element stiffness matrices in a band width form(65)
and
stored as a column vector. The inverse of this matrix is
260
calculated and then multiplied by the column vector of forces
to give the unknown incremental displacements stored in the force
vector space. At any iteration the displacements are found from
the product of the inverted stiffness matrix and the force vector.
When the stiffness of the structure is updated, the new inversion
is sought first and then used for the solution of the subsequent
iterations.
Once the appropriate input data is fed, the program performs
the analysis automatically for a monotonically increasing load
and stops when the predefined criteria are reached.
8.2 FAILURE CRITERION OF MATERIALS •
(a) Steel yields when it reaches a predetermined stress. A
constant yield stress is, however, sustained.
(b) Bond along and perpendicular to the steel is gradually
destroyed due to excessive slip. A constant low value of bond
stress is assumed to exist for a large slip.
(c) Concrete fails by cracking or crushing according
to the assumed criterion of failure in a biaxial state of stress.
The constitutive material relations are given in
Chapter 3.
8.3 CONVERGENCE CRITERIA OF A SOLUTION
At any iteration the principal stresses of an element
obtained from the analysis are compared with the true principal
stresses given by the constitutive relationship. To assess the
accuracy of the results, the difference between the analytical
principal stresses and the true principal stresses (found from
the strains in those directions using initial stress method, see
Chapter 3) is calculated and checked against the true values.
261
Assuming alan and a2an to be the principal stresses of an
element in directions 1 and 2, and alat and a2at are the true
principal stresses in those directions,
alan - alat R1
alat
and
C2an
- a2at R2
a2at
define the ratios of convergence. These ratios are clearly zero
if the material relationship is assumed to be linear. In general
R1 and R2 exist. In order to satisfy the given constitutive
relationship of the element these ratios must be reduced to
negligible values. To obtain the solution of a particular load
increment, the material non-linearity of all the elements should
be satisfied.
The following values were assumed as the criteria of
convergence for the material non-linearities.
(a) for concrete material in biaxial state of stress
(box beam), R1
= R2 < 0.04;
(b) for the reinforcement represented by bar elements
(uniaxial tension members) R1
4 0.02;
(c) for the reinforcement represented by rectangular
elements (I-beam) R1
= R2 < 0.02;
(d) for bond elements R1 = R
2< 0.05.
262
For the cracked concrete elements the convergence criteria
is given along and perpendicular to the crack direction. When
a concrete element cracks in any iteration, the true stress
perpendicular to the crack direction must be zero, i.e. Ulat
= O.
The stiffness of this element was, however, formed earlier for an
uncracked element. Stresses developed therefore in the element
perpendicular to the crack direction. These stresses should
eventually vanish, i.e. 6lan
0. The material relationship of
this element is assumed to be satisfied in an iteration when
alan - alat 4 0.15 N/mm2
or 20 lb/in2
If the element cracks in the second direction or crushes, the following
relation must be satisfied in addition to the above relation.
62an - a
2at 4 0.15 N/mm
2 or 20 lb/in
2
When the stiffness matrix of this element is updated it will
reflect the existence of the crack(s) and its direction, i.e. no
stress will develop perpendicular to the crack direction, or
Elan == 0
at
62an = a = 0
2at
263
8.4 DESCRIPTION OF THE METHOD OF ANALYSIS
Mixed procedures using the initial stress method were
chosen to deal with the material non-linearities. The stiffness
matrix of the structure is updated according to the following
criteria.
(a) When the number of iterations exceeds a
prescribed number.
(b) When the number of cracked elements exceeds a
given number.
(c) When a crack has transversed the whole depth
of the section of the member.
Updating the structural stiffness matrix is quite arbitrary,
with more internal iterations a convergence to the true solution
may be possible. The above criteria were, however, aimed at
obtaining higher accuracy of the results and a more realistic crack
pattern. The speed of convergence of a particular solution
depends on many factors namely:
(1) The geometrical shape of the constitutive relation
of the materials (bond, steel and concrete). A smooth curve
representing this relationship results in a rapid convergence.
(2) The magnitude of the load increment. The smaller
the increment, the more rapid the convergence for the solution of
that increment. However when the load is divided into small
increments, the overall computation time is increased.
(3) Progressive cracking of concrete elements in a particular
load stage. More iteration is necessary in order to obtain an
equilibrium state for the structure at a stage of loading where a
large number of cracks are formed.
264
In order to follow more carefully the effect of initial
cracks on the behaviour of the member, two or three small load
increments were applied subsequent to the initial cracking load.
The stiffness matrix of the structure, however, is not updated
for such load increments unless the conditions previously listed
are reached.
For a careful analysis of the crack propagation (in
members with bond-slip relationship) the concrete elements were
only allowed to crack one at a time, i.e. only one element cracks
(largest principal stress) at any iteration. The released forces
are applied and the stresses will be redistributed in the next
iteration in which the next element may crack and so on.
8.5 FAILURE OF THE STRUCTURE
The structure is assumed to fail in the following modes:
(1) When the structure cannot obtain a state of equilibrium,
i.e. the calculated residual stresses or unbalanced forces (as a
result of the constitutive relation of the materials or due to
yielding of the steel, cracking or crushing of concrete) are large
and do not converge to the predefined criteria within a reasonable
number of iterations.
(2) When the stiffness matrix of the structure becomes
nearly singular (loss of the stiffness of the nodes) resulting
in large nodal displacements.
265
8.6 SEQUENCE OF THE OPERATION
To form the element stiffnesses for the first load
increment the initial tangent modulus of the constitutive relation
was used. In the subsequent iterations the tangent modulus for
steel and concrete and the secant modulus for bond were evaluated.
The following steps are carried out in the main routine
of the program:
(1) Assemble the stiffness matrix of the structure (no
crack is allowed to form until the final solution'of this step
is reached).
(2) Apply the nodal load (external loads or unbalanced
forces) and find the solution of this iteration (incremental
displacements). Add the new increment to the previous displacements
to find total displacements. If the behaviour of the structure
indicates a mode of failure STOP.
(3) Find the incremental strains and stresses from
incremental displacements. Add to the previous values to find
total strains and stresses. Calculate principal stresses and
their directions. Check the concrete principal stresses against
cracking (or crushing).
(4) If the formation of cracking is allowed (this is
controlled by step END later in the program and allows the
formation of a new set of cracks only when the material non-
linearity of all the elements is satisfied. During the process
of cracking such a control is withdrawn) release the nodal forces
of all previously cracked elements since the structural matrix
was last updated and crack the new element.
266
(5) Compare the total stress obtained from the analysis
with the true stress from the constitutive relationship. Calculate
the residual stresses and the ratios of convergence. Convert
the residual stresses into the unbalanced nodal forces. Calculate
the present properties of all the elements.
(6) If no new crack is formed go to Step 8.
(7) If a new set of cracks has transversed the whole
section or the number of cracked elements exceeds the prescribed
number, go to Step 1 to update the stiffness matrix. If there
are more elements to crack, go to Step 2 to crack the element in
the next iteration.
(8) Check the material non-linearities of all the elements
including all previously cracked elements against the convergence
criteria.
(a) The convergence criteria is not satisfied for all
elements. Check the number of iterations, if it does not exceed
the prescribed number, go to Step 2 and perform the next iteration.
If it exceeds the prescribed numbers, go to Step 1 and update the
stiffness matrix.
(b) The convergence criteria is satisfied for all the
elements. Check to see if any uncracked element has reached
the criterion of cracking, if so allow the formation of a new
crack(s) and go to Step 2. If no more elements appear to have
reached the state of cracking, print output of this load increment.
(9) If the total applied load has not reached the ultimate
load, add the new load increment and go to Step 2 to analyse the
structure for this increment. If the ultimate load is reached,
STOP.
267
8.7 A BRIEF DESCRIPTION OF THE SUBROUTINES
The program consists of a number of subroutines called
from the main routine or other subroutines. A brief description
of these subroutines is given below.
(1) Subroutine COORDT
This subroutine generates the geometrical properties
of the nodes, numbering of the nodes and elements and the type of
the elements (steel, concrete or bond).
(2) Subroutine SPRING
The stiffness matrix of the linkage elements (bond
elements) is calculated.
(3) Subroutine REFORCE
This subroutine forms the stiffness matrix of the
reinforcing bars connected to the bond elements.
(a) for uniaxial tension members, bar element is used
(one dimensional);
(b) for the main reinforcement of the partially prestressed
beam, rectangular element with constant shear is used.
(4) Subroutine H-V STEEL
This subroutine calculates the stiffness matrix of
the horizontal and the vertical steel bars for which perfect bond
is assumed.
(5) Subroutine RBMAT
The strain matrix of the rectangular concrete or
steel elements are formed here.
(6) Subroutine DIRECT
This subroutine calculates the elasticity matrix of
the rectangular elements in the direction of the principal stresses
268
and transforms it to the global direction.
(7) Subroutine DBMAT
This subroutine calculates the stress matrix of a
rectangular element,including cracked,crushed or yielded dlements
(it calls DIRECT and DCRMAT).
(8) Subroutine RECTST
This subroutine forms the stiffness matrix of the
concrete rectangular element (it calls RBMAT and DBMAT). There
are three types of these elements:
(a) Simple rectangular elements (uniaxial tension members).
(b) Rectangular elements with constant shear (bending
meMbers).
(c) Rectangular composite elements(it calls H-V STEEL to
add the stiffness of the steel bars).
(9) Subroutine DCRMAT
This subroutine is called from CRACK during the process
of cracking of an element and from DBMAT during the updating of an
element stiffness matrix. It calculates the material property
of the cracked elements.
(10) Subroutine RELEASE
This subroutine calculates the residual stresses
and the unbalanced nodal forces due to the non-linear relationship
of the materials (including cracking, crushing or yielding).
(11) Subroutine CRACK
This subroutine cracks the given concrete element
and releases the nodal forces of the new and previously cracked
elements until the stiffness of these elements is updated.
269
(12) Subroutine BOUND
The displacement boundary conditions of the nodes
given by the input data is introduced into the structural stiffness
matrix.
(13) Subroutine SOLVE
This subroutine solves the given simultaneous equilibrium
equations by the Cholesky method to find the unknown nodal
displacements. The basis of the method is as follows:
(a) Find the inversion of a given non-singular matrix.
The matrix [ K ] is first expressed as the product of a lower
triangular matrix [ L ] and an upper triangular matrix [ U ], so that
[ =[ L 1[ u
If the matrix is symmetric, U = [ L 3 T.N Once the two triangular
matrices [ L ] and [ U ] are calculated, the inverse of the matrix
is determined from
[ K ]-1 = [ U ]-1 [ L ]-1
The inverse of a lower triangular matrix [Li is also a
lower triangular matrix. Similarly the inverse of an upper
triangular matrix will be another triangular matrix. The inverse
of these triangular matrices are easily determined.
(b) Solve the equations for a given force vector. Once
14 [ K j is determined the unknown displacements can be calculated as
[ o ]=[Ku
-1 [ F ]
270
where [ F ] is the vector of nodal forces
[S ] is the vector of unknown displacements
When the structural stiffness is updated the calculations of
both parts (a) and (b) are necessary to obtain the solution. For
the solution of an internal iteration, the calculations of part (b)
are required only.
8.8 COMPUTER TIME FOR THE EXECUTION OF THE PROGRAM
The computer time consumed for the solution of a member is
proportional to number of load increments-and size of the structure
(number of nodes and elements). The time required for the solution
of a load increment depends on the loading history, magnitude of the
increment and criteria for convergence and updating the structural
stiffness matrix.
The total computer time for a given member is the sum of
the time consumed by iterations (the stiffness matrix is kept constant)
and the time to update the stiffness of the structure. Let,
M = total number of elements
N = total number of the unknown displacements
NB = maximum semi-band width of the structural stiffness
matrix.
1. Time for each iteration: This time is divided into
two parts:
(a) To solve the equilibrium equations which are the
product of the already inverted stiffness matrix and the vector of
nodal forces (external or unbalanced).. This is proportional to
N x NB.
271
(b) To calculate stresses, strains and unbalanced forces
for each element. This is proportional to the total number of
elements M.
2. Time for updating the stiffness matrix: This consists
of two parts:
(a) To update the element stiffness matrices, which are .
proportional to M.
(b) To invert the updated stiffness matrix of the
structure, this is proportional to N x NB2 .
8.9 INPUT AND OUTPUT OF THE PROGRAM
The required input data includes:
(1) The geometrical property of the structure, number of
nodes, elements etc.
(2) Material properties of concrete, steel and bond.
(3) Boundary conditions of the structure.
(4) Convergence criteria and updating criteria.
(5) Load increments and the ultimate load.
(6) Output control codes.
The output data at the end of each load increment includes:
(1) Total displacements corresponding to the total load.
(2) Total strains, stresses and principal stresses and
the directions for each element.
(3) The unbalanced nodal forces.
The flow diagram for the main program is given in
Fig. 8.1.
YES YES
Number of cracks equal to the pre-s92121Lfre.„:32
:'rope;ation NO of cracks trans-
versed t'e vhole section?
S COP NO
HO
YES
PRINT OUT-PUT
Material new element nonlinearity
to crack? satisfied for all elements?
272
FPROGRAC BOND
INPUT DATA Read physical and geometrical proncrties.
and the first load increment.
STIFFNESS ASSENBLY BLOCK Set up element stiffness and store in structural stiffness.
introduce boundry conditions.
Solve equilibrium equations,Find incrementaldisplacements.strains and SOLUTION BLOCK
stresses. Find totals. Calculate principal stresses.
I Set force vector to zero.
STOP YES [Does the structure indicaET7i.1
failure mode? NO
YES Examine concrete for cracking?
NO
Release concrete elements already cracked in this stage.
I Crack new concrete element.
Check all elements for material nonliearitv and update the stresses according to the constitutive
Find unbalanced stresses and convert into unbalanced nodal forces.
Has any crack arpeared in this stage ?
YES
YES NO
element NO
cack? YES
YES Add the new load increment
IITERATION BLOCKS Keep the stiffness matrix GO TO SOTUTION SLOCK
'!"l;umber of internal iterations reached
the prescribed number?
NO
constant]
rUPDA.TING BLOCK Update the stiffness matrix
GO TO STIFFNESS A;717?LY 'LOCK
NEXT ITERATION
NEXT STAGE
FLOW CHART OF THE MAIN PROGRAM
FIG. 8.1
273
CHAPTER 9
CONCLUSIONS
9.1 GENERAL CONCLUSIONS
(1) The two-dimensional reinforced concrete model which
was developed and compared with the experimental data can be usefully
employed to study the propagation and the shapes of the tensile cracks
in similar members with different loading conditions.
(2) To deal with the material non-linearities of an analytical
model, mixed procedure using initial stress method is recommended.
For reversal behaviour of a material the secant modulus approach
(initial stress method) can be successfully employed.
(3) The analytical results can be improved if more knowledge
of bond-slip relationships(and its reversal behaviour) become
available.
(4) The formula derived for the concrete biaxial state of
stress (as compared with the test results) can be used for the
constitutive relationship of concrete in conjunction with the
initial stress method.
9.2 CONCLUSIONS FROM THE ANALYTICAL RESULTS
From the analysis of tension members, the following
conclusions may be drawn:
1. Concentric members:
(a) The formation of small internal cracks around the
reinforcement affects the shape of secondary cracks with relatively
large extension, and both define the shape of the primary cracks.
Hence the calculation of the primary crack width is far from reality
if these internal cracks are ignored.
274-
(b) The results of the analysis indicate that the
majority of secondary cracks form before the corresponding primary
cracks. The width of the secondary cracks, unlike that of primary
cracks, is such that they are not visible at the time of formation.
The results suggest that the appearance of secondary cracks, as •
reported by the experimentalist, can be regarded as the widening
of these cracks which are formed earlier.
(c) Once the primary cracks are formed, the variation in
the distribution of the steel stress remains relatively constant.
(d) Primary cracks and some secondary cracks (depending on
their width and extension) when formed, will cause a reversal in the
surrounding bond stress and slip.
(e) Horizontal cracks are initiated at an early stage
of loading at the level of reinforcement, and later extend nearly along
the whole length of the members.
2. Eccentric Members:
(a) The initiation and propagation of cracks in these
members are affected by the member length and the eccentricity of
the reinforcement.
(b) For a member of the same depth as the eccentricity
is increased more transverse cracks will develop at the bottom face
(either initiating from there or from the reinforcement level). As
the eccentricity of the reinforcement is reduced, the type of crack
propagation approaches that of concentric members (more extension
of horizontal cracks at the level of reinforcement as well as more
internal cracks).
275
(c) The formation and extension of horizontal cracks
and transverse cracks (primary or secondary) are interrelated.
Formation of one before the other will control the formation
and extension of the other. This interrelation is also seen for
concentric members.
3. The analytical results of the shape and the spacing
of primary cracks and the distribution of steel stress in tension
members were in agreement with the available experimental data.
The following conclusions may be drawn from the analysis of
the flexural members
1. The partially prestressed beam:
(a) After the formation of the first flexural crack the
initiation and extension of each subsequent crack (from the soffit)
is governed by the extension of preceding cracks. A series of
flexural cracks therefore develop in the member. The formation of
new cracks between the first set of cracks depends on the spacing
of these cracks. If they initiate from the soffit, especially
during the early stages of post-cracking load, they are likely to
propagate similarly as the previous cracks. These cracks affect
the width of the previous cracks at the soffit and at the level of
reinforcement. As the spacing of the cracks reduces most of the
new cracks initiate at the level of reinforcement; they may reach
the soffit depending on the spacing of the surrounding cracks and
the cover of the reinforcement. The propagation of such cracks
forming at the level of reinforcement at later stages is limited;
they remain internal and only affect the width of the surrounding
cracks at the level of reinforcement.
276
(b) At any cracked section, concrete tends to separate
from the top of the reinforcement. High tensile stresses can
therefore develop there which may result in the formation of
horizontal cracks at the level of reinforcement.
(c) The variations of the analytical crack width is due
entirely to the sequential propagation of cracks. Assuming that
the continuous increasing experimental load can be simulated by
the stepwise increasing load in the analysis, the similarity between
the variations of the analytical and the experimental crack width
suggests that the variations of the crack width in the experiment
are as much due to continuously propagating cracks as to the
random nature of concrete strength and bond. Furthermore, the
analytical crack width shows that once the crack pattern is
established, the coefficient of variation of the crack width
remains sensibly constant.
(d) The formation, spacing and width of the flexural cracks
and the mid-span deflection obtained by the analysis were in good
agreement with the experimental data.
2, Prestressed box beam:
The results of the two-dimensional analytical model such as
the orientation of the inclined cracks, the mid-span deflection and
the position cf the maximum compressive stresses were comparable
with the experimental measurements.
277
9.3 RECOMMENDATIONS FOR FUTURE WORK
(1) Similar models can be used to study the non-linear
behaviour of reinforced concrete beams under shear. This may
include:
(a) The effect of shear span-effective depth ratio on
the development of inclined cracks, horizontal splitting cracks
and the interrelation between these cracks near the support.
(b) An assessment of the relative contribution of dowel
action, aggregate interlock and bond resistance to the shear
resistance of the member at increasing load.
(c) The shear resistance of the member with or without
web reinforcement.
(2) The problem of geometric non-linearity (for which the
equilibrium equation must be written with respect to the deformed
geometry) can be incorporated into analytical models using similar
techniques as those employed for material non-linearities.
(3) Plane stress finite elements of higher order could be
employed for a similar analysis. However, such a choice must be
made with an evaluation of the advantages that could be gained as
regards the progressive cracking of the member.
(4) The experimental members were represented by two
dimensional analytical members. In some cases, such a model
may not be a satisfactory approximation of the actual three-
dimensional body for which the progressive cracking is considered
equally significant in the third directibn.
278
REFERENCES
1. SHAH, S.P., WINTER, G.
"Inelastic behaviour and fracture of concrete". ACI Journal,
Vol. 63, No. 9, Sept. 1966, pp. 925-930.
2. POPOVICS, S.
"A review of stress-strain relationship of concrete".
ACI Journal, Vol. 67, No. 14, March 1970, pp.243-2480
3. DESAYI, P., KRISHNAN, S.
"Equation for the stress-strain curve of concrete".
ACI Journal, Vol. 61, No. 3, March 1964, pp.345-350.
4. SAENZ, L.P.
Discussion of "Equation for the stress-strain curve of
concrete" by Desayi and Krishnan. ACI.Journal, Vol. 61,
Sept. 1964, pp.1229-1235.
5. KUPFER, H., MILSDORF, H.K., RUSCH, M.
"BOhaviour of concrete under biaxial stresses". ACI Journal,
Vol. 66, No. 8, Aug. 1969, pp.656-666.
6. BRESLER, B., PISTER, K.S.
"Strength of concrete under combined stresses". ACI Journal,
Vol. 55, No. 3, Sept. 1958, pp.321-345.
7. GOODE, C.D., HELMY, M.A.
"The strength of concrete under combined shear and direct stress".
Magazine of Concrete Research, Vol. 19, No. 59, June 1967,
pp.105 -112.
279
8. McHENRY, D., KARNI, J.
"Strength of concrete under combined tensile and compressive
stress". ACI Journal, Proceedings Vol. 54, No. 10,
April 1958, pp.829-840.
9. NEWMAN, K., NEWMAN, J.B.
"Failure theories and design criteria for plain concrete".
International Conference on Structures, Solid Mechanics
and Engineering Design in Civil Engineering Materials,
Southampton University, April 1969, Paper 83.
10. KUPFER, H.
"Das verhalten des betons unter mehrachsiger Kurzzeitbelastung
unter besonderer bertich-sichtigung der zweiachsigen beans-
pruchung". Deutscher Ausschuss ear Stahlbeton, HEFT 229,
Berlin 1973.
11. HILSDORF,H.
"The specification of biaxial strength in concrete"
(Die Bestimmung der Zweiachsigen Festigkeit von Betons ),
Proceedings, Vol. 173, Deutscher Ausschuss fur Stahlbeton,
Berlin, 1965.
12. BUYUKOZTURK, O., NILSON, A.H., SLATE, F.O.
"Stress-strain response and fracture of a concrete model in
biaxial loading". ACI Journal, Proceedings Vol. 68, No. 8,
Aug. 1971, pp.590-599.
13. LIU, T.C.Y., NILSON, A.H., SLATE, F.O.
"Stress-strain response and fracture of concrete in uniaxial
and biaxial compression". ACI Journal, Vol. 69, No. 5,
May 1972, pp.291-295.
280
14. LIU, T,C.Y., NILSON, A.H., SLATE, F.O.
"Biaxial stress-strain relations for concete". Journal
of the Structural Division, ASCE, Vol. 98, May 1972, pp.1025-
1034.
15. GILKEY, M.J., CHAMBERLIN, S.J., BEAL, R.W.
"Bond between concrete and steel". Bulletin 147, Iowa
State College, Ames, Iowa, 1940.
16. CLARK, A.P.
"Comparative bond efficiency of deformed concrete reinforcing
bars". ACI Journal, Proocedings Vol. 43, No, 4, Dec. 1946,
pp.381-400.
17. CLARK, A.P.
"Bond of concrete reinforcing bars". ACI Journal, Proceedings
Vol.46, No. 3, Nov. 1949, pp.161-184.
18. MAINS, R.M.
"MeasUrement of the distribution of tensile stresses along
reinfOrcing bars". ACI Journal, Vol.23, No. 3, Nov. 1951,
pp. 225-252.
19. MATHEY, R.G., WATSTEIN, D.
"Investigation of bond in beam and pull-out specimens with
high-yield-strength deformed bars". ACI Journal,Proceedings
Vol. 57, No. 9, March 1961, pp.1071 -1090.
20. FERGUSON, P.M., BREEN, J.E., THOMPSON, J.N.
"Pull-out tests on high strength reinforcing bars". ACI
Journal, Proceedings Vol. 62, No. 8, Aug. 1965, pp.933-950.
281
21. FERGUSON, P.M., THOMPSON, J.N.
"Development length for large high strength reinforcing
bars". ACI Journal, Vol. 62, No. 1, Jan. 1965, pp.71-94.
22. PERRY, E.S., THOMPSON, J.N.
"Bond stress distribution on reinforcing steel in beams
and pull-out specimens". ACI Journal,Proceedings Vol. 63,
No. 8, Aug. 1966, pp.865-875.
23. FERGUSON, P.M.
"Bond stress - The state of art". ACI Journal, Vol. 63,
No. 11, Nov. 1966, pp.1161-1190.
24. REHM, G.
"The basic principals of the bond between steel and concrete".
Translation No. 134, Cement and Concrete Association,
London, 1968, 66 pp.
25. LUTZ, L.A., GERGELY, P.
"Mechanics of bond and slip of deformed bars in concrete".
ACI Journal, Vol. 64, No. 11, Nov. 1967, pp.711-721.
26. BRESLER, B., BERTERO, V.
"Behaviour of reinforced concrete under repeated load".
Journal of Structural Division, ASCE, Vol. 94, ST6, June 1968,
pp.1567-1589.
27. NILSON, A.H.
"Bond stress-slip relations in reinforced concrete". Report
No. 345, Department of Structural Engineering, Cornell
University, Dec. 1971.
282
28. GOTO, Y.
"Cracks formed in concrete around deformed tension bars".
ACI Journal, Vol. 68, No. 4, April 1971, pp.244-251.
29. ACI Publication SP-20
"Causes, mechanism and control of cracking in concrete".
Detroit, Michigan, 1968.
30. NAWY, E.G., LOTT, J.I.
"Control of cracking in concrete structures". ACI Journal,
Dec. 1972, pp.717-752.
31. CLARK, A.P.
"Cracking in reinforced concrete flexural members". ACI
Journal, Proceedings Vol. 52, No. 8, April 1956, pp.851-862.
32. HOGNESTAD, E.
"High strength bars as concrete reinforcement, Part 2.
Control of flexural cracking". Journal, Portland Cement
Association Research and Development Laboratories, Vol.4,
No. 1, Jan. 1962, pp.46-63.
33. KAAR, P.M., and MATTOCK, A.H.
"High strength bars as concrete reinforcement, Part 4.
Control of cracking". Journal, Portland Cement Association
Research and Development Laboratories, Vol. 5, No. 1,
Jan. 1963, pp.15-38.
34. WATSTEIN, D., MATHEY, R.G.
"Width of cracks in concrete at the surface of reinforcing
steel evaluated by means of tensile bond specimen". ACI
Journal, Proceedings Vol. 56, No. 1, July 1959, pp.47-56.
283
35. WATSTEIN, D., SEESE, N.A.
"Effect of type of bar or width of cracks in reinforced
concrete subjected to tension". ACI Journal, Proceedings
Vol. 41, No. 4, Feb. 1945, pp.293-304.
36. BROMS, B.
"Stress distribution, crack patterns and failure mechanisms
of reinforced concrete members". ACI Journal, Proceedings
Vol. 61, No. 12, Dec. 1964, pp.1535-1556.
37. BROMS, B.
"Stress distribution in reinforced concrete members with
tension cracks, Part 1". ACI Journal, Proceedings Vol. 62,
No. 9, Sept. 1965, pp.1095-1108.
38. BROMS, B.
"Crack width and crack spacing in reinforced concrete members,
Part 1". ACI Journal, Proceedings Vol. 62, No. 10, Oct. 1965,
pp.1237-1255.
39. BROMS, B.
"Stress distribution in reinforced concrete members with
tension cracks, Part 2". Supplement, ACI Journal, 1965.
40. BROMS, B.
"Crack width and crack spacing in reinforced concrete members,
Part 2". Supplement, ACI Journal, 1965.
41. BROMS, B.
"Technique for investigation of internal cracks in reinforced
concrete members". ACI Journal, Proceedings Vol. 62, No. 1,
Jan. 1965, pp.35-43.
284
42. BROMS, B., LUTZ, A.
"Effect of arrangement of reinforcement on crack width
and spacing of reinforced concrete members, Part 1". ACI
Journal, Proceedings Vol. 62, No. 11, Nov. 1965, pp.1395-1409.
43. BROMS, B., LUTZ, A.
"Effect of Arrangement of reinforcement on crack width and
spacing of reinforced concrete members, Part 2". Supplement,
1965.
44. GERGELY, P., LUTZ, L.A.
"Maximum crack width in reinforced concrete flexural members".
Causes, mechanism and control of cracking in concrete, SP-20,
ACI publication, Detroit, 1968, pp.87 -117.
45. NAWY, E.G.
"Crack control in reinforced concrete structures". ACI
Journal, Vol. 65, No. 10, Oct. 1968, pp.825-836.
46. BASE, G.D., READ, J.B., BEEBY, A.W., TAYLOR, H.P.J.
"An investigation of the crack control characteristics of
various types of bar in reinforced concrete beams". Research
Report No. 18, Part 1, Cement and Concrete Association,
London, Dec. 1966, 44P1).
47, BEEBY, A.W.
"An investigation of cracking in slabs spanning one way".
Technical Report TRA 433, Cement and Concrete Association,
London, April 1970, 33pp.
285
48. ILLSTON, J.M., STEVENS, R.F.
"Internal cracking in reinforced concrete". Concrete,
Vol. 6, No. 7, July 1972, pp.28-31.
49. ILLSTON, J.M., STEVENS, R0F.
"Long-term cracking in reinforced concrete beams".
Proceedings of the Institution of Civil Engineers, Vol. 53,
Dec. 1972, pp.445-459.
50. ODEN, J.T.
"Finite element applications in non-linear structural
analysis". Proceedings of the symposium on application
of finite element methods in Civil Engineering, ASCE,
Nashville, Nov. 1969, pp.419-456.
51. ARGYRIS, J.H.
"Elasto-plastic matrix displacement analysis of three-
dimensional continua". Journal of Royal Aeronautical
Society, Vol. 69, Sept. 1965, pp.633-636.
52. NILSON, A.H.
"Finite element analysis of reinforced concrete".
Ph.D. Thesis, University of California, Berkeley, 1967.
53. Franklin, M.A.
"Non-linear analysis of reinforced concrete frames and
panels". Report No. SESM-70-5, Dept. of Civil Engineering,
University of California, Berkeley, March 1970.
54. MARCAL, P.V., KING, I.P.
"Elastic-plastic analysis of two-dimensional stress systems
by the finite element method". Int. Journal Mech. Scio,
Vol. 9, No. 3, 1967, pp.143-155.
286
55. ZIENKIEWICZ, 0.C., VALLIAPPAN, S., KING, I.P.
"Stress analysis of rock as a 'no tension' material".
Geotechnique, Vol. 18, No. 1, 1968, pp.56-66.
56. ZIENKIEWICZ, 0.C., VALLIAPPAN, S., KING, I.P.
"Elasto-plastic solutions of engineering problems, initial
stress, finite element approach". Int. Journal for Numerical
Methods in Engineering, Vol. 1, Jan. 1969, pp.75-100.
57. ZIENKIEWICZ, 0.C., VALLIAPPAN, S.
"Analysis of real structures for creep, plasticity and
other complex constitutive laws". Conf. on Materials
in Civ. Eng., University of Southampton, 1969.
58. HRENNIKOFF, A.
"Solution of problems of elasticity by the framework method".
Journal of Applied Mechanics, Vol. 8, 1941, A169-A175.
59. McHENRY, D.
"A lattice analogy for the solution of stress problems".
Journal of the Insitution of Civil Engineers, Vol. 21,
No. 2, 1943-1944, pp.59-82.
60. TURNER, M.J., CLOUGH, R.W., MARTIN, M.C., TOPP, L.J.
"STiffness and deflection analysis of complex structures".
Journal of the Aeronautical Sciences, Vol. 23, No. 9, PP, 805-
823, Sept. 1956.
61. CLOUGH, R.W.
"The finite element method in plane stress analysis".
Second conference on electronic computation, ASCE, Pittsburgh,
Sept. 1960, pp.345-378.
287
62. WILSON, E.L.
"Matrix analysis of non-linear structures". 2nd conference
on electronic computation, ASCE, Pittsburgh, Sept. 1960,
pp.415-428.
63. DOHERTY, W.P., WILSON, E.L., TAYLOR, R.L.
"Stress analysis of axisymmetric solids utilising higher
order quadrilateral finite elements". SESM Report 69-3,
Struct. Eng. Lab. University of California, Berkeley, 1969.
64. WILSON, E.L., TAYLOR, R.L., DOHERTY, W.P.,GHABOUSSI , J.
"Incompatible displacement models". Proc. O.N.R. conf.
on numerical methods, urbana, Illinois, Sept.1971, pp.43-67.
65. ZIENKIEWICZ, O.C.
"The finite element method in engineering sciences".
2nd edition, McGraw-Hill Book Co., Inc., London, England, 1971.
66. FENVES, S.J., PERRONE, N., ROBINSON, A.R., SCHNOBRICH, W.C.
"Numerical and computer methods in structural mechanics".
Academic Press, New York and London, 1973.
67. NGO, D., SCORDELIS, A.C.
"Finite element analysis of reinforced concrete beams".
ACI Journal, Vol. 64, No. 3, March 1967, pp.152-163.
68. NGO, D., FRANKLIN, H.A., SCORDELIS, A.C.
"Finite element study of reinforced concrete beams with
diagonal tension cracks". Report No. UC SESM 70-19, Dept.
Civ. Eng., University of California, Berkeley, Dec. 1970.
288
69. VALLIAPPAN, S., NATH, P.
"Tensile crack propagation in reinforced concrete beams
by finite element technique". International conference
on shear torsion and bond in reinforced concrete, Coimbatore,
India, Jan. 1969.
70. CERVENKA, V.
"Inelastic finite element analysis of reinforced concrete
panels under in-plane loads". Ph.D. thesis, University
of Colorado, 1970.
71. VALLIAPPAN, S., DOOLAN, T.F.
"Non-linear stress analysis of reinforced concrete". Journal
of the structural division, ASCE, Vol. 98, No. ST4, Proc.
Paper 8845, April 1972, pp. 885-898.
72. SCHNOBRICH, C., SALEM, M.H., PECKNOLD, D.A., Mohraz, B.
Discussion on "Non-linear stress analysis of reinforced
concrete". Journal of the structural division, ASCE, Vol. 98,
No. ST1O, Oct. 1972, pp. 2327-2328.
73. LUTZ, L.A.
"Analysis of stresses in concrete near a reinforcing bar
due to bond and transverse cracking". ACI Journal, Title
- No. 67-45, Oct. 1970, pp.778-787.
74. NAM, C.H., SALMON, C.G.
"Finite element analysis of concrete beams". Journal of
the structural division, ASCE, ST12, Dec. 1974, pp.2419-2432.
289
75. HAND, F.R., PECKNOLD, D., SCHNOBRICH, W.C.
"Non-linear layered analysis of reinforced concrete plates
and shells". Journal of the structural division, ASCE,
ST7, July 1973, pp.1491-1505.
76. SUIDAN, M., SCHNOBRICH, W.C.
"Finite element analysis of reinforced concrete". Journal
of the structural division, ASCE, ST1O, Oct. 1973, pp.2109-2122.
77. COLVILLE, J., ABBASI, J.
"Plane stress reinforced concrete finite elements". Journal
of the structural division, ASCE, ST5, May 1974, pp.1067-1083.
78. LIN, C., SCORDELIS, A.C.
"Non-linear analysis of reinforced concrete shells of general
form". Journal of the structural division, ASCE, ST3,
March 1975, pp.523-538.
79. TAYLOR, M.A., BROMS, B.
"Shear bond strength between coarse aggregate and cement
paste or mortar". ACI Journal Proceedings Vol.61, No. 8,
Aug. 1964; pp.939-957.
80. DESAYI, P.
"A method for determining the spacing and width of cracks
in partially prestressed concrete beams". Proc. Instn.
Civ. Engrs., Part 2, Vol. 59, Sept. 1975, pp.411-428.
81. DESAYI, P.
"Some studies on the flexural cracking of partially
prestressed concrete beams". University of Leeds, research
report, 1974.
290
82. EDWARDS, A.D., LOVEDAY, R.W.
"Structural behaviour of a prestressed box-beam with
thin webs under combined bending and shear". Concrete
Structures and Technology Section, Department of Civil
Engineering, Imperial College, July 1975.
83. CEDOLIN, L., DEI POLI, S.
"Finite element non-linear analysis of reinforced concrete
bidimensional structures". Technical report No. 40, I.S.T.C.,
Politecnico Milano, Sept. 1974, 66PP.
84. CLOUGH, R.W.
"The finite element method in structural mechanics".
Stress analysis, John Wiley and Sons Ltd., 1965, pp.85 -119.
85. "High tensile steel wires and strands for prestressed concrete".
Richard Johnson and Newphew (Steel) Ltd., Forge Lane,
Manchester 11, England, 1967, 74PP.
86. CEB, "International recommendations for the design and
construction of concrete structures".' FIP Sixth Congress,
Prague, English edition, June 1970.
87. ACI, "Building code requirement for reinforced concrete
(ACI-318)", Sec. 1801(c), 1971.
88. CP11O, "The structural use of concrete", Part 1, Design,
materials and workmanship, 1972.
89. COOK, R.D.
"Concept and applications of finite element analysis".
John Wiley and Sons, Inc., 1974.
291
90. PRZEMIENIECKI, J.S.
"Theory of matrix structural analysis". McGraw-Hill Book
Company, 1968.
91. TIMOSHENKO, S., GOODIER, J.N.
"Theory of elasticity". 2nd edition, McGraw-Hill Book Company,
1951.
105 154
LONG MEMBER T-RC2-3
NO.OF NODES 161 NO.OF ELEMENTS 154
•1101■ ■11601M
Linkage Elements Bond
3
Linkage elements Bond
Steel Bar Elements
4in.
8x.5
16 'ill 22 x .727 i
Concrete Fixed in X Direction 161
7 70 ch
6
5
i 4 0
6
5
4
\ Steel Fixed in Y Direction
Fixed in X and Y Directions
SHORT MEMBER T-RC3
NO.OF NODES 63
NO.OF ELEMENTS 56
3 Fixed in X Direction
Linkage Arrangement Similar to that of Member T-R02-3 (Above)
Fixed in X and Y Directions
Steel Bar Elements Concrete Fixed in Y Direction
Fixed in X and Y Directions
R-
U)
Fixed in Y Direction
APPENDIX 1 FINITE ELEMENT IDEALISATIONS OF CONCENTRIC MEMBERS
.. ~of It ,
10 13 <'! ~ ...... !- 11
10 12 t'! .... .... ....
10
~~ 11 .......
9
&n(l) -110· I---"'": . ...
I
8
(I)",," .. 9
7
~~ 8
C'i<'!A~ 'r-1-
-~ 'l! ''5 3
Fixed in 3 10",," 2 10.
2
10(1) cq. 1 ..
..-- N (J)(j)
5
4ih.
8.11.5
FINITE ELEMENT IDEALISATIONS OF SHORT ECCENTRIC MEMBERS
~ "I.;'
104
hi..
'"'"
NO.OF NODES 104 NO.OF ELEMENTS 108 41n.
r er·5 ~ 10
~m ~
1 1 108 ~, ~
104 13
::Il11 ~
APPENDIX
-ID l2Ef
2
MEl
(: R Sl(T-RE3) • =2.30in.)
('\1,1 112 MEMBER S3(Ecc. =1.15in.)
. - f- .- f--- . 1- - 1--'-
- .- -f- ,- '-I-
...
" 1-10 k
AND
ME (
11 R S2 I"! , I'"
• =1.75~n.) Fixed in X Di~ection
" f---
....- . 1 .~ U1
-0 ~
I.-. ('\I
il"
:~ Fixe
k Dir~ ~ /
/ ,",,~7
II)
"""
d in X ction
3 ~ 31 ~
~ 2
21 I I I ~
~ 1
-li- I l' ~97
'" ' .... I -1s" Bar Elements .~; ___ . __ --1( V 6
(\,) \..0 \J4
15x533 t I
N ..:
N. ,.-.
N ..-
In n
0)
to a. ' Lo _ 10
in u:
-10
13
11
195
12
.
11
- 8
—7
10
9
8
-----H ct4 .-7
3
2
1
----------- -. -- -- —
—
3
2
1
15,033
8192
ck
I t
195
It
N
NI
(I
0.
to
„..
eg:
cl
O
i
12
-11
10
13
12
11 -
9
8
10 - e
4 42 ..g 9
Ck
<1101
4
-3
-2
i
4
3
2
1
192
N
FINITE ELEMENT IDEALISATIONS OF LONG ECCNTRIC MEMBERS APPENDIX 2
NO.OF NODES 192
NO.OF ELEMENTS 195
8.00 n.
8.00 In. C
181
MEMBER Ll(Ecc. =2.30in.)
MEMBER L3(Eco. =1.15in.)
na Boundary Conditions \JO
AND Linkage Arrangement Similar to
those of Short Members
APPENDIX 3
E E 0 O
31 50MM•
12■
V(101.) 0.X3Vtl j x1 5U
11
10.
9
12 132
131 11
, 130 10
129 9
128
7 8 27
126 0 54 1 32a=i=
Fixed in Y Direction
M 9 0 0M.
4x80 4x60 440 9x20 4
132
N
-
o 0
-
t... .4. m
o er
M
I:-
0- c4 Cr ur
130
- 129
- 128
127
126
132
131
372
131
130
129 369
128
127 126
12 12,3 -
FINITE ELEMENT IDEALISATIONS OF PARTIALLY PRE-STRESSED I BEAM NO .OF NODES 384
NO. OF ELEMENTS 372
384
382
375
373
77
375 (-1 3
(c--1, 5
2
Fixed in X Direction
123
APPENDIX 3
LINKAGE ARRANGEMENT Concrete Elements
Concrete Elements 900'
21>c42,857
1
0 N o
2
..,... 1-
° v
,:i .., I-
—131
-130
cc•.;
:-126u"L-127 0
121
132W
125 -
132 372
131
130 -129
129 -128
128
127 126 .. „. _ 121
366 igi
FINITE ELEMENT IDEALISATION OF
SECOND MODLE THE MEMBER WITH ELEMENTS OF EQUAL SIZE IN THE FLEXURAL SPAN NO. OF NODES 384
NO. OF ELEMENTS 372
E 0 co
077 x375 373
'Support, Fixed in Y Direction
M
FINITE ELEMENT IDEALISATION OF THE PRE-STRESSED BOX BEAM NO. OF NODES 459 NO. OF ELEMENTS 400
247.
50 , 2
4750 ,
247.
50 ,
24
7.50
; 106
47,4
7 8
6 54 - •
6
•
5 93
- 5 .
- 4 •
4 3',
3
- 3 • a.
2 362
2 •
1 1 1 •
5xZ18 5x92 10x77.50 29x,8017 1 --
459
456
454 ixed in X Direction
E E
C)
452
451
oad Point
APPENDIX 4
APPENDIX 5
5.A DERIVATION OF THE STIFFNESS MATRIX OF THE ELEMENTS
(1). Stiffness matrix of a simple rectangular element
Let:
Cl = t/3 (Dl.b/a + D6.a/b)
C2 = t/4 (D2 + D6)
C3 = t/6 (-2D1.b/a + D6.a/b)
C4 = t/4 (D2-D6)
C5 = t/3 (D4.a/b + D6.b/a)
C6 = t/6 (D4.a/b 2D6.b/a)
C7 = - (Cl/2 + C3)
C8 = - (C5/2 + C6)
C9 = t.D3
C10 = t.D5
Cll = C9.b/a
C12 = C10.a/b
The upper diagonal terms of the 8 x 8 symmetric matrix
Ki are given as below:
K(1,1) = Cl + C9/2
K(1,2) = C2 + (Cll + C12)/3
K(1,3) = C3
K(1,4) = C4 + C12/6 - C11/3
298
299
K(1,5) = - (CZ + c9)/2
K(1,6) = - C2 - (c11 + c12)/6
K(1,7) = C7
K(1,8) = - C4 + c11/6 - C12/3
K(2,2) = C5 + c10/2
K(2,3) = - C4 + C12/6 - C11/3
K(2,4) = C6
K(2,5) = K(1,6)
K(2,6) = - (c5 + clo)/2
K(2,7) = C4 - C11/3 + C12/6
K(2,8) = C8
K(3,3) = Cl - c9/2
K(3,4) = - C2 + + c12)/3
K(3,5) = K(1,7) , K(3,6) = K(2,7)
K(3,7) = (- ci + c9)/2
K(3,8) = C2 - (c11 + C12)/6
K(4,4) = C5 - c10/2
K(4,5) = K(1,8) , K(4,6) = K(2,8) , K(4,7) = K(3,8)
K(4,8) = (- c5 + clo)/2
K(5,5) = K(1,1) , K(5,6) = K(1,2) , K(5,7) = K(1,3) , K(5,8) = K(1,4)
K(6,6) = K(2,2) , K(6,7) = K(2,3) , K(6,8) = K(2,4)
K(7,7) = K(3,3) , K(7,8) = K(3,4) , K(8,8) = K(4,4)
The integration involved in the calculations is given at the end.
(2) Stiffness matrix of a rectangular element with
constant shear
The procedure for the calculation of the stiffness matrix
of this element is similar to that of simple rectangular elements.
The terms in the third row of the strain matrix LB 14.15, see
300
Chapter 4) associated with the shear strain are given constant
values by substituting the co-ordinate of the element centroid,
1 1 x = -iand y = The strain matrix of this element is then
written as [B] =
-(1-Y)/a O. (1-y) /a 0 Y/a 0-y/a 0
0 -(1-X)/b
0 -x/b 0 Xib 0 -(1-X)/b
- 1 - - 3. - / % 2b 2a 2b 2a '2b 2a 2b 2a
Similarly the stiffness matrix in a given increment using the same
elasticity matrix ( [B] , see Chapter 4) will be
1 1 iirr -1r - [ K J a.b.t. f I [Bj LDJLBJ axdy
o o
where a, b and t are the length, the height and the thickness of
the element. After multiplications and integrations of the above
expression over the element area and letting:
Bl
B2
=
=
t/3 Dl. b/a
t/3 D4. a/b
B3 = t/4 (D2 + D6)
B4 = t/4 (D2 - D6)
B5 = t/2 D3
B6 = t/2 D5
B7 = t/4 D6.a/b
B8 = t/4 D6. b/a
B9 = t/4 D3.b/a
B10 = t/4 D5.a/b
301
The upper diagonal terms of the 8 x 8 symmetric stiffness matrix
of this element [K1] are given as below.
K(1,1) = B1 + B5 + B7
K(1,2) = B3 + B9 + B10
K(1,3) = -Bl + B7
K(1,4) = B4 - B9+ B10
K(1,5) = -(B1/2 + B5 + B7)
K(1,7) = B1/2 - B7
K(1,8) = -K(1,4)
K(2,2) = B2 + B6 + B8
K(2,3) = -B4 - B9 + B10
K(2,4) = B2/2 - B8
K(2,5) = -K(1,2)
K(2,6) = -(B2/2 + B6 + B8)
K(2,7) = -K(2,3)
K(2,8) = -B2 + B8
K(3,3) = Bl - B5 + B7
K(3,4) = -B3 + B9 + B10
K(3,5) = K(1,7) , K(3,6) = -K(2,3)
K(3,7) = -81/2 + B5 - B7
K(3,8) = -K(3,4)
K(4,4) = B2 - B6 + B8
K(4,5) = -K(1,4) , K(4,6) = K(2,8) , K(4,7) = -K(3,4)
K(4,8) = -B2/2 + B6 - B8
K(5',5) = K(1,1), K(5,6) = K(1,2), K(5,7) = K(1,3) , K(5,8) = K(1,4)
K(6,6) = K(2,2), K(6,7) = K(2,3), K(6,8) = K(2,4)
K(7,7) = K(3,3), K(7.8) = K(3,4), K(8,8) = K(4,4)
302
The integrations involved in the derivation of these terms are
given at the end.
(3) Stiffness matrix of the vertical contained reinforcement
(composite element)
Let V = AS.b to be the volume of the bar
and P = E62 / Q - E6 and R - E62
4a 4ab 4b
Then the upper diagonal terms of the stiffness matrix are
given as below:
K(1,1) = ( E1 2 R)V 3a
) K(1,2) = ( (1-m2abE2 + Q)V
K(1,3) = E12 + R)V 3a
K (1,4) = ( mE2 n)v
2ab
K(1,5) = ( - 2 R)V 6a
K(1,6) = ( -1 2abME2 Q)V
K(1,7) = ( E12
R)V 6a
K(1,8) = ( (1-m2ab)E2 + Q)V
m )2E4 K(2,2) = ( (1-
+ P)V b2
K(2,3) = K(1,8)
m K(2,4) = (m(1-)E4 P)V b2
K(2,5) = -K(1,2)
El
303
K(2,6)
K(2,7)
K(2,8)
K(3,3)
K(3,6)
K(4,4)
K(4,5)
K(4,6)
K(4,7)
K(5,5)
K(6,6)
K(7,7)
=
=
=
=
=
=
=
=
=
=
=
m(1-m )E4
K(3,5)
K(3,8)
K(5,7)
K(6,8)
K(8,8)
= K(1,7)
= K(1,2)
= K(1,3),
= K(2,4)
= K(2,2)
K(5,8) =-K(1,8)
( P)V b2
-K(1,8)
(1-m 2E4 ) ( + P)V b2
K(l,l) , K(3,4) = K(1,6) ,
K(1,4) , K(3,7) = K(1,5) ,
m E 2 4 ( + P)V b2
- K(1,4)
2 m E 4 ( + P)V b2
-K(1,6), K(4,8) = K(2,6)
K(1,1), K(5,6) = - K(1,6),
K(4,4), K(6,7) = - K(1,4),
K(1,1), K(7,8) = K(1,2),
(4) Stiffness matrix of the horizontal contained reinforcement
(composite element)
Let V = As.a to be the volume of the bar. P, Q and R
have the same meanings given for vertical reinforcement. The upper
diagonal terms of the stiffness matrix are given as below:
)2El n K(1,1) = ( (1-2 + R)V a
K(1,2) = ( (1-n)E2+ Q)V 2ab
2El ) K(1,3) = ( (1-n + R)V
a2
304
-n K(1,4) = ( (1 2ab
) E2 Q)V
K(1,5) = ( n(12n) E1 R)V a2
K(1,6) = - K(1,2)
K(1,7) = ( n(1-n)E1 R)V a2
K(1,8) = - K(1,4)
K(2,2) = ( E4 + P)V 3b2
K(2,3) = K(1,8)
K(2,4) = ( E42 - P)V
K(2,5) = ( - nE2 Q)V 2ab
K(2,6) = ( - E42 P)V 6b-
nE2 K(2,7) = ( - Q)V tab
K(2,8) = ( E4 + P)V 3b2
K(3,3) = K(1,1), K(3,4) = -K(1,2), K(3,5) = K(1,7)
K(3,6) = K(1,4), K(3,7) = K(1,5) , K(3,8) = K(1,2)
K(4,4) = K(2,2) , K(4,5) = -K(2,7) , K(4,6) = K(2,8)
K(4,7) = -K(2,5), K(4,8) = K(2,6)
n22E1 K(5,5) = ( + R)V
a
K(5,6) = -K(2,5)
K(5,7) . ( 2E1
% -7--- a
K(5,8) = K(2,7)
+ R)V
6b
305
K(6,6) = K(2,2) , K(6,7) = -K(2,7), K(6,8) = K(2,4)
K(7,7) = K(5,5) , K(7,8) = K(2,5) , K(8,8) = K2,2)
(5) The integral expressions resulting from the
multiplications involved in the stiffness matrix
of rectangular elements
1 1 1 1 _ _ (1-i)
2 dxdy = I I (1-x)
2 dxdy 1/3
o o o o
1 1 _2 _ f f x dxdy o o
1 1 - - - =ffy
2dxdy =
1/3
O o
1 1 1 1 (1-x)(1-Y)dXdY = I I XYdXdi = 1/4
o o o o
1 1 1 1 f f R(1-i)didi = I I (1-X)YdXdi = 1/4 o o o o
1 1 1 1 . f f (1-i)YdXdi = I f (1-x)xdxdY = 1/6 o o o o
For rectangular elements with constant shear terms the
following additional expressions were involved:
1 1 1 1 I I (1-i)dXdY = I I (1-X)dXdY = 1/2 o o o o
1 1 _ _ I I xdxdy o o
1 1 = I I YdRdY = 1/2
O 0
The following integral expressions, in addition to the
above expressions, were required for the derivation of stiffness
matrix of the reinforcement contained in the rectangular elements
with constant shear.
306
1 2 - I (1-y) dy =
1 I
- (1-x)
2 dx
0 0
1 1 I (1-y)dy = I (1-X)d7c
1 1 _ I (1-y)YdY ( 1-Tc)XdX = 0 0
1 1 2 x - = I y2 dy = I dx = 1/3
0 0
1 _ 1 _ = I ydy = I xdx =
1/2
= 1/6
5.B TRANSFORMATION MATRIX FOR STRAIN, STRESS, AND THE
ELASTICITY MATRIX
Let [ e ]p, [ a ] and ED I represent the strains,
stresses and the elasticity matrix in the principal directions, and
[ e ] , [a] and [ D ] represent the strains, stresses
and the elasticity matrix in the global direction so that,
LOlp = ED [6] and
Ea] = ED1X [ 6 ]X
Assuming 0 to be the angle between the principal direction 1
and the x axis (taken positive in anticlockwise direction) and
C = CosO, S = Sine
(a)
[ C] and [ x are related as (91):
307
C c ]p = [R ] C E J x (Al)
and
C ]x [R ]
p (A2)
where C2
S2 SC
E R ] - S2 C2 -SC
-2SC 2SC C2-S2
C2 52
-SC
= S2 C2
SC
2SC -2SC C2-S2
(b) By the property of invariance of internal strain
energy
rer Ca p = [E]T [ a L
(T signifies the transpose of the matrix)
from equation (Al)
[e ]T [C OX R
L
hence
(A3)
308
and
[a] = [11-1 ]T [a]
(A4)
(c) In the principal direction
[alp = CD ],[e
from equation (Al)
EGT]p =[D]p [R] EeL
substituting [a] from equation (A4) and premultiplying
by [ R ]T
R T [D]„, [RI [et;
or
D ]x C lx
where
[D]x =
E R ]T
[ D ]p [ R ]
(A5)
is the constitutive law to be used in the global co-ordinate
system. The elasticity matrix in the principal stress direction
ED] p for the steel and the concrete materials is given in
Chapter 3.