THE BEHAVIOUR OF REINFORCED CONCRETE COLUMNS UNDER LATERAL LOADS DUE TO EARTHQUAKES
A thesis submitted for the degree of
Doctor of Philosophy in the Faculty of Engineering of
the University of London
by
Morteza Zahedi, D.I.C., M. Sc.
Imperial College of Science & Technology, London. January 1972
2
ABSTRACT
Two investigations are carried out in this work; the
study of the behaviour of a reinforced concrete column under lateral
load and the study of its behaviour under dynamic loading due to
earthquakes. In the first study, the results of 44tests on reinforced
concrete columns are analysed, and a method is presented for
determining the lateral load-deflection behaviour of a column under
monotonic loading to failure. It is shown that the force-deflection
relationship of a column, computed on the basis of the orthodox
method of integrating the curvatures along the column's length under-
estimates the deflections, especially after the yielding stage. The
behaviour of the hinge zone of a column during the post-crushing stage
is studied, and it is concluded that the hinge length varies linearly
with the neutral axis depth ratio and depends on the level of the
concrete strain. The hinge rotational capacity relies directly on
the crushing strain of the bound concrete. In the second analysis, a column is idealized as a single-
degree-of-freedom system with a load-deflection characteristic similar
to that observed in the tests. The system has a degrading stiffness
characteristic. The responses of the system to six different earth-
quake records are analysed, and the parameters affecting the failure
of the system are studied. It is shown that, the earthquake resistant
capacity of the system depends upon its reserved energy capacity and
is almost independent of its initial natural period of vibration. The
prediction of the resistant capacity of the system, on the basis that
the energy it absorbs during an earthquake is almost the same as
that absorbed by the corresponding similar elastic system, is in most
cases on the safe side. The limitation of this approach is shown and
an empirical relationship for predicting the earthquake resistant
capacity of the system is given. Finally, a study is made on the ductility requirement of
a degrading stiffness system with an elastic-perfectly plastic load-
deformation envelope diagram. The results are compared with those
of an ordinary elasto-plastic hysteresis system. The application of
the above energy approach for predicting the ductility requirement
• is also shown.
3
ACKNOWLEDGEMENTS
The author wishes to express his thanks to the
following: Dr. A. C. Cassell, who supervised this work, for
his valuable guidance and assistance.
Professor S. R. Sparkes for permission to pursue
the work at Imperial College and for his encouragement.
Dr. N. N. Ambraseys and Dr. C. W. Yu, for their
valuable discussions. The Ministry of Science and Higher Education of the
Iranian government, for their financial support during this work. The technical staff of the Structures Laboratory, in
general, and Messrs. P. Guile, J. Neale, and J. Tytler, in particular, for their help in preparing and performing the tests.
My colleagues Mr. J. K. Ward and Miss S. Zaboli
for their assistance in preparing the thesis.
The staff and post-graduate students of the Structures Department for their valuable discussions.
Miss M. C. Sheedy and Miss E. Niblock for typing the thesis, and Miss J. Gurr for the photography.
4
CONTENTS
Page
ABSTRACT
ACKNOWLE
CONTENTS
NOTATION.
CHAPTER 1,
1.1
1.2
1.3
1.4
1.5
2
3
4
8
11
11
13
14
18
21
DGE MENTS
INTRODUCTION AND REVIEW
General Introduction
The Object of the Present Work
The Basic Element Behaviour
Column Behaviour
Review of Previous Work
1.5.1 Moment-Rotation Characteristic,
of a Reinforced Concrete Member
1.5.2 C.E.B. Recommendations and
Related Works
1.5.3 Works of Other Investigators
CHAPTER 2, EXPERIMENTAL WORK
2.1 Introduction
2.2 Details of the Previous Tests
2.3 Present Series of Tests
2.4 Details of the Specimen
2.4.1 Dimensions
2.4.2
Casting and Curing
2,4.3
Concrete Mix
2.5 Test Apparatus
2.6 Loading System
2.7 Instrumentation and Measurement
2.8 Test Procedure
2.9 Test Results
21
25
31
38
38
39
41
42
42
42
43
43
45
45
47
49
5
Page
66 CHAPTER 3, THEORETICAL WORK 3.1 Introduction 66
3.2 Bending Moment-Curvature-Axial Load (M-P-P) Relationship 67
3.3 Assumptions 69 3.4 Material Properties 70
3.4.1 Concrete 70 3.4.2 Steel 72
3.5 Computing Procedure of M-P-P 73
3.6 Force-Deflection Characteristic for the
Rising Branch of the M-P-P Diagram 76 3.6.1 Exact Solution 76
3.6.1.1 Outline of the Computer Program 78
3.6.2 Simplified Solution 82
3.7 Force-Deflection Characteristic in the
Falling Branch of the M-P-P Diagram. 85
3.7.1 Hinge Rotational Stiffness (Analysis of the Experimental Data) 86
3.7.2 The Effect of Different Parameters on KM-4 89
3.7.3 Falling Branch of the M-P-P Diagram 92
3.7.4 Hinge Length 95
CHAPTER 4, COMPARISON OF ANALYTICAL AND
EXPERIMENTAL RESULTS 121
4.1 Introduction 121
4.2 Maximum Bending Moment, "M 121 max.
4.3 Force-Deflection Diagram, "Exact Solution" 125
4.4 Rotation Due to the Base Beam Deformation 127
4.5 Simplified Solution 131
4.6 Comparison of Other Tests 134 4.7 Equivalent Rectangular Stress.Block 135
6
Page
CHAPTER 5, DYNAMIC ANALYSIS OF A COLUMN 195 5.1 Review of Relevant Work 195 5.2 Behaviour of a Reinforced Concrete Column
Under Cyclic Loading 202
5.3 Idealization of the Force-Deflection
Characteristic 205
5.3.1 Envelope Diagram 206
5.3.2 Reversal Path Starting from a
Point on the Envelope Diagram 206
5.3.3 Reversal Path Starting in a Loop 208
5.4 Dynamic Analysis 209
5.4.1 Range of Parameters 211
5.4.2 Earthquake Ground Motion 211
5.4.3 Method of Solution 213
5.5 Earthquake Spectra 214
5.6 Analysis of the Response 216
5.6..1 Effect of Stiffness Degradation 216
5.6.2 Failure Analysis 218
5.6.3 Reserved Energy Capacity of a
System and Its Earthquake Response 222
5.6.4 Variation of Acceleration
Spectrum, Sa, for R = 1 224
5,6.5 R based on the Actual Elastic
Spectrum 227
5.6.6 Ductility Requirement for a
Degrading Stiffness System
with S = O. 0 229
7
Page
CHAPTER 6, DISCUSSION AND CONCLUSION 287 6.1 Summary and Discussion 287
6.1.1 Lateral Load-Deflection Behaviour
of a Column 287 6.1.2 Response to Earthquake Loading 293
6.2 Conclusions 301
6.3 Recommendations for Future Work 302
BIBLIOGRAPHY 309
FIGURES, TABLES AND PLATES Page to Page Figures 1-1 to 1-8 35 37 Figures 2-1 to 2-8 52 59 Figures 3-1 to 3-26 102 120 Figures 4-1 to 4-33 161 193 Figures 5-1 to 5-57 234 286 Figures 6-1 to 6-5 304 308
Tables 4-1 to 4-7 138 160 Table 5-1 233 11■1••
Plates 1 to 6 60 65
8
NOTATION
The symbols used in this work have the following
meanings unless they are defined otherwise in the text. The
notation for Chapters 5 and 6 is given separately.
A Steel total cross-sectional area ..9 Asti Asc Tension, Compression steel cross-sectional area
A" Lateral reinforcement cross-sectional area
b, b1 Width of the concrete cross-section and core section, respectively
d Effective depth
Depth of the compression steel
d1 Height of the concrete core section
do Neutral axis depth
D, Dr D2 Deflection, at the first crushing stage, maximum deflection (at F = 0.0)
Ec Concrete initial modulus of elasticity
Es Steel modulus of elasticity
ec, eoco eco Concrete strain, strain corresponding to the maximum stress in f - e diagram, strain in the outermost fibre of the concrete section
es Steel strain
F Lateral force
fc Concrete stress
ft Concrete 6 x 12 in. cylinder strength
cu Concrete 6 x 6 in. cube strength
fcu Concrete 4 x 4 in. cube strength
fd fss, f su Steel stress, yield level, ultimate level
fs Compression steel stress
Y Yield level in lateral reinforcement
s ft Concrete tensile strength 2h Height of a fixed-ended column
9
Km-.9 Hinge rotational stiffness during post-crushing stage
Lp Hinge length
M, MY Mu Bending moment, at yield, at ultimate
n Neutral axis depth ratio (n in Chapters 5 and 6 is used as the critical damping ratio)
P, Pu Axial load, ultimate load
Pb Balance load
Lateral reinforcement volumetric ratio
S Lateral reinforcement spacing, initial stiffness (S in Chapters 5 and 6 has other definition)
Z Shear span
P, P3e Pu Curvature, at yield, at ultimate
4' 3, u Rotation, at yield, at ultimate
t Height of the concrete section
10
The following symbols are used in Chapters 5 and 6:
a Acceleration equivalent to lateral load yield level, f /M
C Damping coefficient
E, Ed To Earthquake amplification factor
F, Ft A constant proportional to earthquake intensity
F(X), f(x) Lateral load
F3? y
Lateral load yield level
g Acceleration of gravity
2h Height of a fixed-ended column
K Initial stiffness
M Mass of the system
n Critical damping ratio (nd in Chapter 6 is neutral axis depth)
P Axial load
q' qd qd qe Ductility factor
R, R Ratios of E/Eo and E/Eo or qoh and qo/q
S Ratio of the gradient of the falling branch of the force-deflection diagram to the initial stiffness in the diagram
Sa! Sv, Sd Acceleration, velocity, displacement response spectrum
Sa Acceleration spectrum deduced from the non-linear system analysis
Sam Average, maximum §a, SI, SI Earthquake intensity corresponding to Sa and Sa T Initial natural period of vibration
t Time
W, Wo Reserved energy capacity
Dead load
2(0 Earthquake acceleration
Circular natural frequency
11 CHAPTER 1
INTRODUCTION AND REVIEW
1. 1 General Introduction
Research efforts in recent years have shown a large
discrepancy between the results of elastic analysis of structures
subjected to strong motion earthquakes and the forces that typical
building codes specify for design. The code specifications are
favoured by the fact that they produce economical designs which,
in most cases, have successfully withstood severe earthquakes.
This discrepancy cannot be reconciled even by a reasonably large
amount of damping in the structure and by its uncalculated reserve
strength. The remaining gap is commonly explained by the energy
dissipation in structures through the inelastic deformation of the
frame members and non-structural elements, which is produced
by an earthquake of moderate intensity.
During an earthquake, a certain amount of energy is
transmitted into the structure from the ground. The amount
depends upon the characteristics of the ground motion as well as
the structure itself. Some of this energy is stored by the structure
in the form of strain energy, and some is dissipated through inelastic
deformation of the members. The stored energy is later dissipated through hysteretic damping of the members, friction forces at the
joints, and a part of it is radiated back to the ground. On this
account the safety of a structure, as far as earthquake loading is
concerned, depends upon the amount of energy it can absorb and
dissipate through different forms.
It is due to this fact, that the design of earthquake resistant
structures requires each member to be ductile enough so that it can
maintain the energy capacity demanded by an earthquake. This
point is particularly important in the case of modern types of
buildings where non-structural elements are reduced to a minimum,
12
A proper design procedure for this purpose needs a thorough
knowledge of the behaviour of various elements in their inelastic
range, as well as their performance under dynamic loading.
These two subjects have been the main theme of research in this
field, during the past decade, in different institutes all over the
world.
In the field of reinforced concrete structure, it is now
known that although concrete material by itself is brittle, in the
sense that it cannot tolerate strains beyond a limited level, it is
capable of bearing a large strain when properly confined within
reinforcements (1, 2, 3, 4). A compression strain of the order of
0. 030 has been observed in some tests (1). On this account,
reinforced concrete members show a considerable amount of
ductility.
The parameters affecting the ductility of reinforced
concrete members vary in beams and columns. In under-rein-
forced beams the presence of compression steel increases the
ductility (5, 6, 7, 8), but in over-reinforced beams the crushing
strain of the concrete governs the ductility. The concrete crushing
strain depends upon the amount of confinement provided by the shear
reinforcements in the beam. Other parameters, such as the ratio
of the steel area to the cross-sectional area and the concrete and
steel strength, also influence the member' s ductility (5, 8). In
columns, the main parameter is the axial load on the column. The
behaviour of a beam-column member, under the combined action of
axial load and bending moment, is the subject of the first part of this
thesis. A discussion on the behaviour of such a member will be
given in this chapter.
In the inelastic analysis of a structure under dynamic
loading due to an earthquake, it is a tradition to represent the
behaviour of each member by a simple elasto-plastic or bi-linear
13
model. In these models, the structure preserves its initial stiffness
in all unloading and reloading excursions, no matter how much it has
suffered from inelastic deformation. This performance may be true
for steel structures, but it is definitely not for concrete structures.
It has been shown in various experiments (9, 10, 11) that reinforced
concrete members gradually lose their stiffness as the extent of
damage in the members, expressed by their inelastic deformation,
continues to grow. In other words, the stiffness of a member
degrades with inelastic deformation. For this reason the above
models are no longer valid, and a more realistic model should be
used. So far, only a few models have been introduced on this basis
(12, 13), and a new model will be introduced in this report.
In the case of beam-column members, accounting for
inelastic deformation adds a new problem to the already complex
one of the dynamic analysis of the member. The small displacement
assumption which is usually adopted for the analysis of the structure
is no longer valid, and the effect of the gravity loads on the response
of the member should be considered. It is obvious that, if the inelastic
drift in the columns continues to grow, the gravity loads will eventually
become the dominant load and the column will collapse. In assessing
the degree of safety of such a member, consideration of the gravity
load effect is necessary. In the analysis of the response of a column
to earthquake type of loading, which will be presented in Chapter 5,
this effect, as well as the degrading of the column's stiffness, will be
considered.
1. 2 The Object of the Present Work
The aim of the present work was broadly mentioned before.
It is an investigation into the behaviour of a reinforced concrete
column under lateral loading due to earthquakes. On this account
two investigations are made.
14
In the first part, an attempt is made to find a method to
predict the lateral force-deflection characteristic of the column
under monotonic loading up to its failure. This task is done by
analysing the results of the tests previously carried out on laterally
loaded columns at Imperial College (15). The emphasis in these
tests was put on the descending part of the characteristic curve, as
in the present work. A series of five tests on larger scale columns
was carried out by the author to check the adequacy of the method
for full size columns.
In the second part, the column is idealized as a single-degree-of-freedom system with the load-deformation characteristic
similar to that of the column. A simple idealization is made to
portray its behaviour in the unloading and reloading stages. This
was based on observations made in the tests previously carried out
at Imperial College (39). The response of this one-degree-of-freedom
system to the earthquake type of loading is studied. The records of
six actual earthquakes were used as data.
Finally, an attempt is made to find a correlation between
the factors governing the collapse of a column and its energy absorption
capacity. In this way, a more realistic estimate of the safety of a
column under earthquake loading is sought.
1.3 The Basic Element Behaviour
The behaviour of a reinforced concrete member, under the
combined actions of axial load and bending moment, may be investigated
by studying the moment-curvature-load (M-p-p) relationship at a
section of the member. Figure 1-1 shows the (M-P-P) diagram for
a typical cross-section of a column, with symmetrical reinforcements,
under a constant axial load. The following features of these graphs
show the different stages a section undergoes.
(1) Point A represents the start of a noticeable crack
on the tension side of the critical section. The
15
variation of the bending moment up to this point is
nearly linear, and at this point there is a sudden
change of gradient in the bending moment, which is
more significant in members under low axial load.
This implies a sudden reduction in the stiffness of
the member due to cracking.
(2) Point B represents the stage at which the tension
steel yields, i. e. the yield bending moment. This
point marks the transition from a mild non-linearity
to that of a strong one, as shown on the diagram.
The bending moment, after this point, does not
increase at the same rate as before because the
change in the bending moment resistance of the
section, hereafter, is mainly due to the increase in
the lever arm of the concrete compression force.
(3) The appearance of the first crushing of the concrete
on the compression side is shown by point C which,
in the absence of any noticeable work-hardening in the
steel and/or with sufficient axial load on the section,
is very close to the maximum point on the diagram,
1. e. the maximum bending moment capadity of the
section. The difference between the curvature at
this stage and that at the yield moment, point B, is
an index of the ductility of the section and the ductility
of a member is directly related to it. The ratio
between the curvatures at these two stages is referred
to as the ductility factor.
The value of this factor depends mainly on the axial
load. As the axial load on the section increases, the
yielding of the tension steel is attained at a higher
16
bending moment level, i. e. point B gets closer to
C and the ductility of the section decreases. The
axial load level at which these two points coincide,
i. e. the yielding of the tension steel occurs simul-
taneously with the crushing of the concrete on the compression side, is called the balance load Pb.
For axial loads higher than this the concrete crushes
before the steel yields and the plateau BC does not
exist, figure 1-1(c).
The variation in the ductility of the section at the
first crushing stage with the axial load is better seen
on the axial load-bending moment (P-M) and the axial load-curvature (P-P) interaction diagrams shown in
figure 1-2. The curve PQR shows the interaction between the axial load and the bending moment at the
first crushing stage, point C. Point Q represents
the balance axial load which corresponds to the highest
bending moment capacity of the section. In parts
PQ and QR the concrete crushing precedes the steel
yielding and vice versa respectively. QS represents
the bending moments corresponding to the yield level
for P <Pb. On the P-P diagram the curves, P1 Q1 Ri
and Q1 Si show the curvatures corresponding to these
two stages.. The difference between the curvatures at these stages, for different levels of axial load, is seen clearly. In a member under bending action alone,
the ductility factor has the highest value, and as the load approaches the balance load, it decreases until
it becomes negligible.
17
(4) After Point C, figures 1-1(b), 1-1(c), the concrete
on the compression side continues crushing or spoiling,
depending on the axial load, and due to the shift of the
concrete stressed zone at the section and the decrease
of the lever arm of its resultant force, the bending
moment resistance of the section gradually falls off.
Eventually the whole cover on this side of the section
is crushed or spoiled, and only the concrete core
remains to resist the external forces.
The behaviour of the section after point C depends
upon several parameters but mainly upon the axial
load and the crushing strain limit of the concrete core.
The latter is a function of the degree of the confinement
provided at the section. The behaviour of the steel
bears some significance here. Should it show any
work-hardening the drop in the bending moment would
be less severe and, in the absence of any considerable
axial load, the bending moment may even increase.
However, with sufficient lateral reinforcement at the
section, the ultimate failure of the section is post-
poned considerably, in the range of the axial loads
used in practice, and the ductility of the section is
increased quite appreciably. The section finally
fails due to the crushing of the concrete core or the
buckling of the compression reinforcements or even
the rupture of the tensile steel. The failure due to
the shear force acting on the member should also
be mentioned.
18
In figure 1-2 typical interaction diagrams between
P, M and p , at failure stage are shown, curves
PTN and P1 T1 N1. The details of the construction
of this diagram will be given in Chapter 3. The
point to be noticed here is the considerable increase
in the curvature and therefore the ductility of the
section for a relatively small drop in the bending
moment capacity of the section, when P is less than
Pb.
The energy absorption capacity of the section is
increased quite considerably compared with that of
the first crushing level, figures 4-2 to 4-22, but
since the section is unstable - the increase in
deformation is accompanied by the decrease of the
bending moment - the use of this part of the diagram
is limited to a certain category of structures,
or structures under special loading conditions.
This point will be discussed in the following section.
1.4 Column Behaviour
A typical lateral force-deflection diagram of a cantilever
column under constant axial load is shown in figure 1-3. The points
A, B, C and D correspond to the cracking, yielding, first crushing
and ultimate stage of its critical section, as described previously.
The behaviour of the column up to the point A is virtually linear and
elastic. Non-linearity starts after the appearance of the first major
crack or cracks at the critical zone, and deformation in the column
gradually becomes concentrated on this zone. The degree of the
concentration depends mainly upon the bond between the concrete
and steel. In sections with poor bond, deformation increases
rapidly.
19
With the steel yielded, after the point B, the deformation
at the critical zone becomes more pronounced and the column,
hereafter, resembles an elastic strut with a resisting hinge at its
base. The length of this hinge depends on the properties of the
cross-section, axial load and bending moment gradient in the column.
In any case it is larger than that which could be predicted from the
moment-curvature relationship of the cross-section.
As the deformation of the column increases, the critical
zone gradually loses its flexural rigidity and the lateral load approaches
its stationary point. In long columns this point is reached before
the critical section attains its maximum bending moment capacity,
but in short columns these two points usually coincide. After this
point the column becomes unstable.
When the critical section passes its maximum bending moment
capacity, point C, the bending moment resistance at this section decreases,
as was described previously. This cannot be followed with unloading
of the column over its whole length because the overall deformation
of the column is still increasing. Test observations show that, at
this stage, the deformation in the hinge zone is increasing, while it
is decreasing in the rest of the column length. It means that the
hinge zone follows the descending branch of the moment-curvature
curve, CD, while the rest of the column is being unloaded along an
appropriate path. Since the hinge deformation is greater than the
loss of deformation due to the unloading, the overall deflection in
the column is increased.
During this process, the energy released from the unloading
of the region outside the hinge is absorbed by this zone which is
deforming continuously. Experimental evidence (16) shows that if
the energy released is less than the energy absorbed in the hinge this
process will continue. If otherwise, the excess of the released energy
causes a sudden disintegration of the concrete in this region, and the
load falls off immediately with virtually no increase in the deflection.
20
In the absence of sudden failure, the column deformation continues
with the decrease in the lateral load until the critical zone fails in
one of the modes described previously.
The descending branch, of the force-deflection or moment-
rotation diagram of a member, has been obtained in different experi-
ments (14, 15) with the deflection control testing procedure. However,
the member is basically unstable in this region and the applicability
of this branch in the analysis of structures is arguable. In a deter-
minate structure under a static type of loading, the formation of a
hinge transforms the structure into a mechanism and it collapses
almost immediately. Thus the consideration of this branch here
is irrelevant.
The case for indeterminate structures is different. When
one or more hinges in a structure enter the falling branch of the
moment-curvature diagram, there will be a redistribution of forces
and moments in the structure which may allow the loading to be
increased until the ultimate load capacity of the structure is attained.
In other words, the structure may be still hardening while a few hinges
are softening (17). In this case, if this branch of the diagram is
ignored, a considerable error in the ultimate load capacity estimation
may arise. The situation is different with structures under dynamic
type of loading, in which the intensity and the direction of the forces
change very rapidly, as in earthquake loading. Here a structure
does not have the opportunity to deform excessively in one direction,
leading to instability of its members and to total collapse, before
the change in the intensity or the direction of loading may return
the structure to its stable position. Under such conditions, the
structure collapses due to either a great shock, which deforms it
right to the end of its capacity without any unloading, or by the
accumulation of damage in the members, expressed in terms of
their plastic deformation. In the field of earthquake design, in
21
which the energy absorption capacity of the structure is the main
criterion, the contribution of this descending branch of the force-
deflection diagram is considerable and a true evaluation of the
safety of the structure cannot be made without considering it.
1. 5 Review of Previous Work
The reinforced concrete column, as a major structural
unit in buildings has always been under investigation for various
reasons. Most investigators have dealt with the strength and
stability of the column under different conditions of loading, restraint
or slenderness. The works done by Broms and Viest (18), Chang
and Ferguson (19), Pfrang and Siess (20), Cranston (21), and the
experiments by Breen and Ferguson (22, 23), are to be mentioned
on these subjects. These works are not discussed here. The
works which concern the lateral load-deformation behaviour of the
column under static type of loading are discussed here and those
related to the dynamic behaviour are dealt with in Chapter 5. With
reference to the load-deformation behaviour of a reinforced concrete
member, the main approach is that introduced by Professor Baker (26).
In this review, after a brief discussion on this approach, the works
related to its application are examined. Works by other investigators
will be discussed later.
1.5. 1 Moment-Rotation Characteristic of a Reinforced Concrete
Member
The classical method of determining the deformation of a
member under bending and axial load is based on the moment-
curvature relationship at a point in the member. This relationship
is derived as a unique function, of the member's cross-sectional
properties as well as the axial load acting on it. The influence of
the axial load is due to the non-linear characteristic of the concrete.
On this account, all cross-sections with the same properties and
22
axial load follow one relationship, immaterial of the location of
the section in the member and the loading condition it is subjected
to. Moreover, the Bernoulli assumption that "plain sections
remain plain after deformation" is usually adopted in the derivation.
However, the results obtained by this method show certain short-
comings in practice.
Experimental evidence (24) shows that the basic idea of
the uniqueness of the M-P relationship is true only in the period of
load history before the cracking of the concrete. After cracking,
the M. diagram shows a marked non-linearity and each section
follows a relationship which is different from the others, It is
thought that the distribution and the gradient of the bending moment
significantly influence the behaviour at a section.
On the other hand, the Bernoulli assumption certainly does
not hold in the cracked region, particularly when the steel yields.
This results in a severe underestimation of the critical zone's length
and consequently in the member's deformation. In a compression
hinge, one in which the concrete crushes before the tension steel
yields, the concrete cover crushes over a considerable length on
both sides of the critical section, even when the bending moment
distribution diagram of the member shows a peak. In the tension
hinge, the length over which the steel is yielded is far greater than
that which is predicted by the Air-p relationship. The length depends
upon the quality of the bond between concrete and steel, and the
possibility of inclined cracking, which is more severe in the members
under greater shear force.
Considering these deficiencies of the M-P approach, an
alternative would be to derive a relationship between the bending
moment, distributed in a certain form along the member, and the
rotation produced between the two ends of the member, i, e. the
integrated curvature along the member. In this derivation the
23
the properties concerning the member's deformation, such as the
end condition, type of loading, bending moment gradient and dis-
continuities due to cracking and crushing of the concrete, can be
considered as well as those belonging to its cross-sections. On
this basis the M-4 relationship will be the characteristic of a member
under certain conditions and not of a cross-section.
A typical M-4 diagram for a member and the corresponding
M-P diagram for one of its sections, are shown in figure 1-4. In
the works yet to be discussed, nearly all the investigators have dealt
with the rising branch of the M-P diagram and for convenience in
design, they have idealized the corresponding M-4 diagram by a bi-
linear characteristic shown also in the figure. However, there are
other idealizations for the M-9J diagram such as the tri-linear
idealization (1) which is also shown.
In the bi-linear idealization recommended by the European
Concrete Committee, C.E.B. (25), the rotation of the member at
two stages, yielding (L1) and ultimate (L2 ), are given, figure 1-4.
The yielding stage here is when either, the tension steel yields, or
the concrete strain in the outermost fibre of the section exceeds the
strain corresponding to the maximum stress in the concrete stress-
strain relationship. This point is, in fact, when the M-9J diagram
changes to a strong non-linearity. The ultimate limit is where the
descending branch of the 11/1-fb diagram starts.
The member is assumed to behave with linear elasticity
up to the ultimate moment with the flexural rigidity of the section
of
EI = (p ) yield
and at the ultimate an extra rotation at the hinge is considered. The
rotation of the member at limit L2' is therefore
42 = 41 +
24
where el is the elastic rotation of the member given by assuming
the constant flexural rigidity mentioned above, and 44 is the plastic
rotation of the hinge. 0 is usually expressed in the form of
M u— . L (1-1) 4p = (9!)u
Lp is called the "hinge length" and represents the length over which
so-called plasticity has developed, figure 1-5.
As seen on the curvature distribution diagram, the actual
curvature is not much different from the idealized one for values less
than Py, but it is quite different in the hinge zone. It is due to the
inadequacy of the actual M-0 relationship to predict the true defor-
mation which occurs in the hinge zone, as was discussed before.
Considering that ec2
'u ec1 - nld
and that Mu is not much different from M i.e. Mu/My =s-̂ 1, the
0 relationship may be written as:
ec2 eci
d p n2 nl Lpd
where ecr ec2' and n1d, n2d are the concrete strains at the outer-
most fibre of the section and the neutral axis depths at the limits Li
and L2 respectively. Ignoring the small difference between n1 and
n2, this relationship can be further simplified:
ec2 ecl =
(1-2) n2 d
This relationship shows more clearly the parameters affecting the
hinge rotation. ecl is nearly always constant, but ec2' the
crushing strain of concrete, varies according to the degree of
25
confinement of the concrete and the strain gradient at the section.
It is L which is influenced by the grade of concrete, the steel
properties, the amount of confinement, the neutral axis depth and
the bending moment gradient in the member. These parameters
have been the subject of various investigations, some of which
are discussed here.
1. 5. 2 C E. B. Recommendations and Related Works
The first published recommendations for the hinge
rotation are given by Professor Baker in relation to the Simpli-
fied Limit Design method (26).
These are:
(i) For tension hinges 9p = 0. 01/n2
(ii) For compression hinges 8p = 0. 01
For well bound concrete
and 9 = 0. 001
For unbound concrete
which, for normal ranges of ec1 and ec2, gives a hinge length
equal to the effective depth of the section.
In 1962 the Institution of Civil Engineers (27) recommended
a more comprehensive relationship for the inelastic rotation in the
hinge zone. This recommendation was later adopted by the C.E.B.
and further works were carried out in conjunction with it. The
bi-linear moment rotation, as described, was suggested with a
parabolic-rectangular stress-strain curve for the concrete, and
an elasto-plastic one for the steel. These characteristics with
their corresponding limits, L1 and L2, are shown in the figure 1-6.
The suggested plastic hinge rotation was:
(i) For tension hinges - ac2 @c1 L (1-3)
n2d - (ii) For compression hinges ec2 ecl L (1-4)
where Lp =K1.K2 . K3 dZ d
(1-5)
26
The value of ec2' the crushing strain of the concrete, was
recommended to be 0. 0035 for the unbound and 0. 012 for the
well bound concrete. K1, K2 and K3 were three constants
depending on the type of steel, the axial load on the section
and the grade of concrete represented by its strength,
respectively. These constants were to be determined in
further research. However, they were recommended initially
as
K1 = O. 7 - 0. 9
for a steel yield level of 40 to 80 Kips/in
K2 = 1 4- 0. 5 P/P u (1-6)
K3 = O. 6 - 0. 9
for a concrete cube strength of 6, 000 to 2, 000 psi.
Pu is the ultimate axial load capacity of the section and P the one
acting upon it. The hinge length, on this account, varies within
0. 4d and 2. 4d.
Subsequent to this recommendation, several series of
tests have been carried out on beams and columns in different
European countries under the auspices of the C.E.B. The results
of these tests have changed the overall form of the initial recom-
mendation for the hinge length. These works will be discussed
here but, before doing so, the programme of testing suggested by
the C.E.B. and adopted in all these tests, will be outlined.
In the programme,-the specimen is loaded up to 60 per cent, of
its ultimate load in four stages and within one hour. Then, every
fifteen minutes, the applied load is increased by five per cent.
In this way, the whole test takes approximately three hours.
Bremner (28) studied the effect on the parameter K1 of
the type and proportion of the steel in the section. He performed
a series of sixteen tests on 6 x 8 x 80 in. simply supported beams
under mid-span loading. The reinforcements were of different
27
proportions and of mild steel or cold worked bars. The concrete
strength was kept constant in all the tests, at nearly fct u = 5000 psi.
He found that the hinge length varies proportionally with n2, the
neutral axis depth ratio at the limit L2. Thereby, he introduced
Ki as a linear function of n2.
K1 = K1 n2
where K1 is a constant whose value was found to be
K i = 2.0 for mild steel
K1 = 3.55 for cold worked steel.
Substituting K1 in equation (1-5) yields
p z =( ec2 - c1) .17 K2 K3 u (n.)4 (1-7) 1
This equation, with further substitution for the constants and
z/d from his test data, results in
= 2. 0( ec2 ec1) (1-8)
for mild steel. This form of expression for the hinge rotation
has been confirmed by other investigators, as will be seen later.
With regard to the effect of the steel ratio, he observed
that the increase in the steel ratio reduces the ultimate curvature
and increases the plastic hinge length. The overall effect is the
reduction of the hinge rotational-capacity. The same conclusion
has been deduced by other investigators (8).
Amarakone (29) examined the effect of the axial load on
the hinge rotation, i. e the parameter K2. He tested twelve
specimens with the same dimension and under the same condition
as those of Bremner, except for the axial load. The neutral-axis
depth at the critical section was kept constant during the test by
changing the axial load in relation to the lateral load. In this way,
the effect of neutral axis depth could be isolated and studied. This
parameter was varied in different tests within the range of n = 0.4
and n = 1.6. The hinge length was found to be a parabolic function
ftc
P'; ft ,
--S1 is the lateral reinforcement parameter.
= 0.0017 1 - c2 0 + 0.14 qb + 1. 3 (1-10) 26 x n2
28
of n. He therefore suggested that the value of K2 in equation
(1-5) should be:
K2 = n2 Lid
where L is the length of the member. The choice of Lid as the
constant is arbitrary since only one length of column was tested.
The hinge length, L , estimated by the new expression is some-
times very large: it can even be greater than the column's length.
This relatively large estimation of L may be due to the under-
estimation of the concrete crushing strain at the ultimate limit.
Amarakone observed that the ultimate crushing strain
of confined concrete decreases as the neutral axis depth increases.
Taking the results of Bremner's tests into account he concluded
that
= 0.0015 (1 + 1 3 ) (1-9) c2 n2 The effect of concrete strength on the ultimate crushing strain had
previously been studied by Hognestad, Hanson and McHenry (30),
and the influence of lateral binding had been examined by Chan (1).
Amarakone, using the results of their work, finally derived an
expression for ec2 as
where qb =
In the light of the new information obtained from the above-
mentioned tests and those carried out in other European countries,
a new recommendation was made by Professor Baker and An.iarakone (31)
in 1964. In this recommendation, improvements were made on the
stress-strain relationship for concrete in bending, the ultimate
crushing strain and the hinge rotation. The stress- strain curve for
bound concrete is shown in figure 1-7 in which
29
I . ,0. 0.8+ o. l) f,
c
(1-11)
(1-12)
(1-13)
c ' n2
ec2 = O. 0015
For unbound concrete
= 0.0035 ec2 and the hinge rotation
4)13 = 0.8 (ec2
1 c
[1.45 + 1.5P" + (0.7 - 0.1P") 1 -- — n2
was recommended as:
- ec1 ) K1 K3 Zid
--7, 10‘*
where K1 and K3 are the same as the original recommendation,
equations (1-6), and eel = 0.002.
This new relationship for the hinge rotation was later
confirmed by Chinwah (32) in a series of tests on columns in which
he followed the same procedure as Amarakone. Chinwah found no
appreciable change in the hinge rotation at the sections with different
percentage of steel area. Remembering that here, the compression
and tension steel have the same ratio, this result is not contradictory
with that seen in Bremner's test. He also confirmed the obser-
vation made by Amarakone, that under a high axial load, the hinge
length may be very large.
The effects of lateral confinement on the strength and
ductility of the concrete have been studied by several investigators
(1, 3). It is confirmed that both the strength and ductility increase
with increasing confinement. It has also been observed that the
ductility of concrete increases under eccentric loading in comparison
with concentric loading. The microscopic observations (33) made
on the behaviour of the concrete under concentric and eccentric
loading, show that in the latter case mortar crackings between
aggregates develop less than that in the former one, for any strain
higher than 0.0017. Thereby, the stress in the falling branch of
the stress-strain curve, decreases less rapidly in the eccentric
loading than the concentric one.
30
The effects of these two parameters on the concrete stress-
strain relationship have been studied by Sargin and Handa (34), and
Soliman (2), and their corresponding effects on the rotational capacity
of the hinge in the beams and columns, have also been examined by
Soliman (35).
Regarding the stress-strain relationship, Soliman observed
that the binding does not increase the strength and the ductility in
the case of eccentric loading as much as it does in the case of con-
centric loading. The confinement effect is increased appreciably
by reducing the binders' spacing, but alterations in their size or
type have less effect. His proposed concrete stress-strain curve
is shown in figure 1-8. The results of his tests concerning the
effects of lateral binding, axial load, and bending moment gradient
in the beam or column on the hinge rotation, are interesting. They
are summarized as follows:
(1) The increase in the bending moment gradient increases the
hinge rotational capacity. This is due mainly to the increase thus
produced in the deformation capacity of the section, i, e. the crushing
strain of the concrete, rather than the hinge length itself. This
result confirms observations made by Mattock (36) and Corley (37).
(2) Lateral binding increases the crushing strain of the concrete
and consequently increases the rotation of the hinge.
(3) Axial load decreases the hinge rotation but increases the
hinge length.
With these observations he suggested the hinge rotation to be
where
and
{4p= 2.5 ( ec2 - ecl)
c101 =.002
ec2 = 0.0031+ 0.8 [
(1-14)
1 - n2 + 0.5-1-n2 (1+n2)2 (1-15)
This value is for sections under combined compression and bending.
For sections under bending only, it is nearly 1.67 times this value.
Ab A" (S -S) = ( 1.40 — --0.45) A Ac 's
qn ( 1.16) . S + 0. 0028S2 . B
31
As i s seen, the bending moment gradient effect Z/d is considered
in ec2 and not in -p. The parameter q" characterises the lateral
binding of the section as a whole and also takes into consideration
the unbound part. It is given as
where A" and S are the binders' cross-sectional area and their
spacing.
So = 10 in.
Ab and Ac are the areas of the bound and unbound concrete
in the section under compression. They are shown, together with
the parameter B, in figure 1-8.
9p corresponds to a bending moment of 0.95 M on the
descending branch of M-0 curve.
The overall conclusion deduced from the above works, is
that the major parameters affecting the rotational capacity of a hinge
are:
(1) The ultimate concrete crushing strain, influenced mainly by
the lateral confinement and, to a lesser extent, by the strain gradient
in the section and the bending moment gradient in the member.
(2) The neutral axis depth, influenced mainly by the axial
load. The hinge length is directly proportional to the neutral axis
depth.
1.5.3 Works of Other Investigators
There are many other reports on the subject under dis-
cussion, but most of them concern the behaviour of beams and
their hinge rotational capacity. Furthermore, many of them have
dealt with some aspect of the problem qualitatively rather than
quantitatively. Among these works, a relatively large group of tests
has been reported by Mattock (36) and later by Corley (37). These tests
32
are on simply supported beams under mid-span loading, and in
them various parameters, such as bending moment gradient, size
of cross-section, tension steel ratio, concrete and steel strength
and lateral binding, have been studied.
Among these parameters, the lateral binding and the
moment gradient in the beam have been found to have the greatest
effect on the crushing strain of the concrete and the hinge rotation.
The overall conclusions drawn by Corley are:
(1) The maximum concrete compression strain is related to
the shear span and the lateral reinforcement by
ec = 0. 003 + 0.02 b/Z + (Pii f" sy/20)2 (1-17)
(2) The hinge length is also related to the shear span by
L = d/2 + 0. 2Z/ \fir (1-18)
In the recommendation made by Sawyer (41) for the limit
design of reinforced concrete members, the moment-curvature
diagram is idealized as a bi-linear one with the elastic and
ultimate limits taken at the points corresponding to, Me =0.85Mu
and Mu . Mu is the bending moment at the first crushing stage
and it is determined with an assumed concrete crushing strain of
0. 0038. With this idealization, the hinge length at the ultimate
limit is recommended as
Lp = d/4 + O. 0'75 Z (1-19)
In all the above works the moment-rotation relationship
at its rising branch has been studied, and the falling branch has
been ignored for the reasons given previously. As was discussed
earlier, the bending moment at the section does not decrease very
rapidly in beams or sections under low axial load, especially when
the steel shows considerable work-hardening or the section is well
bound. For this reason, the critical zone in the member undergoes
33
a considerable amount of deformation before there is a noticeable
decrease in the bending moment. However, in the case of columns
under a practical range of loading, the bending moment starts
decreasing after the first cm shing in the concrete appears. The
moment-rotation curve shows here a distinct falling branch which
is usually ignored.
Yamashiro and Siess have obtained the falling branch of
the curve in their tests (14) on simply-supported beams under mid-
span loading with, and without, axial load.
These beams had mostly 6 x 12 in. rectangular cross-
sections and 12 ft. spans between the supports. At mid-span,
they had a stub 12 in. in width and 6 in. in height on each side, to
simulate a beam - column connection. The tests were carried
out under constant axial load and incremental transverse loading,
up to the first crushing of the concrete. They were then continued
with incremental deflection at mid-span until the specimen collapsed
due to the extent of the damage. Each test took 4 - 6 hours. The
bending moment at the face of the stub due to the transverse and
axial loads, and the corresponding rotation, were recorded during
the test. In their theoretical work, they studied the stages corres-
ponding to yield, first crushing of the concrete, and the ultimate or
collapse position. At each stage, special assumptions regarding
the behaviour of the concrete, were made. The collapse point was
assumed to have been reached when the bending moment in the crushed
section, without cover on the compression side, attained its maximum
in the moment-curvature diagram, or the load in the compression
steel reached its buckling load. In deriving the moment-curvature
relationship, no limit for confined concrete strain has been considered.
The results of the analysis of their tests, and those carried
out by McCollister and Burns on similar beams, form a series of
empirical relationships for the evaluation of rotations at the above
34
stages. These relationships contain the effect of the bond between
concrete and steel, plastic rotation at cracked sections, and diagonal
cracking at the critical sections.
At the yield stage, an extra rotation, other than the elastic
deformation in the member, has been introduced at the critical
section. This extra rotation is directly proportional to the steel
strain and the bond stress, and inversely proportional to the length
of the member, i. e. the loss of bond between steel and concrete,
and the bending moment gradient in the member, have significant
effect on the deformation at this stage. The hinge rotation, at the
stage of first crushing of the concrete, has been found to be a
parabolic function of the plastic strain developed in the tension
steel. At the ultimate stage, the whole deformation of the member
has been assumed to be concentrated at the hinge zone, the length
of which has been arbitrarily chosen as half the depth of the section
(6 in. ). Further on, a concentrated rotation has been considered
at the critical section, the amount of which has been found to equal
the tension steel strain at this stage. A comparison of some of
their tests with the theory developed in this work, is given in
Chapter 4.
Bailey carried out a series of tests on fixed-ended columns
under lateral loading (15), His experiments will be discussed in
Chapter 2. He attempted to predict the force-deflection diagram
of the column, by assuming the hinge zone to follow the descending
branch of the moment-curvature diagram of the cross-section.
He used Krishnan's (38) stress-strain curve for concrete, with
special consideration for the confined concrete in calculating the
moment-curvature relationship. No limit for concrete strain,
confined or unconfined, has been considered. With the aid of a
computer program, he tried different lengths for the hinge zone,
to get a reasonable agreement with the experimental results. He
found the hinge length to be between 3 - 5 in. for his specimen tests
(4 in. square cross-section), depending on the axial load. The
higher the axial load the longer the hinge length.
Yielding First Crushing Ultimate
R1 N1 Curvature,
P . A
xia
l Lo
ad
Ax
ial Lo
ad
NS
P
R Bending-Moment, M
35
M
Curvature, 0
Curvature,
(a)
(b) Pa ( Pb
(c) Pc) Pb
Figure 1-1 Moment-Curvature-Axial Load Diagram for a Cross- section
Figure 1-2 L oad- Moment, Load-Curvature Interaction Diagrams
unstable
a) U to O
Deflection, D
Figure 1-3 Lateral Force-Deflection Diagram
Bi-linear idealization M
My IL
T ri- linear
( E y
(a)
(b)
Figure 1-5 Bending Moment & Curvature Diagrams
Idealized
Actual (MulMy)•O
36
Oc Curvature,
(a) Moment-Curvature Diagram for a cross-section
°Y.91 1t2tation, ft
(b) Moment-Rotation Diagram for a member
Oy
Figure 1-4
fc _1 fc
o.8fc-
ec ece
ecs ecf
Pailabolic
37
fc fs Cold Worked Steel
/.0.901 L2
/ d L2
/ I 1\111d Stec-1
/ I
ec es ec2 0.0
Concrete Stress-Strain Relationship
= 0.002 ec1 ec2
= 0.0035 For unbound concrete
ec2 = 0.012 For well bound concrete
Figure 1-6 C. E. B. Recommendation, 1962 (25)
1 -f- = ( 0.8 + 0.1 ) f' I c ' n2 c
Pa Fabolic I= 0.002 I ecl I I e c2 is given in equation (1-12) I ), ec
ecl ec2
Figure 1-7 Bound Concrete Stress-Strain Relationship, Baker and Amarakone, 1964 (31)
Steel Stress-Strain Relationship
fc
fc
Tc =0.9fd(1+0.05q") ece=0.55r 10-6
ecs=0.0025 (1+q")
ect =0.0045 (1+0.85q")
qn is given in equation (1-16)
B = b1 or 0.7 n d1 whichever is greater.
Figure 1-8 Soliman - Yu Stress-Strain Relationship for Concrete.
38
CHAPTER 2
EXPERIMENTAL WORK
2.1 Introduction
The present programme of tests on the laterally loaded
reinforced concrete column started in 1965 as a result of obser-
vations made on the damaged buildings in Skopje after the earth-
quake on 26 July 1963 (40). It was observed that a number of
buildings, five or more storeys high with open plan ground floor,
were still standing after having undergone a considerable amount
of residual deformation. Some of the columns at ground level had
been swayed up to approximately seven degrees with the vertical.
This showed that the reinforced concrete columns had the capacity
to undergo a relatively large deformation in practice.
To study the behaviour of a column under lateral loading
a special rig was designed by Bailey (15). He carried out a series
of tests which were later extended by others. In these tests a fixed-
ended column, figure 2-1(a), was subjected to a cyclic lateral
deformation while supporting a constant axial load. The lateral
deformation was usually extended well beyond the stable range.
The tests were carried out under a deformation-control procedure
in order to prevent the collapse of the column due to instability.
In most of these tests, the column was incrementally deformed until
the resisting horizontal force became null, i. e. the column was
supporting only the axial load and the bending moment it produced.
The direction of the loading was then reversed and the same
procedure followed.
This procedure continued until the column collapsed due
to the excess damage at its critical sections. The rate of
deformation was kept constant during the test. A typical force-
deflection diagram obtained in the tests is shown in figure 2-1.
39
Among the various parameters which influence the behaviour
of the column under these conditions, attention has been concentrated
mainly on the axial load and the longitudinal reinforcement ratio.
Two different heights for the column and a number of different rates
of loading have also been examined, but more investigation is
needed to arrive at a concrete conclusion for the effects of these
parameters.
In the present series of tests the same object is followed,
but with a different size of column and different end conditions.
The details will be discussed later. Since nearly all the previous
tests will be analysed in this work, herewith is a brief description
of the details of these tests.
2. 2 Details of the previous tests
The specimen was a 4 x 4 in. rectangular cross-section
column, with a length of either 48 or 60 in. Each end was cast
into a reinforced concrete beam 25 in. long, the cross-section being
the same as that of the column. These beams were kept fixed to the
rig during the test and provided the required fixity for the column's
ends. The details of the specimen and the arrangement of the
reinforcements are given in figure 2-2(d).
In these tests, the applied axial load was within the range
of 5 - 19 tons and in most cases it was under the balance load,
figure 3-24. The longitudinal reinforcements were mild steel bars
with a gross ratio of 0.3 to 5%, and the stirrups were 1/8 in.
diameter mild steel bars with 4 in. spacing. In three of the tests,
the spacing was reduced to 3 in. and to 1 in. and no considerable
change was noticed in the results, figure 3-17.
The rate of applied deformation at the head of the column
was mostly within the range of 0.1 to 0.5 in. per minute. A few
tests were performed with a higher rate, in the order of 10 in. per
minute. It is difficult to distinguish the effect of this parameter,
40
if any, in this range, in these tests, figure 3-18. A more systematic
series of tests is needed to clarify this point.
The general view of the testing rig is shown in plate 1.
It consisted of two main parts.
(1) A fixed frame, with the upper end of the column fixed to it,
transmitted the force and the bending moment of this end to the
ground. Also, a calibrated load-cell attached to it was used to
measure the horizontal force transmitted through the column to
this frame.
(2) A loading frame consisting of three interconnected parts:
(a) A main frame free to move horizontally in order
to guide the other two parts and transmit the moment
forces to the ground.
(b) An inner frame, free to move vertically, had the
lower end of the column fixed to it.
(c) A trolley, beneath these two parts, transmitted
the axial load reaction to the ground, figure 2-3.
The cyclic loading was performed by pushing the loading frame
backwards and forwards with two long lapped rams. The testing
procedure was in some ways similar to that done in the author's
tests and will be described later.
Bailey (15) tested thirty six columns of 4 x 4 x 60 in
with the main variables and the programme of loading as des-
cribed in 2.1. From this group, nineteen tests will be
analysed in this work. The details of these tests are given in
table 4-1 under the serial number "B".
Neal (39) carried out a series of four tests on the
columns under cyclic loading and with different amplitudes of
deflection. The specimens were the same as Bailey's and the
axial load was ten tons for all of them. The object was to study
the behaviour of the column during the unloading and reloading
41
stages, information on which is of vital importance for the dynamic
analysis of the column. These tests give a qualitative idea of the
behaviour of the column at these stages. They have been used as
the basis of the idealization made in Chapter 5 in order to study the
response of the column to an earthquake type of loading. They will
be discussed in detail later.
Koprna (73) continued the tests on columns of different size.
Like Neal, he performed the cyclic loading with limited amplitude
for some of the columns. All his tests will be examined here, listed
in table 4-1 under the serial number "K".
2.3 Present series of tests.
The size of the specimen in the previous tests was relatively
small and the application of its results to a column in a practical
range was arguable, unless they were confirmed by more tests on
columns with a more realistic size. Besides, a few minor points
had to be investigated. Regarding these tests, the concrete cover
in their cross-sections was relatively thick (o.75 in. in a 4 x 4 in.
cross-section), and since the spalling of this part of the section
affects the behaviour of the column in the post-crushing stage, its
effect on the results needed to be examined. The longitudinal
reinforcements in the column were spot welded to those in the end-
beams for the sake of anchoring. Bearing in mind that welding
changes the mechanical properties of the material around the welded
zone, its probable effect had to be checked, since the critical sections
of the column were adjacent to these zones.
It was decided to design a new test apparatus to simulate
nearly the same procedure of testing and the same condition for the
column, as in the previous rig. This was done because the previous
test rig was not designed to withstand the axial load required for a
column of greater size. On this account, the specimen and the
testing layout, shown in figures 2-4, 2-5, were agreed upon, and
42
with cost in mind, a 6 x 3 in, cross-section was chosen for the
column. The column is hinged at one end and cast into a reinforced
concrete beam at the other. A cyclic bending moment is applied at
the beam-end of the column by rotating the beam around a pin, fixed
at the base of the column, alternately in opposite directions. In
this way, the column behaves as a pin-ended column under the action
of an end bending moment, figure 2-2(a). The applied bending
moment and the corresponding rotation are measured.
2.4 Details of the Specimen
2.4.1 Dimensions
Five 6 x 8 x 45 in. columns were tested under four different
axial loads. The details of the specimens and the arrangements of
the reinforcments are shown in figure 2-2(b), followed by the other
details in table 4-1 under the serial number "Z". The tests Z1
and Z2 were identical, except that in Z1 the longitudinal reinforce-
ments were spot welded to those of the beam, whereas in Z2 they
were anchored at the bottom of the beam. In the other three tests,
these reinforcements were welded to preserve the similarity between
these tests and the previous ones. The main reinforcements were
welded to a one inch thick plate at the head of the column. The
reinforcements were mild steel bars and in the 'coupon( test they
showed a considerable amount of work-hardening and a very small
perfect plastic plateau. Six 'coupon' tests were done for each batch
of reinforcements and their average'values are given in table 4-1.
2.4.2 Casting and Curing
The specimens were cast vertically with the free end on the
ground. They were vibrated by a 2 in. diameter poker vibrator
during the casting. The shuttering was removed after twenty four
hours and the specimens were cured under wet hessian in the ambient
atmosphere of the laboratory for twenty eight days. Then they were
tested. Four 4 in. cubes were cast for each specimen and were
43
cured under the same conditions as the column. These cubes were
tested on the same day as the column. The average concrete cube
strengths are given in table 4-1.
2. 4. 3 Concrete Mix
The concrete mix used in these tests was the same as in
previous tests. The cement was Ordinary Portland Cement and the
ratios between the different ingredients were as follows:
Water/Cement ratio (by weight)
0. 63
Aggregate/Cement ratio 4.7
Coarse/fine aggregate ratio 1.25
The maximum size of the coarse aggregate was 3/8 in.
2. 5 Test apparatus
In this series of tests, the columns were tested horizontally.
The general view of the test rig and the loading system are shown in
plates 2, 3. The rig consisted of, at its base, a steel beam with an
interconnecting small frame on both sides, and at its head, two long
ties.
The steel beam was an 8 x 12 in. hollow cross-section
60 in. long. It was fastened to the concrete beam so that the
required bending moment could be transmitted to the column's base,
figures 2-4, 2-5. Two small solid beams (32), welded to the base
of this beam, were used to connect the beam to the small inter-
connecting frame on each side of it. Each frame was made of a
solid beam (S1) and two steel tubes (TU), through which a 1 in.
diameter high tensile steel bolt was passed. These bolts connected
the beams S1 to the beams S2 and, by being pre-tensioned to nearly
15 tons, pre-stressed the frame and maintained an almost rigid
connection.
Each beam Si was supported at its mid-span by a 1.5 in.
diameter steel ball-bearing, which in turn, rested on a solid steel
T-sectioned support. These supports were fixed to a rigid concrete
44
block fixed to the strong floor of the laboratory. The position of the
ball-bearings was such that the axis passing through their centres
coincided with the centre-line of the cross-section of the column's
base. This was the axis about which the whole system of the steel
beam and its connected frames rotated. In this way, the system
could rotate and provide the required rotation at the column's base,
while transmitting the axial load in the column to the supports.
The other end of the column was attached to the head of
the loading ram through a 1. 5 in. diameter steel ball-bearing. It
could, therefore, rotate freely. Although the existence of the
friction forces between the ball and the head of the column disturbed
the ideal free-rotation condition, the fixity thus produced had little
effect in comparison with the applied bending moment at the base;
hence, the assumption of free rotation is justified. The head of the
ram was kept in position by two 1 in. diameter steel ties. These
ties were long enough to provide the necessary freedom for the head
of the ram to move backwards and forwards according to the move
ment of the column. The shear force transmitted through the column
was taken by these ties and, as will be discussed later, the applied
bending moment was obtained by measuring the force in the ties.
Before the test, the ties were pre-tensioned to 5 tons to prevent
them from buckling during the course of cyclic loading. The self-
weight of the system was transmitted to the ground by three steel
ball-bearings under the steel beam and one ball under the column
near to its head. By adjusting the position of these balls, the
horizontal level of the system was maintained. The contact points
were made as smooth as possible to minimize the friction forces.
The stability of the column in the free-bending direction
was maintained by a frame (F) fixed to the ground and attached to
the column through two steel balls, figure 2-4.
45
2.6 Loading System
The axial load was applied to the head of the column by a
50 ton ram controlled by an Amsler load maintaining cabinet,
figure 2-6(a). The required bending moment at the base was
maintained by two opposing lapped rams acting at points 24 in.
away from the centre of rotation.
One of them was a 10 ton ram connected to an Amsler
loading accumulator maintaining a constant pressure. This ram
acted as a dummy and provided a constant force on the system
during the test. In order to raise the load in this ram gradually
to the level of the accumulator's load, the ram was also connected
to a load maintaining cabinet. This cabinet was used to unload the
ram when the column failed.
The capacity of the second ram was 20 tons. It was
connected to an Amsler Hydro-pacer, through which the load was
controlled such that a pre-set rate of displacement was maintained
during the test. This was done by a displacement transducer
attached to the end of the beam, figure 2-6(a), transducer 1. This
ram was the active one, by which the loading and unloading of the
system was operated.
The purpose of the dummy ram was to make the cyclic
loading possible. It is clear that, when the load in the active ram
is higher than that in the dummy, the system rotates in one direction,
and when it is lower, the system rotates in the opposite direction.
2.7 Instrumentation and measurement
There were two main measurements in the test; the
rotation of the base beam and the bending moment applied at the
column's base. The rotation was measured by recording the
displacement of a point, at the end of the beam and at the level of
the centre of rotation, with a displacement transducer, figure 2-6(a),
transducer 2.
46
The bending moment was found by measuring the horizontal
load transmitted by the column and t aken by the ties at the head of
the column. It was done by measuring the strain in two calibrated
dynamometers attached to the ties, figure 2-6(a). The dynamo-
meters were made of steel bars of 1 in. diameter and 24 in. long,
with an electrical resistance strain gauge on each. The strain
gauges, with two fixed electrical resistances formed a four arm
wheatsone bridge as shown in figure 2-6(b). With this arrangement,
the two dynamometers had an opposing effect in the bridge circuit,
and the output of the circuit was proportional to the difference
between the forces in the ties. Proportionality existed because
the ties had identical lengths and cross-sections, and the dynamo-
meters showed the same force-strain properties in the calibration
tests. As mentioned previously, the ties were pre-tensioned up to
5 tons before applying the load and moment. For any force trans-
mitted from the column to these ties, the tensile force in one of them
would be reduced while in the other it would be increased, the
difference being equal to the column's force.
The measured deflection and the force were recorded by a
Kent x-y plotter during the test. It was calibrated before the test.
The rotation measured in the test is partly due to the
elastic deformation of the column, partly to the rotation at the
cracked zone, and finally, to a lesser extent, to the deformation of
the concrete base beam itself. The deformation of the steel beam
and framing are small enough to be ignored. It was desirable to
separate the proportion of these three causes in the total deforma-
tion measured. Observations in the previous tests showed that the
main tension crack formed at a distance of one half of the full depth
of the cross-section, from the base of the column. Therefore, the
rotations of two cross-sections were measured at the heights of 1 in.
and 5-1,0in. from the base, with respect to the steel beam. The first
47
was to give the deformation of the concrete base beam, and the
second of the hinge. This was done by measuring the relative dis-
placements of the ends of two bars, fixed to the column at these
heights, with four strain gauge deflection gauges. The recordings
were taken by a Solartron digital data logger at time intervals.
These measurements were taken only in the rising part of the
moment-deformation curve, before the crushing of the concrete
began.
2.8 Test procedure
The column was placed in the rig, and after the initial
preparation, it was fastened to the steel beam. To ensure full
contact between the two beams, a gap of nearly 3/16 in. in width
between them was filled with "plycol" rapid hardening filler. The
beams were fastened together with six bolts of 1 in. diameter, and
in order to prevent any separation between them due to the high
bending moment, the bolts were pre-stressed to nearly 8 tons.
After the calibration of the measuring instruments, a small
axial load was exerted on the column to absorb the slack in the system.
Then, the ties were set in their positions and were pulled from both
sides until the dynamometers recorded strains equivalent to tie-
loads of about 5 tons. The head of the column was checked con-
tinuously to see that it remained in line with the centre-line of the
assembly.
With all the preparations complete, the full axial load was
applied to the column and the preparation for applying the bending
moment at the base commenced. The loads in both the dummy and
the active rams were increased simultaneously to the level of the
loading accumulator. This was done by gradually increasing the
load in the dummy ram through the corresponding loading cabinet.
Since the Hydro-pacer was being controlled by the displacement
transducer, any rotation in the base beam due to the loading of the
48
dummy ram, was compensated for by the reaction load in the active
ram. In other words, both rams were loaded simultaneously, and
there was no rotation in the base beam. When the load reached the
level of the accumulator, it was connected to the dummy ram and
the load was kept constant thereafter. This load was 10 tons in all
the tests except Z5 where it was 7.5 tons.
The test was started by loading the active ram through the
Hydro-pacer. The rate of displacement was equal to 0.2 in. per
minute in all the tests. As the rotation of the base was being
gradually increased, the required bending moment initially increased
gradually and, after exceeding the maximum bending moment capacity
of the section, it began to decrease. The process stopped when the
following relationship was satisfied:
The applied bending moment = h
where P, 43 and h. were the axial load, measured rotation, and
the height of the column. This point is where the bending moment
capacity of the cri tical section is equal to the bending moment produced
by the axial load alone, in a similar cantilever column under lateral
loading; that is, when the lateral load becomes null. The loading
was then reversed and the column was gradually unloaded, after which,
it was deformed in the opposite direction. The same process was
repeated until the afore-mentioned point was reached, when the
loading was reversed again. The cycle was continued in this manner
until the column collapsed due to an excess of crusing and spalling
of the concrete at the critical zone, and subsequently, due to the
buckling of the reinforcements or due to sudden slip occurring at
this section from shear failure of the remaining concrete. At this
stage the test was terminated and the loads were removed by operating
on the corresponding cabinets.
49
2.9 Test results
A typical curve for the moment-rotation relationship
obtained in the test Z4 is shown in figure 2-7. The results of the
other tests are shown in figure 3-13 and figures 4-19 to 4-22 in the
form of the lateral force-deflection curve for the corresponding
cantilever columns.
As mentioned previously, specimens Z1 and Z2 had identical
properties, except that in Z1, the reinforcements in the column were
spot welded to those in the beam. They showed similar results
except that the maximum bending moment capacity in Z2 was higher
than that in Z1, by nearly 8%. The initial stiffnesses were the same.
Their cube strengths were nearly the same and their reinforcemeht
belonged to the same batch. The main tension crack in Z1 was
formed 5 in. above the base, and Z2 had one 4 in. and another 8 in.
above the base. Considering that the weld position was nearly 7 in.
below the critical section, and it was only spot welding, it is hard
to believe that this fall in the bending moment capacity has much to
do with the welding effect. This should be interpreted as a variation
in the experimental results.
In all the tests, the concrete crushing at the critical zone
started when the maximum bending moment was reached. However,
the rate of crushing or spalling was not t he same: it was higher in
the specimen under higher axial load. In Z1 and Z2, where the axial
loads were relatively high, the spalling was sudden. In Z1, almost
the whole cover was spalled over a length of approximately 8 in. from
the base. It was obvious that the compression reinforcements were
slightly bent. However, this was not so in Z2. The effect of this
excess of spalling in Z1 can be seen in the gradient of the bending
moment in the falling branch of its M-e characteristic, figure 3-13.
The gradient here is higher than that of Z2. In the other tests,
spalling was gradual, but by the end of the first quarter cycle nearly
50
the whole cover was being shed. In Z5, with the lowest axial load,
the spalling was slight for a long time, while the full bending moment
capacity was being sustained at the critical section. However, near
the end of the quarter cycle, a sudden spalling occurred and the
cover was shed, figure 3-13.
To illustrate the behaviour of the column at different stages
and the mechanism of its collapse, the observations made in test Z4
are described here, figure 2-7. At the end of the first quarter
cycle, as explained above, the concrete cover on the compression
side of the critical zone was shed and the compression reinforcements
were slightly bent. On the tension side, the tension crack was wide
open at a height of nearly 5 in. By reversing the direction of loading,
the column was gradually unloaded with an initial stiffness less than
that seen in the loading path. At the time of zero applied bending
moment, an equivalent deflection of 1.9 in. for the head of the column
remained in the column.
As loading continued, the previous tension crack gradually
closed and a new tension crack formed in the opposite side. It was
observed that before the previous tension crack was fully closed, a
number of longitudinal cracks appeared in the concrete along the
reinforcement at that side. These were probably due to the bending
of the reinforcements, caused by the large permanent deformation
remaining in them from t he previous stage.
The maximum bending moment at this stage was slightly
less than that in the previous one. The same result was observed
in the other tests. The maximum difference occurred in Z2 and
was as high as 11%. In Z5, both the maximum bending moments
were the same. Bearing in mind that the spelled side of the
critical section at this stage is under tension, and does not con-
tribute to the resisting bending moment at any time, this decrease
in the bending moment capacity, is probably due to the loss of bond
51
between the concrete and steel, and the general deterioration of the
concrete itself in the compression side.
As the coluran passed its maximum bending moment capacity,
the crushing and spalling of the concrete began, and continued until
the whole cover was removed by the end of this quarter cycle. The
loading was reversed again and the same process was continued.
This column could not survive more than 1.5 cycles (approximately),
and collapsed when the concrete at the critical zone was crushed
completely, and the reinforcements buckled.
The columns Z1 and Z2 did not finish their full cycle before
collapsing by compression and shear failure respectively. Column
Z3 behaved almost the same as Z4, and Z5 continued until the end of
the third cycle. As the load-cycle continued, concrete crushing
penetrated into the core of the section, and finally the column collapsed
by shear failure.
For columns Z3, Z4 and Z5, the deformation of the concrete
beam and the rotation at the cracked sections measured in the test,
are shown in figure 2-8. In the early stage of loading, the rotation
due to deformation of the cracked zone is seen to vary almost linearly
with respect to the column's total deformation. However, as the
load increases, this part deforms at a greater rate than the rest of
the column due to the formation of the cracks. As the bending moment
passes the level corresponding to the yielding of the steel, nearly all
the increase in the total deformation is due to the deformation which
occurs in this zone, and the elastic deformation in the rest of the
column is nearly constant. The effect of the axial load, which is
the main variable in these tests, on the load deformation character-
istic, is clearly seen in figure 3-13. As far as the rotational capacity
of the hinge is concerned, the gradient of the bending moment in the
falling branch is seen to be greater in the case of the sections under
higher loads, resulting in a relatively small rotation in the hinge
before failure. The variation of the hinge characteristic with the
axial load will be discussed in detail in Chapter 3.
· ~p=loton
Fur
~I / -....t
" ..cl I N
(a)
-1.0
B26 Fton
Ii)
1'/1.0
0.6'1 ----:-T---==-~---.----
10 0
0.2
~
-0.5 0.0 0·5 1·0
-0.2
Figure 2-1 Force-DefJection Diagram for Colun1n B26 vi t'"
(d)
53
3/4
6"
CO
to
NJ
S.6"
Welded
d=174
Welded
6,8 xlin. Plate Pp
(a)
d=1"/4
C
4br 6" 60'br 48" 374
Figure 2-2 Details of "l3" & "N" Series Specimens
0 0
Elevation
1
Li •
Plan (without frame 2)
Figure 2-3 Loading Frame (not to scale) (1 5)
F
o o
r- .. B 8
30" 3D"
'I Cl II I II II II 'I II II II II II f,
II II I· II
J I II I, I
II II II II " II II II I I
-N
I f 21." , I
Fjgure 2-4 The Test Apparatus Ar'rang-enlent
Solid Steel T-Sectjon !
10X1/2 In.
I.x2 In.
5, 3Y2x31J2x12in. Solid Steel
Section B-B
1 Steel Ball Bearing
D:i·5"
1" Rod
52 31/2 x 3112 ,,1" 1J2 in.
Solid Steet
SecUon c-c
Figure 2-5 Details of the Test Apparatus
50 Ton Ran
Loading Cabinet 2
Accumulator
t- 07
C
10 Ton Ram Hy d ro - pa r. e K/Z7/////2/
20 Ton. Ram
['/ //////// /1
////////////i/.
Tran
sduc
er 1
Dynamometers
Loading Cabinet 1
Dynamometers
Fixed Resistance
57
(a) Loading System
(h) Arrangement of Dynamometers
Figure 2-G Loading Arrangement and Measurement Positions
0 2 E-.
pr.3oton Z4
P. D h h
3 D = 44.h, in.
1 2
Figure 2-7 Bending D.Tornent-Dotal ion Diagram for Z4
8 4
12 3 x 103 20 8 12 $ x 103 20 0
Figure 2-8 Measured Rotations
4 8 0.
/ ..,-- ......---
z * .----
/j
i •
/1
I ? Z5
1.0
)i 0.8
2 0.6
0.4
0.2
0.8 C
0.6
0.4
1.0
0.2
0.2 y = 8 in.
■••
F
1.0
.0.8
---.. 0.6
0.4
2
Z3
12 103, Radian 0.0
431 42 413
Z4 y = 10 in.
60
PLATE 1 ,GENERAL VIEW OF THE TESTING RIG IN PREVIOUS TESTS(15)
61
PLATE 2, GENERAL VIEW OF THE TEST APPARATUS
62
PLATE 3 , GENERAL VIEW OF THE TEST APPARATUS
63
PLATE I. , FAILURE STAGE OF SPECIMEN 'Z1'
_
111
Yr
11111-1111
64
PLATE 5, FAILURE STAGE OF SPECIMEN 'Z3'
65
PLATE 6, FAILURE STAGE OF SPECIMEN 'Z5'
66
CHAPTER 3
THEORETICAL WORK
3. 1 Introduction
The behaviour of a reinforced concrete column under non-
axial leading was discussed in general in Chapter 1. Here, it will
be studied in more detail and a method will be presented to predict
the lateral load-deformation characteristic of the column. The
method deals with the case of a column under monotonically increasing
lateral deformation to the final failure and does not cover the case
for the cyclic loading. The method is presented in two stages.
The first stage considers the load-deformation behaviour
of the column in the rising branch of the M-P-P diagram. An
exact solution for the beam-column problem is carried out and the
results are compared with the experimental ones. The solution is
based on the integration of the general differential equation of the
column with the aid of a computer program. It is shown that this
solution considerably underestimates the deformation of the column
mainly because of the inadequacy of the M-P-P relationship alone
to deal with the inelastic deformation occurring at the hinge zone.
The above solution is simplified in a later procedure, and
in addition to that, the inelastic rotation at the hinge is considered.
For the hinge rotation at this stage the relationship, developed by
the investigators and discussed in Chapter 1, is used. The results
are compared with those of the tests.
In the second stage the characteristic of the column in the
falling branch of the M-p-P diagram is investigated. The test
results are analysed and the characteristic of the hinge rotation is
obtained. Later on a reasonable path for the hinge zone is assumed
on the M-P-P diagram and the properties of this path are related to
the experimental results. Finally, from these results, an empirical
relationship is derived for the hinge rotation at this stage.
67
Before dealing with these subjects in detail, the derivation
of the M-P-P relationship and the related topics are discussed.
3. 2 Bending Moment-Curvature-Axial load (M-P-P) relationship
The M-P-P relationship of a cross-section is derived by
satisfying equilibrium and compatibility at that section. The
equilibrium condition for a section, shown in figure 3-1, is
written as
C c +C s -T s -P=0 (3-1)
where
Cc is the resultant of the compression forces in the concrete, and
is given by dn
Cc =b1 fc (Z) dZ (3-2)
in which fc (Z) is the concrete stress at the level Z.
Cs and Ts are the resultant forces in the compression and the tension
reinforcements.
These are (3-3)
T s = fs . Ast
The tensile strength of the concrete has been ignored in equation (3-1).
The stresses in the concrete layer and the reinforcements,
fc (Z), f' and fs are determined by the corresponding strains and the
stress-strain laws of the materials. With the assumption of a
linear strain distribution across the section for the compatibility
condition, the strains can be calculated in terms of two variables
such as ft, and Z. Consequently, the stresses can also be expressed
in terms of these two variables. The resultant forces Cc, Cs and Ts can then be derived as functions of P and dn, i, e.
cc = cc dn)
Cs = C s (P, dn)
(3-4)
T s s = T (g), an)
C = f' . A s s sc
68
Finally, the equation (3-1) is written as a function of these two
variables and P, the axial load. It is shown as
g p, dn, P) = 0 (3-5)
The bending moment of these forces with respect to the centre line
of the section is written as
d ' M = C -.(1 dn + (T + C ) d c 2 —t s s (3-6)
where cc dn is the length of the lever arm of the concrete resultant
force, Cc, from the neutral axis. It is given by
bfdn
Z fc (Z) dZ dn C
(3-7) c
Substituting equations (3-4) and (3-7) in (3-6) gives the bending moment
as a function of P and dn only, i. e. it can be written as
M = M (p, dn) (3-8)
Equations (3-5) and (3-8) are the two basic relationships between
M, p, P, and dn. The parameter dn can be eliminated and a
relationship between M, p, P can be derived as
f (M, p, P) = 0 (3-9)
The complexity of this equation depends upon the stress-strain laws
of the concrete and steel and its derivation may be very complicated.
In most cases a numerical solution is inevitable. In the present
work the stress-strain relationship for concrete is a function of the
neutral axis depth itself, and an analytical derivation is very cumber-
some, if not impossible. A numerical procedure has been adopted
based on an iterative process with the aid of a computer program.
The details of this procedure are given later. Here the assumptions
made in the analysis and the materials' constituent properties are
discussed.
69
3.3 Assumptions
(1) The strain distribution at the cross-section is linear.
The degree of the validity of this assumption was discussed previously.
It is satisfied at the early stage of loading but loses its validity when
the concrete starts cracking. A more appropriate assumption would
be a bi-linear strain distribution characterized by the tension steel
strain as - es = F(1 n n -) e
c (3-10)
where ec and n represent the strain of the concrete at the outermost
fibre and the neutral axis depth ratio respectively.
F, the strain compatibility factor, depends upon the amount of bond
between the concrete and steel as well as the loading condition.
Experimental evidence (43) shows that the value of F is high at the
time of cracking but it gradually approaches unity if the bond is
rather poor. However, there are some reported results which show
the opposite trend (42). As the value of the strain compatibility
factor, F, is not quite clear, it is taken as unity which represents a
linear strain distribution.
(2) Concrete has no tensile strength
The tensile strength of concrete is nearly one tenth of its compressive
strength. It is therefore clear that this assumption makes the M-P-P
diagram more flexible at the stage immediately before cracking, but
the discrepancy would not be much.
(3) The stress-strain relationship of compression concrete is
unique, i, e. in the case of any reversal of the strain no separate path
is considered.
In the present programme of testing that the axial load is applied
first and the bending moment later, a certain part of the cross-
section becomes unloaded in due course. As the maximum strain
that these fibres experience is not very high, their position on the
stress-strain curve is not far from the relatively linear part of it.
70
Their reversal path would therefore be very close to the initial curve,
and the assumption is well justified.
(4) The stress-strain relationship of the steel reinforcement is
known. It is the same in compression and tension, and for any
reversal in the strain a new path is followed by the stress.
(5) The concrete of the cover starts crushing when the strain
in the outermost fibre reaches 0. 0035. This limit increases as the
fibres get closer to the core of the section. This point will be dis-
cussed in the text.
(6) In the calculation of the deflections, the shear deformation
in the column is ignored.
3.4 Material Properties
3. 4. 1 Concrete
The stress-strain relationship of concrete has been
subjected to a lot of research, leading to a number of different
relationships. Most of these relationships, however, explain a
particular aspect of concrete behaviour under special conditions,
among which are the effect of the concrete strength, concentric and
eccentric loading, lateral confinement, rate of loading, and for long
term loading include the effects of creep and shrinkage. The
behaviour of concrete under repeated loading, concentric and
eccentric, has been studied extensively (44, 45) and a few relation-
ships have been put forward. The relationship used in the present
work is the one which was recently introduced by Sargin and Handa (34).
To the author's knowledge it is the most general relationship which has
been introduced so far. Here, the version of it which concerns short
term loading is given, figure 3-3.
f =K f' AX + (D - 1).X
2 2
c 3 c 1 + (A - 2) X + DX (3-11)
71
where ec X= — e oc
E . e c oc A 113 . f' c
D = 0.65 - 5 x 10-5 f'
(3-12)
Ec is the initial modulus of elasticity of the concrete given as
Ec = 60000 rf-- psi. (3-13)
Ec in the original version of the formula was given as 720000but
all the recent reports (2, 46) indicate a lower value for the constant
as given above.
eoc is the strain corresponding to the maximum stress in the
relationship. It is given as
pl f f" l 1
e = 0.0021 1 + 0.25115 + 0.154 (1 0.7 S —) oc dn
K3 is the ratio between the maximum stress and the 6 x 12 in.
cylinder strength. It is expressed in the form of
(3-14)
[K3 = 1 + 0.007
pl? et
sy
n + 0.015 (1 - 0.25§
1 )
13 (3-15)
where f' is in pounds per square inch.
These relationships, as they stand, are used for the confined core of
the cross-section. For the concrete in the cover of the cross-
section the terms corresponding to the confinement effect are
ignored.
The cylinder strength of the concrete is assume d to be
= 0.85 f cu
where fcu is the 4 in. cube strength of the concrete, available for
72
the tests analysed here. This ratio is acceptable (47) since the
strength of the concrete in these tests was rather high (feu = 7000 psi. ).
A plot of this relationship for concrete under concentric and eccentric
loading and with different degrees of binding is shown in figure 3-5.
The diagrams indicating the behaviour under eccentric loading are
the characteristics of fibres on the concrete cover and core under
monotonically increasing bending moment, i. e. the eccentricity of
the load is not constant. The effect of eccentric loading on the
ductility of the concrete is seen clearly by comparing curves 1 and 2.
The effect of binding is shown by the curves 2, 3 and 4. Both
strength and ductility of concrete are affected by the binding. In
the case of concrete with a strength of 5000 psi. the increase in
strength is nearly 7% and 14% for shear reinforcement volumetric
ratios of 0. 92% and 1.84% respectively. However, the increase
in the ductility is quite considerable.
3.4.2 Steel
The steel reinforcements were of mild steel and they showed
an elasto-plastic characteristic with a strain-hardening effect in the
coupon tests, figure 3-4. This characteristic is used in the analysis,
the strain-hardening part of the curve being represented by the
following third-order polynomial
- e (f su f ) (3-16) f
s = f
su - (e
e su - e
s )3 su sy su sh
The parameters are defined in the figure 3-4.
The corresponding data for the work-hardening range is
not available for the previous tests. As an elastic-perfectly plastic
characteristic is assumed in the analysis for these tests, their
results may have some shortcomings.
The modulus of elasticity is assumed to be 30 x 106 psi.
The yield strength and other data is obtained from the coupon tests.
The yield strength of the lateral reinforcements in the previous
73
. 2 tests was assumed to be 16 ton/in. .
Should any unloading of the steel occur, the reversal paths
are considered to be elastic and parallel to the initial path. As
the unloading in the steel is not much, the Bauschinger effect is
not considered.
3.5 Computing procedure of M-P-P
As discussed previously, the procedure adopted in this
analysis is an iterative one in which a strain distribution is assumed
at the cross-section, and then, the equilibrium condition is checked.
Should it not be satisfied, the strain distribution is changed and the
process is continued until equilibrium is attained. For a linear
distribution, the strain at any point on the cross-section is fixed
by specifying two parameters. In the iterative process one of these
two parameters is specified at the start and the other is assumed,
and then changed during the course of the iteration.
In this program the two parameters are the curvature and
the strain at the centre of gravity of the concrete section. The
former is specified and the latter is assumed. The steps taken in
the process are as follows:
(1) The cross-section is divided into N strips, figure 3-2,
and the area of each strip, a(i), and the distance y(i) of its centre
of gravity from the reference axis ox are calculated and stored.
The location y(j) of the reinforcements, and their cross-sectional
area As(j) are also stored.
(2) A value for the curvature, P, is specified.
(3) The strain eo at point 0, the centre of gravity of the
concrete section, is assumed. With p, eo and the strain dis-
tribution known, the strain at any section can be found from
ec(i) = eo + P • y(i)
(3-17)
The neutral-axis depth is calculated.
74
(4) The stress fc(i) at any strip, corresponding to ec(i), is
found from the stress-strain law. The stresses in the reinforce-
ments, fs(j), are calculated and a check is made on the reinforce-
ment strains. If any reversal in strain has occurred, the correct
path is found and the stresses calculated accordingly.
(5) The sum of the internal forces on the section is found
from
N K Pi => fc(i) • a(i) +> f s(j) • As(j)
i=1 j=1 (3-18)
where K is the number of reinforcements.
(6) Let c)<= PT P (3-19)
If c<= 0, the equilibrium condition is satisfied and eo is the correct
strain at the point C for the specified curvature. The process of
iteration is therefore terminated and the bending moment of the
forces acting on the cross-section is determined from
N K M => fc(i) a(i) • Y(i) +> fs(j) • As(i) • Y(i) (3-20)
j=1 For further calculation the process starts at step 2.
If o(#0, the equilibrium condition is not satisfied and a new value for
eo should be chosen.
Let the new value of the strain be
eo eo + Leo (3-21)
If L eo is the correct change for eo then the corresponding resultant
of the internal forces
PI = Pi +L.Pi
should satisfy the equilibrium condition,
Pi + AP' - P = 0
A comparison of this equation with (3-19) results in
AP' = -
(3-22)
(3. 23)
(3.24)
75
The change in P', due to the change in eo, can be found
from equation (3-18). Thus
P' = ?fc(i) ? e (i) i=1 c
K ?f (j) A s(j) , es(j) (3-25) a(i) • 6 ec(i) + 57-s-(7 .
With "c(i) - Ec(i) ec(i)
fs(i) - E es(j) s
(j)
(3-26)
and use of equation (3-17) one can write
ec(i) = A co
and similarly es(j) = a eo
The equation (3-25) can therefore be written as
= 6 e •{) Ec(i) . a(i) + E
s(j) . As(j) (3-27)
i=1 j=1 where Ec(i) and Es(j) are the tangent moduli of elasticity of the
concrete and steel at the strain ec(i) and es(j) respectively. These
values correspond to the value of e0 in the current stage of the
iteration and they can be calculated and stored at step 4.
The required change of strain at 0 for the next stage of
iteration is found, from equations (3-24) and (3-27), to be
eo - of
(3-23) Ec(i.) . a(i) +LE s(j) . A,(D
1=1
With the new value of eo the process is repeated from step 3 and
the iteration continues until equilibrium is satisfied. The process
is then continued from step 2.
76
The number of iterations required for convergence depends
upon a value of 0( 1 being specified and which is considered to satisfy
the equilibrium condition. The process was terminated when
I 0( 1 O. 01 P
it having been found that a greater accuracy did not change the
results significantly. With this precision the number of iterations
was usually less than five and often two. The method, which is
in fact the Rapson-Newton method, does not always guarantee con-
vergence. In this report convergence was not always obtained on
the falling branch of the moment-curvature diagram. When it was
not obtained the program carried out a procedure based on the
Newton bisection method.
Typical M-b-P diagrams for a section under different
axial loads are shown in figure 3-6. The terminal points on the
diagrams represent the crushing stage, i. e. when the maximum
strain in the concrete is 0.0035. The falling branch of M-P-P is
not shown and will be discussed later. The P-M and P-P interaction
diagrams for this section are also shown in the figure.
3.6 Force-Deflection Characteristic for the Rising Branch of
the M-P-P Diagram
3.6.1 Exact solution
The load-deformation behaviour of a column under non-axial
loading is governed by the second order differential equation
d2 (EI) v + Py = P - D + F (h - x) (3-29) x dx
and which is applicable to the column under the loading condition
shown in figure 3-7(a). (EI)x is the flexural rigidity of the cross-
section at the level x. In a reinforced concrete column where
(EI)x varies along its length, an analytical solution of this
equation is impossible and a numerical solution is therefore
inevitable. Among the various numerical solutions for the beam-
77
column problem, two distinct approaches have become very popular
in this field. In both of them the M-P-P relationship is needed
beforehand.
In one of these approaches the deformed configuration of
the column is assumed to follow a pre-determined course, usually
a sine or a cosine curve. The governing equation is then satisfied
at the critical section and the load deformation relationship derived.
This approach makes the calculation much simpler but lacks the
accuracy of the other method. The procedure is reported by Brorns
and Viest (18).
The other approach is based on the procedure originally
proposed by Newmark (48). In this approach the deformed shape
of the column corresponding to a set of loads is proposed. The
bending-moment and the corresponding curvature at different cross-
sections are found and, by double integration of the curvature along
the length of the column, the deflected shape of the column is
determined and then compared with the proposed shape. Should
they differ by more than the required accuracy, the determined
shape becomes the proposed shape in a new round of calculations
and the process is continued until convergence is achieved. At
this stage, the governing equation is almost satisfied at quite a
number of points. The accuracy can be improved by increasing
the number of these points. This approach has been adopted by
Pfrang and Siess (20) in their general solution for restrained
columns, and by Cranston (21) in a similar program.
The method followed here is the same as the latter approach
and a computer program has been developed in which the lateral
force, corresponding to a specified deflection at the head of the
column, is computed.
78
3.6.1.1 Outline of the Computer Program
The process is an iterative one in which the two main
parameters are the deflection at the head of the column and its
corresponding lateral force. The deflection is specified and the
lateral force and the deflections at the other points of the column
are assumed. The assumed deflections remain constant during
the first stage of the iteration in which the lateral force is changed
in order to obtain the specified deflection at the head of the column.
The deflections are then compared with the values calculated in the
last iteration. If they do not satisfy the required condition, they
are modified and the next iteration starts again from the very
beginning.
In the program the column is divided into N segments,
figure 3-7(b), and the co-ordinate at the end of each segment is
recorded, x(i). These end points are referred to as the division
points in the following passage.
At the end of the (K-1)th step of the deflection increments,
the following parameters are known
Dk- 1 Fk- 1
the deflection at the head of the column
the lateral force
the deflection at the division point i. (Note that Yk_ i(N) =
th The K step of the computation is carried out as follows:
(1) D is specified an k i ifid d Y k(i) is assumed at all the division
points. Fk is also assumed. These are arbitrarily extrapolated
from the previous steps by using Dk
Yk (i) = Yk-1(1) k-1
- F = F +
Fk-1 Fk-2 - Dk-1) k k-1 Dk- 1 - Dk-2
(3-30)
79
(2) The bending moment M(i) at the cross-section i is found
from
M(i) = P Dk- Yk(i)) + Fk [ h - x(i) (3-31)
(3) Having determined the bending moment at a section, the
corresponding curvature is found from the M-0-P diagram. In
practice, the related pairs of the bending moment and the curvature
are tabulated in the computer and, for any value of the bending
moment, the corresponding curvature is found by linear interpolation
between the lower and the upper values in the table. However, for
the sake of accuracy a third order polynomial is fitted between any
point in the table and the three nearest points to it, and the inter-
polation is carried out according to this function. The coefficients
of these polynomials are found by the Lagrange interpolation formula
and recorded beforehand. For values of bending moment between
the yield moment and the crushing moment, a linear interpolation
is adopted because the variation in the bending moment is very small.
(4) The deflection at any division point is found by the integration
of the curvature diagram along the column
/Ix Y(i) !(x) dx dx (3.32)
o o
Assuming that there is a linear variation of the curvature between
the two adjacent division points, this integral gives
43(i) =
=
(1 - 1)
7(i-1) +
+ -1 2 [p (1)
- 1) .
p (i _
0 x + jd%
1)1
[ 2
ox
p(i - 1) + pad
(3-33)
p
where z x is the length of each segment of the column. The
deflection of the head of the column is
Dc =Y (N)
?Dc ( )
and as fb is a function of M
P = P (M)
fo 0 F F x
h Ix
80
(5) Let °<= De - Dk (3-34)
t , the, first stage of the iteration (the determination of the
lateral force corresponding to Dk) is complete. If
1 7(i) - Yk(01 ‘ t
for i = 1 —P N (3-35)
Where e is the specified tolerance for convergence, the assumed
deflections for the Kth step are satisfactory. The computation for
this step is terminated and the next step starts. If this condition is
not satisfied then the Yk(i) values are replaced by Y (i) and the process
is repeated from stage 2.
(6) If I of I ›e the assumed lateral force is not correct and an
improvement upon it should be made. The improvement should be
such that the resulting change in De makes c< = 0 in equation (3-34),
e.
dx . dx (3-37)
(3-38) then
?f) dip a m ob ? m or E (3-39)
F ? M F cb,M • ?F
which, by taking equation (3-31) into consideration, yields
3 Ep, m (h - x) ? F
1" , is the rate at which the curvature varies with respect to the
Y-4 bending moment in the M-O-P diagram. It is, in fact, the inverse
of the flexural rigidity of the section at level x, and can be found
for any section in stage 3 during the calculation of the curvatures.
? De 6Dc ? F F = —0e (3-36)
From equation (3-32), for i = N, one can write
(3-40)
31
Substituting equation (3-40) into (3-37) gives
?De G=
( Ep, N )x . ( h - x ) dx . dx o F /h jox (3-41)
This equation and the assumption made at stage 4:for the distribution
of the curvature between the two adjacent points, results in an
expression similar to equations(3-33) in which the following
replacement has been made
9!)(i) replaced by E0 M (i) [ h - x(i)]
o f can be found by substituting equation (3-41) into equation (3-36)
o F = - o/G (3-42)
Therefore, the new assumed Fk will be
Fk = Fk (in the last iteration) + 6 F (3-43)
and the process restarts from stage 2 and continues until convergence
is achieved.
The process described in stage 6 cannot be applied when the
bending moment at the critical section approaches the maximum point
in the M-0-P diagram. This is because the gradient of the M-cb-P
diagram gradually becomes smaller and, consequently, E0,
becomes larger and equation (3-42) becomes ill-conditioned. To
overcome this difficulty, when the bending moment at the critical
section passes the yield moment, another procedure based on the
Newton bisection method is followed.
The tolerance specified for the limit of convergence &
was 1% of the prescribed displacement. With this degree of
accuracy, the number of iterations needed for convergence was
usually about five. However, in the Newton bisection method it
could be more than five.
The force-deflection characteristics obtained by this method
for a column under various axial loads, and for columns with different
82
heights, are shown in figure 3-10. The M-b-P diagrams of the
cross-section of the columns are shown in figure 3-6.
3.6.2 Simplified Solution
The method of solution discussed above is obviously too
difficult to be used in practical work and a simplification is necessary.
As it stands, however, the integration of M-P-P without considering
the plastic deformation which occurs at the hinge zone, results in an
underestimation of the deflection. As will be shown in Chapter 4,
this overestimates the degree of safety. Although the deficiency
may be eliminated by using a modified M-P-P relationship, or by
incorporating an extra rotation for the cracked zone into the cal-
culation, the difficulty in the method of solution still remains.
The simplified method based on a bi-linear idealization
of the M-P-P diagram recommended by C.E.B. (25) makes the
process of calculation much simpler, but it does approximate
considerably the behaviour of the column during the early stage of
loading. Here the principles recommended by the C.E.B. are
considered, but they are applied to the whole range of the M-p-P
curve instead of only the two limits. The assumptions made here
are:
(i) During the stage prior to yielding of the critical section, the
column behaves linearly-elastic with a flexural rigidity of each cross-
section equal to
EI = M/P (3-44)
where M and fare the bending moment and curvature at the critical
section, figure 3-11.
(ii) After yielding of the critical section, the value of EI remains
constant and equal to
EI = (EI)y = (EI) (3-45) 'yield
and an extra rotation, a plastic rotation, concentrated at the critical
section is considered. The magnitude of this rotation is calculated
83
from 4p Oy Lp (3-46)
where Lp = 2.5 n- d
(3-47)
and rand n are the curvature and the neutral axis depth ratio at the
critical section.
This relationship is, in fact, recommended by Solima.n
and given in equation (1-14) in Chapter 1. In that equation, ep
is the plastic rotation at limit L2. By considering
ecl = P1 . n1 d
(3-48)
ec2 = P2 . n2 d
and ignoring the small difference between n1 and n2, L at the
limit L2 will reduce to that given in equation (3-47). Here this
relationship for 1_, is assumed to be true over the range from the
yield moment to the crushing moment.
On the basis of these assumptions the deflection at the
head of the short cantilever column shown in figure 3-7 is
D = h2 for M M
and D 3 (EI) = 1 h2 + h . 4:4p for M) M M
where M and Tare the bending moment and curvature at the
critical section.
It is obvious that with the assumption (i), the deflection
of the column is overestimated when compared with the exact
solution. This is because the properties of the cracked section
are assumed for the whole length of the column. However, as
the exact solution itself underestimates the deflections, the results
would be nearer to the actual ones. A comparison of the results
will be given in Chapter 4.
In long columns the effect of the axial load, known as the
instability effect, is more pronounced. This phenomenon should
84
be noted carefully and accounted for by the use of stability functions,
as is done for linear-elastic columns. At the post-yield stage, the
effect of the hinge rotation should be considered. A full treatment
of this problem for a general beam-column member is given by
Nahhas (49). It is not discussed here. The deflection at the head
of the column, for the simple case of the cantilever column in
question, can be shownto be
D K h+ p + h. 43p (3-49)
where M is the bending moment at the critical section, K is the side-
sway stiffness of the column which includes the instability effect
given by
K - (1 +C) EI 2 m h3 (3-50)
and S, C & m are the stability functionsgivenby Horne and Merchant (50).
In figure 3-10 the force-deflection characteristics of a column
under various axial loads, and columns with different heights, are
shown and compared with the exact solution. The vast difference in
the estimation of the deflection, between the two methods, is clearly
shown. By considering the M-P-P diagram shown in figure 3-6,
it can be seen that the increase in the deflection of the column, after
yielding of its critical section, is virtually nothing in the exact
solution. The long plateau seen in the M-P-I" diagram, particu-
larly for low axial loads, do not produce much deformation due to
the limited and unrealistic hinge length predicted by this method.
The hinge length is predicted more realistically in the simplified
solution. Other than this major difference between the two
methods, the figure shows that the simplified method overestimates
the deflections before the yielding stage.
85
3i 7 Force-Deflection Characteristic in the Falling Branch
of the M-P-P Diagram
The behaviour of the column in the rising branch of the
M-P-P diagram is harmonious in the sense that, an increase in
the bending moment will produce an increase in the deformation at
each cross-section of the column, although the rate of increase will
not necessarily be the same throughout the column. This is not
true for the falling branch of the M-p-P diagram. When the
critical section passes its maximum bending moment capacity, the
bending moment starts decreasing at all sections and there is a
discontinuity in the behaviour of the hinge zone and the area outside
it. The deformation in the former increases continuously while it
decreases in the latter. An observation made from one of the
previous tests is shown in figure 3-12. The curvature in this
diagram is not very accurate because it was measured indirectly
from the bent configuration of the column. However, the figure
does confirm the idea that, at this stage, the hinge zone usually
follows the falling branch of the M-P-P diagram while the region
outside it follows the unloading paths.
At this stage, the deformation at the hinge zone itself is
mainly concentrated at one major cracked section. At this section
the crack is gradually widening and the tension steel, now yielded,
is elongating and spreading its yield over a distance on both sides
of the section, the distance depending upon the bond between the
concrete and steel. With a poor bond, a longer distance for the
yielded steel is expected. On the compression side, the concrete
is crushing or spalling over a considerable distance on both sides
of the section. The spread of crushing is greater for columns
under higher axial load because the amount of compression concrete
at the section is greater and obviously, the concrete in a larger
zone on both sides of the section will be under high stress. This
results in more crushing.
86
The amount of deformation occurring at the critical
section bef ore f ai lure depends mainly upon the behavi our of the
concrete and, especially, its crushing strain. This strain is
rather limited for unbound concrete and, consequently, the cover
on the compression side of the section crushes or spalls rather
quickly. The crushing strain is higher for the concrete core but
depends on the amount of bound provi ded at the section. The
concrete core is under tri-axial pressure due to the effect of the
binders and, as shown by experiments, its ductility and strength
are increased considerably.
In this section, the test results are analysed and the
rotational capacity of the hinges is found. The variation of the
hinge rotational stiffness with the parameters studied in the tests
is also shown. As the concrete cover during this stage of defor-
mation gradually crushes or spalls, the falling branch of M-P-P
diagram based on the uncrushed section cannot be considered. A
reasonable path for this branch of the diagram is assumed, and
later, the properties of this path are empirically related to the
experimental hinge rotational stiffness. Finally, an expression
for the hinge length is derived.
3.7,1 Hinge Rotational Stiffness (Analysis of the Experimental Data)
A typical diagram of the lateral force versus the deflection
at the head of the column, F-D, obtained in the previous tests is
shown in figure 3-9(a). The bending moment at the critical section
is found from
M.F. h+P. D (3-51)
and a diagram of this bending moment against the deflection, M-D,
as found in the "Z" tests, is shown in figure 3-9(b). The falling
branch of the M-D diagram represents the moment rotation
characteristic of the hinge, since all the deformation of the column
during this stage is concentrated there. This statement is true
provided that no unloading occurs in the column outside the hinge
87
region. However, the unloading of this part of the column will
cause a loss in its deflection, as compared with that at the time of
the maximum bending moment. Therefore, the deflection caused
by the rotation of the hinge, corresponding to a drop of 6 M in the
bending moment, is
d =711 +712 (3-52)
instead of being d2 alone. al is the loss in deflection due to the
unloading and as seen in the figure it is very small compared with
d2. For the tests under consideration, its value is calculated on
a simple assumption in order to estimate its relative effect on the
hinge characteristic.
It is assumed that during unloading, the column behaves
linearly-elastic with the flexural rigidity of its section equal to (El)°,
the initial value of the uncracked section, figure 3-8(a). In addition
to that, it is assumed that the length of the hinge zone is small com-
pared with the height of the column, and consequently, the unloading
of the column is assumed to extend over its complete height. The
hinge, therefore, acts as a rotational spring with a softening
characteristic at the base of the column.
With these assumptions, the elastic stiffness of the column
along the unloading path of the F-D diagram is equal to its initial
stiffness which is represented by the initial gradient of the diagram.
Calling this. initial stiffness S, d1 is found from
(3-53)
(3-54)
(3-55)
LIM 1 P+Sh
Similarly, ; is calculated from
71 2 - LM P + a h
where F
a - D
in the AB segment of the F-D diagram and a F and o D are measured
88
from the point A corresponding to the maximum bending moment.
The total deflection 71 is then given by
d= AM( 1 1 , P+ah P+Sh/ (3-56)
and assuming that the hinge is located at the base of the column
= h A 4 (3-57)
where L4 is the increase in the hinge rotation corresponding to a
drop of A M in the bending moment. Substituting equation (3-57)
into equation (3-56) results in
A M , , 1 1 n k KM-4 P+ah P+Shi (3-58)
where KM-43 represents the rotational stiffness of the hinge at this
stage.
The AB segment of the force-deflection diagram proved
to be nearly linear in the tests, especially when the axial load was
not very high, figure 4-2 to 4-22. In fact, the form of the diagram
in this segment depends mainly upon the way in which the concrete
proceeds to crush. If the crushing is gradual, then the loss in the
bending moment resistance is also gradual, and the segment AB is
nearly smooth, but if the crushing is accompanied by a sudden
spalling of a relatively large part of the cover, the bending moment
resistance drops rather rapidly at first, and then decreases
smoothly. This latter case usually occurs in columns under high
axial loads, figure 4-22.
The parameter 'a' is taken as a constant, and in cases
where AB is not quite linear, an average value of 'a' is assumed.
The measured values of 'a' and S obtained from the F-D or M-D
diagrams are tabulated in table 4-2.
The second term on the right hand side of equation (3-58)
is due to the unloading of the column. As previously stated, the
contribution of this term to KM-0 is relatively small. If this term
89
is ignored then KM-Ja is
M-4= h (P + a h)
(3-59)
This equation shows that for perfect plasticity in the hinge (Km..4= 0)
the gradient of AB is
a= -P/h
which is the gradient of the F-D diagram at the point of maximum
bending moment, i. e. the slope of AC.
In table 4-2 K calculated for different tests, is given
with and without taking the unloading term into consideration. The
table shows that the difference between the two values is negligible
for columns under low axial loads, but that it increases as the axial
load is increased. With high axial load, the rate of decrease in the
bending moment resistance is greater and it causes a more rapid
unloading, i. e. greater 711. d2 itself is relatively smaller here.
Thus the effect of d1 is more pronounced in d.
In the following analysis the effect of unloading is ignored
and the corresponding Km_.e. is used. This means that in the falling
branch of 1114)-P diagram, the deflection of the column is found
simply by adding the deflection due to the rotation of the hinge to
that corresponding to the attainment of the maximum bending moment.
3.7.2 The Effect of Different Parameters on KM-4 The main parameter affecting Km...e, is the axial load on the
column. In figures 3-13 to 3-15 the variation of the bending moment
at the critical section with the deformation of the column, D or 4, is
plotted for different cross- sections and axial loads. The average
gradient of the bending moment on the falling branch of these diagrams
is either equal or proportional to K . As seen its absolute value
increases with the axial load. For the 6 x 8 in. sections, this branch
of the diagram falls rather rapidly at first, and then becomes smooth
and falls more slowly. This is due primarily to the sudden spalling
at the first crushing of the concrete, and secondly, to the effect of
90
the steel strain-hardening, The reinforcements used in these tests
showed a considerable amount of work-hardening in the coupon test.
Such behaviour was not observed in most of the previous tests where
the drop in bending moment was rather smooth and uniform.
A plot of the non-dimensional terms Y = 2f' and
X = P/bdf' is shown in figure 3-19. First of all, there is no sign
of grouping between the values belonging to the same cross-sectional
size. As far as any comparison is possible, the scattering of the
points is the same for all of them. Secondly, although the variation
of Y with respect to X is smooth and almost parabolic for the lower
range of axial loads, it varies rather sharply in the higher range.
Knowing that the balance load for all of these sections is between
X = 0.3 to X = 0.4, it appears that Km_o is less sensitive to a
variation in the axial load, provided that the loads are less than
the balance point, and vice versa. This means that in compression
hinges the deterioration of the section is more rapid than in tension
hinges.
A simple parabolic relationship which fits the points in the
range of loads below the balance point is
-43 P 1215 bdf“
bd f'
or P2
= 15 for M-43 bf' bdf' 0.4
The curve representing this relationship is shown in the figure.
For values of P higher than the balance load, a linear relationship
would probably be suitable.
The relatively high gradient of Y, in the range of high axial
loads, may be due to the relatively thick concrete cover used in the
smaller sections. When crushing of the cover occurs in these
sections, a relatively larger proportion of the section is removed
91
and the bending moment resistance decreases more rapidly. As
an example, compare KM-.0 for the case of K11 with the others in
its group, table 4-2, there the concrete cover for this specimen is
0.125 in. in contrast to 0.75 in. for others. For this reason, the
results obtained in this range should be treated more carefully.
In figures 3-14 and 3-15, the BA-.9 diagrams for columns
with different heights are shown. For some of the taller columns,
the values of KM-€4 are higher, for others it is less. This is
verified by comparing B14 with K22 in figure 3-15 and K9 with K15
in figure 3-14. In this respect, a definite conclusion cannot be
drawn.
The effect of steel cross-sectional ratio on KM- is 43 examined in figure 3-16. Once again, there is no definite trend
in the values of KM-$. This result supports the observation made
by Chinwah (32) in which no change in the hinge rotational capacity
was observed due to a change in the steel ratio.
Three tests have been performed with different spacings
of the shear reinforcements. The results of these tests are shown
in figure 3-17. B24 has the least spacing and shows a higher KM-0 which contradicts the observation, made in many tests by different
investigators (2, 3), that the smaller the spacing the greater the
ductility. Here the reduction in spacing may not have had much
effect on the confinement of the concrete in the core, due to the
rather limited volume of the core caused by the thickness of the
cover. However, the effect of this parameter should be clarified
in further tests.
The effect of different rates of displacement on K is M-0 shown in figure 3-18. Although it shows an increase in Km-.9 with
an increase in the rate of displacement, it cannot be relied upon
because the values of KM-43 lie within the scatter of values obtained
in other tests with the same axial load. However, it is unlikely
that the rate of loading, in the range used here, has any significant
92
effect on the behaviour of the member. Dynamic tests on concrete
cylinders under a much higher rate of stressing, 106 psi/ sec, (51)
show a small increase of about 10% in the ductility, but a much larger
increase in strength of about 38%.
The variation in Km_.e. for columns under similar axial loads
is interesting. Reference to table 4-2 shows that a difference of 100c/0
is sometimes found. Some of these differences are due to variations
in the strength of the materials used in the different specimens; they
can also be due to the effect of the above parameters. However, a
major part of this variation should be due to the performance of the
concrete cover during crushing or spalling. Should the crushing
proceed rapidly, the fall in the bending moment would be more rapid
and KM-4 would be greater. In this respect, perhaps it would be
more rational to find an upper and lower bound for the value of
KM-$, consequently for the hinge length. However, the average values given in the table should be considered as the most appropriate.
3.7.3 Falling Branch of the M-P-P Diagram
In this branch of the diagram, the concrete on the com-
pression side of the section crushes or spans, and the depth of the
section is gradually reduced. The behaviour of the section depends
very much on the crushing strain capacity of the concrete. The
unbound concrete on the cover has a limited amount of strain capacity
and usually crushes when the strain reaches a value in the range of
0. 003 - 0. 004. For the sections under combined axial load and
bending, the limiting strain tends to the lower bound, and vice versa.
The bound concrete, however, shows a higher crushing strain capacity
which varies according to the degree of confinement of the section.
A value of 0. 030, nearly ten times greater than that of the unbound
concrete, has been observed in some tests (1). Other parameters,
such as the grade of the concrete, the strain gradient at the section,
and the bending moment gradient in the member, affect this limit.
93
The value quoted above for the unbound concrete occurs
when a noticeable crush appears in the cover. The cover then
gradually crushes, or, if the axial load is rather high, it suddenly
spans under the lateral pressure of the concrete core or the longi-
tudinal reinforcement. The crushing strain of the fibres in the
cover closer to the core is higher than this value. This is due to
continuity between the concrete of the cover and that of the core.
This is verified by the gradual drop in the bending moment capacity
of the section seen in the test results, figuires4-2 to 4-22, as opposed
to a sudden fall in the bending moment, caused by a sudden removal
of the cover.
Figure 3-20 shows IVI-P-P diagrams for the specimens
Z1 and Z5, under axial loads of 50 and 20 tons, respectively.
Point A in the figure represents the stage at which the strain in
the outer fibre of the section reaches the crushing limit of 0.0035.
The curve AB shows the behaviour of the section when crushing of
the concrete is ignored by assuming the crushing strain limit to be
very high. As seen, the curve is very smooth, and for the case of
Z5, it even rises due to work-hardening of the steel. At a later
stage, it falls rather suddenly. The curve AFGD shows the behaviour
of the section when the crushing limit of the concrete is assumed to
be 0.0035 for all the fibres in the cover. On segment AF, the cover
is gradually crushed and in FGD, the cover is completely removed
and the curve coincides with the M-P-P plot O'GDE for the cross-
section without any cover on the compression side. The drop in
the bending moment, as seen, is sudden and quite considerable,
and at point F it is less than the corresponding point on the O'GDE
diagram. This is because the sudden crushing of the concrete
cover causes the neutral axis to shift rather rapidly, which results
in a strain reversal in the steel and a smaller bending moment
resistance. However, the reversal is recovered very quickly and
AFG joins O'GDE at G.
94
The curves AB and AFGD are the upper and lower bounds
of the bending moment capacity of a section for this stage. If a
small amount of the concrete crushes, the curve representing the
behaviour of the section will be closer to AB, and for the opposite
case, it is closer to the curve AFGD. The rate of loading, as well
as the parameters mentioned above, should also influence the
behaviour. In the present analysis, this part of the diagram is
obtained by the following assumptions:
(1) The concrete starts crushing when the compression strain at
the outermost fibre of the section reaches 0.0035.
(2) The crushing strain limit of the bound concrete is obtained
from the following relationship introduced by Soliman (2).
- 4 ec =0.003 1 + 0.8 tr + 1 0.5 n +,n 4. (1 +n)Z (3-60)
The details of this relationship were given in Chapter 1, following
equation (1-15). The dip.grarn representing the relationship,
figure 3-21, shows that the limiting strain is reduced as the neutral
axis depth ratio increases, i. e. as the axial load on the section is
increased.
(3) The crushing strain for the different fibres in the cover varies
linearly between the above two limits.
With these assumptions, the falling branch of the M-P-P
diagram is similar to the curve AC shown in figure 3-20. This
curve joins the curve O'GDE at point E, which means that, at this
point, the concrete strain in the outer fibre of the core section is
that given by equation 3-60. It should be noted that the value of the
ultimate crushing strain, given in the above equation, is the lower
bound of the strains recorded in the tests. Therefore, the bending
moment diagram is expected, in some cases, to extend beyond the
point corresponding to this strain.
95
3.7.4 Hinge Length
In this section the hinge characteristic, found in the
experiments, is related to the properties of the section in the
post-crushing stage. The behaviour of the hinge zone is charac-
terized. by the falling branch of the M-P-P diagram described in
the previous section. The hinge length, as before, is defined as
L = (3-61)
where 4) and p are the hinge rotation and the curvature at the
critical section at any level of loading. is measured from the
yield moment level . L for the stage of loading between the yield
moment and the first crushing of the concrete, has been given before
as L = 2.5 nd in equation (3-47). Here, the purpose is to find a
relationship for L applicable to the post-crushing stage.
Before launching into the main subject, one point should
be clarified.P in the above relationship is found from the experi-
ment, whereas the curvatures are derived from the M-p-r, relation-
ship of the cross-section. This relationship, as previously dis-
cussed, is based on certain assumptions whose application,
especially at this stage of loading, is arguable. However, due to
the simplicity with which they can be applied, they are used readily.
For this reason, the curvatures found in this way may have nothing
to do with the actual "mean" curvature in the hinge area of the
member. As a result, L may differ significantly from the actual
length of the critical zone observed in the experiment. Its value
generally depends upon the assumptions adopted for the derivation
of M-P-P, and for this stage, particularly on the crushing strain
of the concrete. Figure 3-20 shows that, as the falling branch of
the M-P-P diagram approaches the curve AB, representing the
uncrushed section, the change in curvature resulting from a
certain amount of change in the bending moment becomes greater,
and subsequently, L becomes smaller; and vice versa.
96
From the experiments, the relationship between the change
in the bending moment at the hinge zone and the change in the
corresponding rotation, was established as
LM = Km . (3-62)p -9Considering the tendency of the variation of Km_o in the range of
axial loads less than the balance load, figure 3-19, L M seems to be
a parabolic function of the axial load.
L M = C . P2 . .8-9p (3-63)
where C is a constant.
In order to examine the parameters influencing 43 at this
stage, a rough assumption is made for the concrete stress-strain
law so that a relationship for M-P-P can be derived. The section
is assumed to have identical tension and compression steel, all of
which is assumed to have yielded by this stage but without any strain
hardening effect. The stress- strain law for the concrete in com-
pression is assumed to be a polynomial in ec
fc = f (ec)
Reference to figure 3-1 enables the following relationships for the
compatibility and the equilibrium conditions to be written
e = e c nd co
ind P =b f (e c) dZ = g (eco) • nd (3-64)
nd oe. nd = ,b1 z
o . f (ec) dZ = h (eco) nd
Finally, the bending moment resistance of the section is
M = Ts (d d') + 0.5 (d + d') P - {1 - h (e co)] nd. P (3-65)
which, by substituting for 'nd' from the equation (3-64), becomes
97
M = T(d-d') + 0.5 (d + d') P - 1 - h (eco)
P2 (3-66)
g (eco)
or M = C1 + C2 ' P + k (eco) . P2
where C1 and C2 are two constants and g (e co ), h (e co ) and k (eco)
are all functions of eco only.
The change in M due to a change in eco or P is
= P2 . A k) co (3-67)
Comparing this equation with equation (3-63), it is concluded that
641 is independent of P and is a function of eco only. At this stage,
since the neutral axis depth depends mainly on P, AO is also indepen-
dent of n. Therefore L in equation (3-61) should be a linear function
of n. This conclusion confirms the observations made by most of
the investigators and which were discussed in Chapter 1.
The relationship between M and eco in equation (3-66) is
very complicated, despite the simple assumption made for the concrete
stress-strain law. In order to find a linear relationship between M
and a function of eco, for "the actual concrete stress-strain law" used
in the present study, several combinations of parameters were tried.
The most convenient and suitable one was found to be
) = (P - P y ) n (3-68)
which is, in fact, independent of n. This form of equation was
chosen in preference to a direct function of eco because the concrete
on the cover is gradually removed due to crushing and therefore,
eco does not belong to a single fibre.
A typical plot of the bending moment against A.d for the
falling branch of the M-P-P diagram, for various axial loads and
different cross-sections, is shown in figure 3-23. The effective
depth d of the uncrushed section is constant for each section. The
variation of the bending moment is seen to be almost linear until the
bending moment in M-P-P diagram drops very rapidly.
98
At this stage, the section fails. The change in the bending moment
may be written as
M = Km-p. £) d (3-69)
where KM-0 is a constant. Considering that ) is a function of e co,
a comparison of this equation with equation (3-63) shows that Km_p
should be a parabolic function of P. The values of KM-0 for
different columns, obtained from plotting M against )1d, are given
in table 4-2.
Substitution in this equation for M from equation (3-62)
results in
(3-70) or L19 p = A. . d
From the above discussion, "A" should be independent of P. The
values of "A" for different columns are given in table 4-2. The
variation in the values of "A", for each group, is mainly due to the
scattering seen in the values of KM-$. The reason for this was
discussed previously. Despite the scatter, the average values of
"A" for the different groups do not significantly differ from each
other.
The value of "A" appears to be greater for the sections under
low axial load. It gradually decreases as the axial load is increased,
and it seems that it passes through a minimum value and then increases.
Its value for 4 x 4 in. sections under 15 tons, 4 x 6 in. sections under
19 tons, and Z3 under 38 tons, is the lowest value in each group. With
the exception of Z3, these loads are very close to the balance load
of the sections, figure 3-24. In figure 3-25, the average values of
"A" are plotted against the P/bd fi values for the columns.
However, the relatively high value of "A" for the sections
under low axial loads may be due to the following factors:
-0 K 46.0 - . . d p K M--9
99
(1) The crushing strains of the bound concrete for these columns,
found from equation (3-60), are listed in table 4-3. From the
condition set for the application of this equation to beams, i. e.
the crushing strain of the bound concrete in beams is nearly 1.67
times greater than that predicted by this equation, page 30, it is
deduced that this equation underestimates the corresponding strain
in the sections under low axial loads. The increase in this limiting
strain results in a smoother curve for the falling branch of the
M-P-P diagram, a smaller value for Km-p, and a reduction in the
"A" value. The increase in the crushing strain from 0.008, quoted
in table 4-3, too. 010 for the 4 x 4 in. sections under 5 - 6 ton load,
reduces the average value of "A" froml2. 1 to 10.4. A similar
result is seen for K16, The effect of this increase is shown in
figure 4-3.
(2) In the case of 4 x 4 in. and 4 x 6 in. sections, the strain-
hardening of the steel is not considered in the derivation of the
M-P-P diagram, although there is an indication that the steel had
shown some strain-hardening effect (15). This effect causes a
relatively smaller drop in the bending moment with respect to P
and, consequently, results in smaller values of Km_ p and "A".
The effect, however, is more pronounced in the sections under
lower axial loads. The values of "A" for Z5 and Z4 without the
strain-hardening effect are nearly 11 and 9.3 respectively, but the
values do not change for the other sections in this group.
It is concluded from these considerations that the values
of "A" do not differ significantly for columns under different axial
loads. The differences are certainly less than that seen in table
4-2.
The overall average value of "A" is 9.70, but considering
the above points and the values of "A" for 6 x 8 in. and 4 x 6 in.
sections, which are closer to ordinary sizes, the value of "A" is
chosen to be 9.
A= 9.0 (3-71)
(Pc P ) nc = 0.72 Y (p- p ) n
where
(3-75)
(3-. 76)
(3-77)
and L , from equation (3-61), will be
Lp = 9 n d l 1 - 0.72 (p - ) n
or = 9 (1 - y ) n d
100
Equation (3-70) can be written as
.0,44 = 9 d
which is applicable to the falling branch of the M-P-P diagram.
It may be expanded to
e -= 9 ( ) d p pc c
p - pc = 9 [(p-p ) n - (pc - P) nc d (3-73)
in which the values with the subscript c belong to the first crushing
stage. From equations (3-46) and (3-47)
pc = 2.5 (p c - p y ) n c . d
and substituting for in equation (3-73) gives Pc
= 9 (P - Py) n d - 6.5 (pc - Py) nc d (3-74)
(3-72)
or
The variation of L with respect to "n" is shown in figure 3-26 for
various columns. For the stage between yielding and the first
crushing, the ratio L nd is constant and equal to 2.5. After
first crushing, this ratio increases as the damage in the hinge
progresses. The outermost points of the curves represent the
stage where the strain in the outermost fibre of the concrete core
is equal to the crushing strain, as previously explained. At this
stage, L is very close to
(.Pc - py ) ne
L = 6.75 n d
101
The variation of n for the sections under different axial
loads is interesting. For high axial loads, n starts increasing
immediately after the first crushing; this means that, at this
point, the concrete under compression has reached its highest
overall strength, For low axial loads this state occurs later
than the first crushing stage because n continues to decrease
until this state is attained. It then increases,
The range of variation of Lp increases with the axial load.
For Z5 = 20 tons), L varies between 0.65 d and 2. 2 d but for
Z1, Z2 (p = 50 tons), it varies between 1.3 d and 3.9 d. However,
the variation in curvature for sections under lower axial load is
far greater, resulting in a greater rotational capacity for these
sections. Figure 3-21 shows the lateral force deflection character-
istics of a column under various axial loads obtained by this method.
The variation of the bending moment in the falling branch of M-130-P
diagram for each case is also shown in this figure.
e c eoC
K3fc' sy
unloading path
Es1
esy esh esu
102
4' N•A•
•
"0
•
Ts ""-‹
SI
Figure 3-1 Strain Distribution and Forces at a Section
•
a(; )
fs(ii)
Figure 3-2
Figure 3-3 Stress-Strain Figure 3-4 Stress-Strain 'Relation- Relationship for Concrete ship for Steel in Tension and in Compression
Compression
Bound Concrete (Core) Plain Concrete (Cover) Plain Concrete under Concentric Loading
6 P = 30 Ton 2
f = 18 Ton/in. sy 2
f" = 20 Ton/in. sy co Spacing = S
U=1%4.
0 1 2 3 4 5 6 7' Figure 3-5 Stress-Strain Diagrams for Concrete under Compression
104
125 6"
100
•
075 H
a)
0
50 r. •,-■
CU
ft = 5000 psi c 2 f = 16 Ton/in. sy
f" = 16 Ton/in? sy
= 113 Ton
25
2 3 1.0
53 1 Curvature, P x 10, inT
Yield Stage First Crushing Stage
0.75,
0.5
0.25
0.0 50 190 0 Bending Moment, 1 on-in. Curvature, 0 xi.103,in:1
Figure 3-6 Bending Moment-Curvature-Axial Load Interaction Diagrams
eral
Forc
e
M
( b)
D
F
Deflection, D
(b) Force Deflection Diagram
Curvature,
(a) M-P-P Diagram
Ben
din
g M
omen
t
(a)
Figure 3-7
Deflection, D
105
Figure 3-8
(a) Force—Deflection Diagram (b) Bending Moment at the Critical Section-Deflection Diagram
Figure 3-9
106
Exact Solution Simplified Solution
11.48"
6"
O
Properties as given in Fig. 3-6
P = 30 Ton
2.0
1.5 h =60"
CO
0. 0 0.5 1.0 1.5 DeflectioP, 2D, in. 2'5
p.50-ron
1.5=
2 60 ,
E., 2G ri4
1.0— 61.0—
0 a; 1s 0
;-. c..-, 80 o o r.,
(IS S-4 S-4 Cll CD 0.5— "c'z' 0.5— o; 0.0 4
.-.1
0.0 0.5 2D, in. 1.0 0.0
0.5 1.0 2D, in 1'5 (a) Exact Solution (b) Simplified Solution
1.5=
0
h = 60 in.
10
80
0.0
Figure 3-10 .Force-Dellection Characteristics for the Rising Branch of 1\1-0-P Diagram
Oy
(a) PkPb
Curvature, 0
M corresponds to) e =e
sy
My corresponds to
ec
= e Orig. 3- 3) oc
Pb = Balance Load
w' Or CUrvature;
(b) P> Pb Figure 3-11 Simplification of M-p-P Diagram
Ben
din
g M
omen
t M
Mc My
M
Ben
din
g M
omen
t; M
•
O
107
B26
Figure 3-12 CurvaturesCalculated from the Measured Deflections for Column B26
•d
-,°100
z E 0
bx, 75
a")
175
150
125
50
25
:CO
6" P
D r 1
• •
• 0
\ Z 2 (P-z50-1bn) Kii;',/
M
D(
Z1 ( p=50Ton)
Z4(P=30Thn )
Z5(P=20b) ) z3(p=38Ton )
Figure 3-13 Experimental Moment-Deflection Diagrams for 6 x 8 in. Columns "Z"
3.5 1 G5 1.0 1.5 2.0 I 1 - 2.5 3.01 4.0 flection. D. in.
p.,6Ton K16
to
Pao F
4 ', • 1"/ 9
4
•
0 2 4 6 8 10 12 D 2 14 Rotation, (4 = x 10
Figure 3-14 Bending Moment-Rotation Diagrams for 4 x 6 in. Columns
M 25
20
0 15
C) 0
10 -c C.)
5
F:■.igTon ..,.... p_. 15Ton 1.1 .., `../ J-1."
h = 24 in. .•••• •••• ....
.....-- V B33
N‹ B34 N lk —
1 B14 p=16ion
K3 p,..6Ton ______ — ,
....._
P=14 Ton K22
K21 ///.
I ' 4" pi
F D
..]
I, ' " e
o
a
at /
,
0 1 5 6 D 2 7 Rotation. = —h
) x 10 2 3 4
Figure 3-15 Bending Moment-Rotation Diagrams for 4 x 4 in. Columns
•
• •
'1 I I
112
1.1
B18(4 /
B34(4 x1/4"
[0.6
0.5
0.4
03 tD
'ea, 0.2
0.1 P= 15 Ton
1.0
0.9
0.8
0.7
0.9 rq 0.8
0.7
0.
0.
0.
0.
0.
0.
2.0 2 5 Deflection, 2D, in.
B17(4xY)
B15(4 x '8")
B2812 /4 II ) B14(4x1/4")
4"
...1 •
• •
• •
I" = 10 Ton
0.5 1.0 1.5
0.0 0 5 1.0 1.5 2.0 2.5 Deflection, 21), in.
Figure 3-16 Bending Moment-Deflection Diagrams for Columns with Different Steel Ratios
B26(S=3“)
B14(S7-.4
B24(S=1“)
4"
11
o 1/0 4
• 0 P = 10 Ton
05 1.0 15 20 2.5
30 Deflection, 2D, in.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
., 0.1
1.13
Figure 3-17 Bending Moment-Deflection Diagrams for Columns with Different Stirrup Spacing
, •-• -----„_, V=Olin/min)
B26
(V=10 in/mini_ B35
(W5in/min)K1
4"
..
-.7
• 1/4'•
• •
P = 10 Ton 1
08
b.,0 0.7
0.6
W 0.5
0.4
03
0.2
0.1
30 0.0 0 5 1 0 1 5 2.0 2.5 Deflection, 2 D, in.
Figure 3-18 Bending Moment-Deflection Diagram for Columns under Different Rates of Loading
114
Y = K bd211 IV1---0/
g .
g
O
o
x
X o / /
Y=15X2
'''''''' ,
.7.
g •
•
.: / 4
4 .i: 4 in. sections 0 4 x 6 in. ' I x 6 x 8 in. li v ,
• • z,„.../. ---- g....." g v
......,g 0.5 0.6 X = 1)/bdfi
C
10
9
8
7
b
5
4
3
2
1
0
01
02
03
04 0.7
Figure 3-19 Variation of the Hinge Rotational Stiffness with Axial. Load
!--;udi , n C rusnhil); Gradual C' rush tug -No t.ru51iLIg
A
12
6"
71
P = 50 Ton
-Section without Cover at ab
Bending Moment, Ton-in. M
A
Section without No Crushing
0.2
0.4 0.6 0.8 1.0 1.2 1.4 1.6 Curvature, 0 x 103, in. 1
Bending Moment, Ton-in. M
100
Cover at ah Gradual Crushing
Z5
= 20 Ton
1 2 3 4 i 5 1
Curvature, P x 103, in.
Figure 3-20 Falling Branch of the M-0-P Diagram
11
10
0.1 02 03 0-4 05 06 07 08. 9 1 Neutral axis depth ratio, n
Q
Figure 3-21 Bound Concrete Crushing Strain
6 x 8 in. sections (except Z4)
4 x 6 in. sections, 2h = 48 in.
4 x 6 in. sections, 2h = 60 in. & Z4
4 x 4 in. sections, 2h = 60 in.
ec=0.003[1+0.8q"+(1-n)/(0.5+n)+4d/((1+n)z)]
q". (1.4Ab/Ac-0.45)1A(So-S)/(A;S+0.0028BS2)]
5 6 3 7 -1 Curvature, ) x 10 , .
P 2D 1.6
01.2
r.T.:; 1.0
0.8 O
0.6
0.2
P)eflection, 21), in. 0.0
117
25
G"
0 0
Details as Fig. 3-G. P = 113 Ton
Figure 3-22 ]' ')-Pand F-D Diagrams for a Column Under 'Various Axial Loads
B18 40
30
50
K14 B15
20
K16
10 K3
0 0
F(
-------
Z1,Z2
Z3
Z4
Z5
2 3 - 4 5 6 7 0
0
b.0 160
a)
120
80
40
0 8
118
2 3 4 5 6 7 83 Ad= n(P )dx 10
Ad= n( - d x10'
Figure 3-23 M-Xd Variation (Lines do not have the same origin)
4"
/4 CD
Q
178
120
0 H
Ttin 80 0
40
119
For 4 x 4 in. & 4 x 6 in sections:
f = 7000 psi cn
• f = 18 Ton/in. sy
f" = 16 Ton/in. sy
P 160
10 20 30 40 Bending Moment, Ton-in.
6"
cn
Properties as Z3
M 0 40 80 120 160 0 20 40
Bending Moment Ton-in.
Fip.ure 3-24 M-P Interaction Diagrams at Ilse First. Crushing Stage for Various Sections
40
20 14
10
M 60
2
1
a.
4bf)
a)
a)
0.7 0.6 2/ bciP
01 02 0.3 0.4 05 Figure 3-25 'Variation of "A" with Axial Load
0 01 02 03 04 0.5 0.6 07 Neutral Axis Depth Ratio, n
i.
V.-
--r -v-- _Y ---.
o
+ 0
V .
4 x 4 in. sections
4 x G in. 6 x 8 in.
11
ti
0
x v
6 x 8 in. sections
4 x 6 in. sections -----
LP. =6.75nd ./'
/1
k8
V // Z4 /
(K7 I
----*-----i-- -- ___-------
Z1,Z2
------ Lp=2.5nd
7
,----- ------
1-..06 \ ),,--
V / K2I%
/ / I i
--A 5 % .
.7
."-- ------
.7
.----- '—'-----
Figure 3-20 Variation of L D Living Post-Crushing Stage
14
12
10
2
4
3
6
8
4
121
CHAPTER 4
COMPARISON OF ANALYTICAL
EXPERIMENTAL RESULTS
4.1 Introduction
In this chapter, a comparison is made between the analytical
solutions and the experimental results, and the conclusions drawn
are discussed. The results pertaining to the maximum bending
moment capacity of the sections are discussed first, and the
parameters affecting them are shown. This is followed by a com-
parison of the experimental force-deflection characteristics of the
columns with those obtained by the exact and the simplified methods
of solution. Finally, some of the results of Yamashiro's tests (14)
are analysed and compared with the procedure developed in this
report.
The procedure outlined in Chapter 3 for calculating the
bending moment resistance of a section proves too tedious in
practice, due to the complexity of the concrete stress-strain law.
Further on in the chapter, the equivalent rectangular stress-block
for the concrete in compression at a section is found and the corres-
ponding parameters are obtained. Using this procedure, the bending
moment capacity of a section can be found easily, and subsequently,
the force deflection characteristic of a column can be obtained without
any trouble.
4. 2 Maximum Bending Moment, "M tt max.
The measured experimental and the computed analytical
values of Mmax. for different sections are given in table 4-4. The
values corresponding to the analytical results are the bending moment
capacity of the sections at the first crushing stage, i. e. when the
strain of the concrete in compression reaches 0.0035.
The analytical results are usually less than the experimental
results, the agreement between the two groups becoming better as
122
the axial load on the section increases. For 4 x 4 in. sections, the
averages of the analytical results are nearly 88.1% and 101.6% of
the experimental results for the 5 - 6 tons and the 19 tons axial
load respectively. The same trend, more or less, is observed
for the 4 x 6 in. sections. The 'Z' tests show a better agreement
with the predicted results, the agreement being nearly 101% with
a standard deviation of 6. 5%.
The underestimation seen in these results may be attributed
to the following sources:
Strain-hardening of the steel has not been considered in the
computations except for the 'Z' tests. This is due to a lack of the
necessary information concerning the behaviour of the steel in these
tests. This effect, however, only influences the results of those
sections under lower axial load, in which the tension steel under-
goes a relatively high strain. The R values for the 'Z' sections,
without considering the strain-hardening effect of the steel, are
shown in column 6 of table 4-4; as seen, R for Z5 is increased by
8% approximately as a result of this effect. For the higher axial
loads, the influence is less and is never more than 2%. Assuming
a strain-hardening characteristic for steel, as shown in figure 3-4,
with properties of
esh = 3 e sy
e = 15 e su sy
f = 1.33 f su sy
the values of R for K16 and the 4 x 4 in. sections under 5 - 6 ton
axial loads are increased by 5.5% and 3% respectively. This
increase is negligible for the other axial loads.
It was thought that the overall underestimation of Mmax. may be due to the low evaluation of the bound concrete strength in
the stress-strain relationship. As the core concrete has relatively
123
small sizes in these sections, the bound effect may be higher.
However, the results obtained by increasing the concrete strength
did not provide a very favourable answer. The average values of
R for each group, when the concrete strength is raised by 20%, are
also shown in table 4-4. As seen, the R value for the sections
under low axial load (4 x 4 in. sections under 5 - 6 tons) is increased
by 3%, whereas in the sections under high axial loads (same sections
under 19 tons) the increase is as high as 14%. The relatively low
increase in R for the former sections is due to the fact that the
bending moment capacity in these sections is controlled mainly by
the steel resultant forces, and therefore, the concrete strength has
not much influence on it (compare B22 and B30). It is seen that the
increase in the concrete strength does not improve the results for
the low axial load cases, and makes the already good results for
the high axial load cases worse. Because of this, the under-
estimation of Mmax. cannot be attributed to this parameter very
much.
The other parameter which claims some influence on the
bending moment capacity of a section is the tensile strength of the
concrete. The tensile strength of concrete is about 10% of its
compression strength (46). Taking this strength into account and
assuming that, when the tensile stress in the concrete passes this
level the concrete cracks and its contribution to the strength of
the member becomes null, there would be very little increase in
max. although it increases the bending moment capacity during
the early stage of loading. In view of this, it was ignored in the
computations. The experiments by Ferry-Borges and Arga E
Lima (52) show that the concrete surrounding the tension reinforce-
ment provides a stiffening effect even when the strain in the re-
inforcement is several times that of the cracking strain of the
concrete. In other words, the effect of the concrete after it has
124
cracked cannot be entirely ignored as far as the load carrying
capacity of the member is concerned. On this basis, the stress-
strain relationship for concrete in tension is assumed to be linear-
elastic with a modulus of elasticity equal to its corresponding value
in compression, equation (3-13), and with a strength as recommended
by the ACI code of practice (54),
ft = 7.5fc c fi in psi
(4-1)
After cracking it is assumed that concrete tolerates a constant
stress of fl given by
ft = 0.25 f t t (4-2)
figure 4-1(b). This assumption may be a little unrealistic, but it
does help to account for the stiffening effect of the concrete after
cracking. A similar characteristic has been used by Cranston (53)
in his computations and the results show a closer correlation between
experimental and analytical values.
With the above assumptions, the bending moment capacity
of the section is increased, particularly when the axial load on the
section is not high. The values of R corresponding to the com-
putations with these assumptions are given in column 6 of the
table 4-4 for 4 x 4 in. and 4 x 6 in. sections. The increase in
R for K16, under the lowest axial load, and 4 x 4 in. sections under
5 - 6 ton loads are nearly 11% and 7%, whereas for the 4 x 4 in.
sections under 19 ton loads it is less than 1%. On this basis, not
only the total average of R is increased as compared with the R
values in column 5 of this table, but the results are more uniform
and lead to a smaller overall standard of deviation. The corres-
ponding results are given in the table.
The conclusion is that the main cause in the underestimation
of the analytical results is due to ignoring the tensile strength of the
concrete and, to a lesser extent, the steel strain-hardening.
125
However, for the medium range of axial loads, these effects do not
contribute more than 5 - 10% to Mmax.
4. 3 Force-Deflection Diagram, "Exact Solution"
The analytical and experimental diagrams for force-
deflection for the 4 x 4 in. and 4 x 6 in. columns are shown in
figures 4-2 to 4-18, and the bending moment-deflection diagrams
for the 6 x 8 in columns are shown in figures 4-25 and 4-26. For
each column, the maximum lateral force resistance Fmax , and
the deflection at the first crushing stage D1 are compared with the
analytical results in table 4-5. The experimental deflections
recorded in this table belong to the points on the F-D diagrams
corresponding to the maximum bending moment for the critical
sections, Mmax. These points are not very distinct on the diagrams,
particularly in the case of columns under low axial loads where the
gradient of the bending moment in the falling branch is very low.
For this reason, these deflections are subjected to some error.
The agreement between the experimental and analytical
results, as far as Frnax. is concerned, depends upon the accuracy
of the predicted Mmax. and the column's deflection at this stage.
In the previous section, it was shown that the predicted values of
M are generally lower than the experimental values. This max. obviously results in an underestimation of Fmax. However, since
the deflections in the exact solution are grossly underestimated, the
underestimation in Fmax. is, to a certain extent, compensated.
In certain cases Fmax. is even overestimated.
For the 4 x 4 in. sections under 5 - 6 ton axial loads, or
the 4 x 6 in. sections in which Mmax. was relatively low, Fmax. is on average around 90% and 92% of the actual values respectively.
For the former sections under higher axial loads, the agreement is
better, and for columns under 19 ton loads F is overestimated max. by an average of 11%.
Fmax. for 6 x 8 in. columns is grossly over-
126
estimated, 13% on average, due to a large underestimation in the
deflections. However, by accounting for the deflections due to
deformation of the base beam, as measured in the tests, the over-
estimation is reduced to 11%. It is seen that the lack of sufficient
accuracy in the evaluation of the deflections in the exact solution
leads to an overestimation of the lateral resistance of the column,
which is on the unsafe side as far as the design is concerned.
The analytical deflections at the first crushing stage D1
are generally much lower than the test results (column 8 of table
4-5 or column 10 for 6 x 8 in. sections). These deflections are in
fact the elastic components of the overall deflections.. The deflec-
tions due to plastic rotation at the hinges, the deformation of the
base beam, the shear deformation of the column, and finally, the
deformation of the test apparatus, have been ignored. However,
the ratios given in the table indicate the contributions of the elastic
deformations to the overall deformations.
For 4 x 4 in. sections, the average ratio varies between 59%
and 70% for low to high level axial loads; but, by taking into account
the poor agreement in Mmax. obtained for the low axial loads, these
ratios are found to be closer together and about 68%. In 4 x 6 in.
and 6 x 8 in. sections the average ratios are approximately 46% and
38% respectively. The short 4 x 6 in. columns have a smaller
average ratio than the long ones. This trend is not seen in the 4 x 4
in., columns.
Among the other components in the overall deformation
mentioned above, an estimate of the deflections due to the defor-
mation of the base beam can be made: this is considered in the
following section. With this component and the elastic component,
the relative contribution of the hinge rotation and the other com-
ponents are deduced from the overall deformation.
127
4.4 Rotation Due to the Base Beam Deformation
In the analytical computation, the colunin is assumed to be
rigidly fixed to the concrete base beam. This assumption is not
true since the beam, under the action of the axial load and the
bending moment in the column, deforms and produces a rotation
at the column's base. Because of this, all the analytical diagrams
are stiffer than the experimental diagrams during the early stage of
loading, figures 4-2 to 4-22. The deformation of the testing apparatus,
if any, also has a share in the discrepancy. In the 'Z' tests, an
attempt was made to measure this rotation by the simple device
explained in Chapter 2. The measured rotations are shown in
figure 2-8.
In order to calculate the deformation of the base beam in
other specimens, the beam is assumed to behave as a semi-infinite
elastic medium with its base fixed to the steel platform. The load
distribution on the beam due to the bending moment in the column is
assumed to vary linearly as shown in figure 4-23(c). This second
assumption is only reasonable during the early stage of loading before
the concrete cracks. After cracking, the distribution of the load
becomes more concentrated on the concrete compression side and in
the tension steel region of the column's section. More rotation is
expected at this stage than at the early stage of loading. The effect
of the axial load in the computation is ignored because it does not
contribute to the rotation of the base.
The vertical deformation of point D, under the triangular
load distribution shown in figure 4-23(a), can be proved to be (55)
] qa VD ITEa (a
2 - x2) log ad x + x2 d log -1-c, x --(4- 3) -
where d is the depth of the rigid boundary of the elastic medium,
E and 2) are the modulus of elasticity and Poisson's ratio of the
medium, and q and 'a' are the maximum load intensity and the
128
length over which the load is distributed.
The vertical deformation for a point c outside the loading
V - c nEa q [ 2 2 d + x
(a - x ) log - ., E — + a .--)
a+ x 7 n 2 log d - ] qa ( x
(4-4)
The variation of V for the loading conditicn shown in figure 4-23(c)
is given in figure 4-23(d); d and V are assumed to be
d = t = 2a
= O. 2
The average rotation -0 produced in this way is almost equal to
.0 E Ebt2 (4-5)
where M is the bending moment applied to a strip of width b.
The amount of 9 in this relationship can be shown to be independent
of d, the depth of the elastic medium. In this equation, E is
the modulus of elasticity for the concrete and its value depends upon
the stress level and, therefore, upon the axial load on the column.
With the assumption of a parabolic concrete stress-strain law with
an initial modulus of elasticity equal to that given in equation (3-13),
the E value is calculated for each column.
A line representing the equivalent deflection of -9 in the
above equation is shown in figures 4-25 and 4-26, line (1). This
line is very close to the initial slope of curve (2) which represents
the measured base rotation.
Equation (4-5) in terms of the lateral force and deflection
of the column may be written as
6 2 D -Fh (4-6)
Ebt or K F Ebt2
(4-7)
where h is the height of the column. When substituting for M in
region is
129
equation (4-5), the term corresponding to the axial load was ignored.
K is the initial stiffness of the base rotation characteristic.
In order to obtain an estimate of the change in the initial
stiffness of the column due to the base rotation, the experimental
and analytical values are compared in table 4-6. The values listed
in column 6 were calculated using the above equation. The experi-
mental values are not always sufficiently accurate because they were
measured from the force-deflection diagrams. This is particularly
true in the case of the 4 x 6 in. sections where a small error in the
measurement changes the results considerably.
However, the ratios given in column 5 of this table for the
4 x 4 in. sections show that the analytical results are, on average,
nearly 30% stiffer than the experimental ones. This lack of agree-
ment, shown by the figures given in column 8, is almost entirely due
to the base rotation. For low axial loads, the analytical results are
still stiffer even when the base rotation is included. This may be
due to an underestimation of the base rotation by the assumed load
distribution used when assessing 9 from equation (4-5) for this range
of axial load.
The results for 4 x 6 in. and 6 x 8 in. sections are not so
favourable as for the small sections. The analytical values which
include the base rotation contributions are still stiffer than the
experimental ones by an average amount of 28% and 63%. Considering
that, in the case of 6 x 8 in. sections, the measured base rotations
have been used in the computation, this lack of agreement should be
interpreted as the effects of the deformation of the testing apparatus
and the friction forces. In the case of the 4 x 6 in. sections, the
discrepancy may be caused by the deformation of the testing rig under
a heavy lateral force, and a possible underestimation of the base
rotation.
Finally, it should be mentioned that a part of this discrepancy
in the results may be due to the overestimation of the concrete's initial
130
modulus of elasticity given by equation (3-13). However, a small
variation in Ec does not alter the results very much. Figure 4-1(a)
shows the effect of reducing the value of Ec by one-third for the Z3
result. On the other hand, ignoring the tensile strength of the
concrete should have made the results more flexible.
By relying on the results of 4 x 4 in. sections, the con-
clusion is that equation (4-5) gives a reasonable estimate of the
base rotation during the early stage of loading.
The base rotation deviates from the straight line, represented
by equation (4-5), as the bending moment in the column increases.
The deviation is partly due to the change of E and partly due to the
change in the load distribution on the beam. As an example, the
deformation of the base beam at the first crushing stage for the
specimen Z3 is shown in figure 4-24. In this example, the concrete
stress distribution is replaced by its equivalent rectangular stress-
block, as shown in figure 4-24(c). It is also assumed that the steel
forces are distributed uniformly across the width of the section on
a strip with the same width as the reinforcement diameter. The
average rotation under this condition is
= 2. 16/E radians
This value is nearly 20% greater than the value predicted by equation
(4-5), at the first crushing stage, using the same value of E. By
considering the E corresponding to the stress level at this stage 2 2 (E = 1630 ton/in. for the axial load stress only, and E = 780 ton/in.
for the stress level at this stage), 9 will be almost 2.33 times
greater than that predicted by equation (4-5). In test Z3, the ratio
between the 0 measured at M to that given by the initial gradient max. of the diagram is nearly 2.5, figure 4-25. This shows the validity
of this example. This ratio varies between 2 and 3 for the other
tests in this group. On this basis, it is reasonable to assume that
6 Mmax. fi (At Mmax. ) = 2.5 Ebt2 (4-8)
131
the contribution of the base rotation to the overall deflection at the
first crushing stage is nearly 2.5 times greater than that predicted
by equation (4-5)
The values given in column 9 of table 4-5 are calculated
according to this relationship.
In the case of the 4 x 4 in. sections, the relative contribution
of the base rotation effect to the overall deflection varies between
the averages of 11% and 28%, depending upon the axial load. The
total contribution of the elastic deflection and the deflection due to
the base rotation is shown in column 11. The average total con-
tribution is 70% for low axial loads and 98% for high axial loads.
However, if the average total contribution is based on the exact
values of Mmax., the contributions will be between 78% and 97%.
Consequently, the plastic rotation contribution will be between 22%
for low axial loads and 3% for high axial loads.
In the case of 4 x 6 and 6 x 8 in. sections, the total contri-
bution of the elastic and base rotation deflections is found to be
between 74% and 60% of the total deflection after allowing for the
difference between the actual and exact values of M The max.
remaining part of the deflection is due to the plastic deformation at
the hinge and the deformation of the testing apparatus. Unfortunately,
these two components cannot be separated, and therefore no definite
conclusion can be drawn with regard to the relative magnitude of
the plastic rotation.
4.5 Simplified Solution
The diagrams corresponding to this method of solution are
shown in figures 4-2 to 4-22. A summary of the maximum lateral
resistance of the columns Fmax., the deflections D1 at the first
crushing stage and the maximum deflections D2 corresponding to
F = 0.0, are given in table 4-7. These figures show the difference
132
between the result S obtained by the two methods of solution discussed
in Chapter 3. The simplified method predicts larger deflections
for the columns than the other method after the cracking stage.
It can be seen that if the initial stiffness of the analytical and
experimental results are the same, the exact solution results are
stiffer than the actual ones; whereas those obtained by the other
method are closer to the actual ones and are sometimes more
flexible. However, the difference becomes more apparent at the
post-yield stage when the effect of the hinge rotation is considered
in the simplified method - compare the points corresponding to
the first crushing stage on the three diagrams in each figure.
The estimation of Fmax. by this solution is more realistic
than by the exact solution. The agreement with the experiments is
similar to the agreement of the maximum bending moment capacity
given in table 4-4 - compare the average ratios in column 5 with
the average values of R in table 4-4 for each group. As the agree-
ment is slightly less than the M agreement, it indicates an max. overestimation in the deflections. However, the differences are
very small. The conclusion is that, with a good prediction for
, the lateral force resistance can be predicted reasonably max. close to the actual force.
The deflections at the first crushing stage D1 are, on
average, generally in good agreement with the experimental ones
for the 4 x 4 in. sections, although the variation in them is rather
high and sometimes differs from the average by 20%. In the 4 x 6 in
sections, the deflections are overestimated for the low axial loads
and underestimated for the high axial loads. Though the under-
estimation for 1(13 and K18 is due to the poor agreement in Mmax. In the 6 x 8 in. sections, despite the fact that the deflections due to
the base rotations are taken into consideration, the values d D1 are lower than the actual ones.
133
It must be noted that the rotation of the base is not included
in the calculation of the deflections at this stage, and in spite of this,
the agreements are generally good. Therefore, by including the
base rotation, overestimation of the deflections occurs. This over-
estimation is due to the overestimation of the hinge length used in
the computation at this stage, equation (3-47). As previously
mentioned on page 31, this equation corresponds to the hinge
rotation at the point on the falling branch of the M-P-P diagram
where M = 0.95 M max.
Therefore, by using it at the Mmax. stage, it overestimates the
hinge rotation. It is more reasonable to use a coefficient of 2
instead of 2.5 in this expression for the hinge rotation at this stage.
In most cases, the falling branches of the experimental and analytical
F-D diagrams are seen to be almost parallel. Full coincidence is
only achieved if the point corresponding to the first crushing stage
can be predicted accurately. This means that the agreement will
be better when Mmax. and D1 are in good agreement with the test
results. In some of the figures, the diagram corresponding to the
case in which the concrete tensile strength has been considered is
shown. As seen, the results are in better agreement with the tests
due to an improvement in Mmax. As a means for comparison, the maximum deflections D2
are compared with the experimental values in table 4-7, column 11.
In order to cancel the effect of the inadequacy of the Mmax. values,
the —R values are divided by R from table 4-4, colum 5. This
normalizes the maximum bending moments given by the experiments
and analysis. The Rill values are listed in column 12 of the table.
The deflections for the columns under low axial load are
underestimated by nearly 7% on average. The reasons were given
while discussing the parameters affecting the value of "A", page 98.
134
However, it is mainly due to the underestimation of the crushing
strain for the bound concrete. Figure 4-3 shows the effect of
increasing this strain and figures 4-4 and 4-14 show the effect of
steel strain-hardening. For the 4 x 4 in. columns under 14 - 15
ton axial loads where the "A" value was generally lower than the
average value, table 4-2, the deflections are overestimated by
nearly 7% on average and have a standard deviation of 5.8%. The
overall average of the kill ratios is nearly 100% for columns of
this size and about 101% for the 4 x 6 in. and 6 x 8 in. columns.
The standard of deviations of these ratios is about 8 - 10%, as
given in the table. The reasonably good agreement obtained in
the prediction of the maximum deflections, D2, shows the validity
of the expression derived for the hinge length at the post-crushing
stage.
4. 6 Comparison of Other Tests
In Chapter 1 it was mentioned that a similar investigation
on the force-deflection characteristic of the beam-column has been
carried out by Yamashiro and Siess (14), and a summary of their
work was given. In this section, the results of some of their
experiments are compared with the results of the analytical approach
developed in the present investigation.
Figure 4-27 shows diagramatically the specimen and the
testing procedure. In figures 4-28 to 4-31 the experimental bending
moment M at the critical section is plotted against the equivalent
deflection , shown in figure 4-27, for five specimens under
different axial loads. The other two curves in each figure
represent the analytical results. The broken line represents
the deformation of the beam- column only and does not include
the stub deformation. The latter has been calculated by Yama-
shiro for the first crushing and ultimate stages. The other curve
represents the total deformation obtained by adding his calculated
135
values to the deformation of the beam-column.
The agreement between the bending moment capacity of the
sections at the first crushing stage is generally good for all of them.
In the post-crushing stage, there is an underestimation in the
analytical results. For columns J31, J26 and J25, a rise in the
bending moment after the first crushing is not seen in the analytical
results. The rise is due to steel strain-hardening, and the disagree-
ment indicates an underestimation of the tension steel strain in the
analysis. This is the inadequacy of the strain compatibility con-
dition. It could also be due to the shortcoming of the concrete
stress-strain law in its falling branch, Should it underestimate
the ductility of the bound concrete at this stage, the concrete deterior-
ation would precede the steel strain-hardening, and therefore, no
significant rise in the bending moment would be seen due to this
effect.
In J34 the sudden fall in the bending moment immediately
after crushing is due to a sudden spalling. The two diagrams con-
verge as the theoretical crushing proceeds in the analytical approach.
The deflections are generally in good agreement at all stages
of loading. The agreement in the rising branch of the diagrams is
quite good.
4.7 Equivalent Rectangular Stress-Block
The method outlined in Chapter 3 for evaluating the bending
moment resistance of a section is far too tedious to be used in
engineering practice because of the complexity of the stress-strain
relationship of the concrete in compression. A method, which has
been successfully applied in practice, is to replace the compression
concrete stress distribution at a section by an equivalent rectangular
stress distribution, and find the properties of this stress-block in
terms of the parameters involved in the stress-strain relationship.
Referring to figure 4-32(a), the rectangular stress-block
properties are determined so that the resultant force and the bending
136
moment it produces are equivalent to the actual stress-block. On
this basis, the parameters which should be determined are the
average stress distribution oc.fel, and the lever arm of its resultant
force from the compression face of the section, a: dn. With a
concrete stress- strain relationship of
fc = f(ec)
dn
f(i-c • e ) clx co
=
(4-9) f' . dn
do x . f (I. • e co) dx
r= oe. f' . d 2 n
where eco
is the strain in the outermost fibre of the concrete.
If f(ec) is a polynomial in ec, oe and r will be independent
of do and their main variable will be e co. In the case of f(ec)
being defined by equation (3-11), oe and I are functions of fel, dn,
pn f" and S/bi as well as e co . The most important of these para- meters are 1' and 0" , and for an approximate evaluation of 0‹ and sy r , the other two parameters can be assumed to be constant. In
the curves shown in figure 4-32 these two parameters are taken as
Sibi= 1.0
dn = 4.0 in.
f" y = 16.0 ton/in. s However, in order to show the variation of o and I with
dn, these parameters are given for various values of do in figure
4-33.
In applying these constants, it should be noted that they
have been calculated for the bound concrete, and using them for
the whole section results in an overestimation of the bending moment
and
137
in the post-crushing stage. As the cover in the sections normally
used in practice is only a small proportion of the whole section,
this overestimation should not be much.
Having determined 0( and r , the M-P-P diagram can be
found easily, and by applying the simplified method outlined in
Chapter 3, the lateral force-deflection diagram of the column is
obtained. In practice, the evaluation of a few points before the
yielding stage and the values corresponding to the yielding, first
crushing, and the ultimate stages, is sufficient. The part of the
F-D diagram between the first crushing and the ultimate stage can
be assumed to be linear.
At the yielding stage, the tension steel strain is fixed, eco
is assumed, and oe and I are found. Then, by trial and error, eco
is adjusted and M is evaluated. At the first crushing stage, eco is assumed to be 0.0035, and thus oe and I are readily found. At
the ultimate stage, the strain in the outermost fibre of the com-
pression side of the concrete core should be estimated, i, e. the
bound concrete crushing strain. As this strain is a function of n,
equation (3-60), it is assumed and o(, ( , and do are found. Then
the assumed strain is corrected, and the process is repeated. To
begin with, the strain corresponding to the value of n at the first
crushing stage can be tried, since according to figure 3-26, n does
not change very much during the post-crushing stage.
Table 4-1 Columns' Properties
4 x 4 in. and 4 x 6 in. columns
No.
Cross-
Section
in. x in.
Height
(2 h)
in.
Axial
Load
Tons
Concrete
4 in. cube
Strength
psi
Long.
Reinforce-
ment
in.
Yield
Strength
f sy
Ton/in2
Rate
- of
Loading
in. /min.
Remarks
B14 4x4 60 10 7280 4 x 1/4 24.0 0.1
B15 it n
10 . 7380 4 x 3/8 17.5 0.1
B16 ff II
15 6370 4 x 3/8 18.0 0.2
B17 ft II
10 6650 4 x 1/2 17.5 0.2
B18 It ft
15 7100 4 x 1/2 17.0 0.2
B19 It It
19 7400 4 x 1/2 18.0 0.2
B20 It 71 .
12.5 7560 4 x 1/4 16.0 0.2
B21 II tt
19 5110 4 x 1/4 17.0 0.2
B22 it II
5 6930 4 x 1/8 18.0 0.1
B23 II II
15 6580 4 x 1/8 17.5 0.1
• B24 tt If
10 6880 4 x 1/4 17.0 0.1 Spacing = 1 in.
B26 it II
10 7000 4 x 1/4 18.0 0.1 Spacing = 2 in.
B28 II It
10 7200 2 x 1/4 17.0 0.1
B30 ft . II
5 7800 4 x 1/2 17.5 0.1
B31 ft II
5 7430 4 x 3/8 17.0 0.1
B32 IT it
19 6850 4 x 3/8 17.5 0.1
B33 ft 11
19 6590 4 x 1/4 18.0 0.1.
B34 11 II
15 7350 4 x 1/4 17.5 0.1
B35 it It
10 6850 4 x 1/4 17.0 10
Lateral reinforcements mall sections are 1/8" @ 4" except in B24 and B26 -
Yield level of the lateral reinforcements is assumed to be = 16.0 Ton/in2.
Concrete cover in all sections is 0.75 in. thick.
Table 4-1 (continued)
4 x 4 in. and 4 x 6 in. columns
No.
Cross-
Section
in. x in.
Height
(2 h)
in.
Axial
Load
Tons
Concrete
4 in. cube
Strength
psi
Long.
Reinforce-
ment
in.
Yield
St sength
f sy
Ton/in2.
Rate
of
Loading
in. /min.
Remarks
K1
K2 4x4
4x1.3 60 n
10
10 6720
7390 4 x 1/4
4 x 1/4 22.5
30.0
C•si 0 0 0 L
O
If) L
O If) L
C) 1
0 L
O
LO L
O L
O L
O L
O If)
VD
• •
•
• •
•
•
• •
•
•
•
•
•
•
•
•
CD
C
D
0 C
D
CD
• K3 4x4 II
6 7400 4 x 1/4 24.2
K5 ft n
14 6940 4 x 1/4 22.8
K6 n n
14 7610 6 x 1/4 28.9
K7 4x6 n
19 7500 4 x 1/4 22.6
K8 n n
14 8670 4 x 1/4 22.4
K9 n It
10 7390 4 x 1/4 22.3
Kll 4x4 n
14 7620 4 x 1/4 19.1 Concrete cover = 0.125 in.
K13 4x6 48 19 7840 4 x 1/4 19.3
K14 n ft
14 6720 4 x 1/4 20.1
1(15 n n
10 7080 4 x 1/4 17.5
K16 n n
6 6500 4 x 1/4 21.5
K17 II II
19 6750 4 x 3/8 14.4
K18 II It
19 7400 4 x 3/8 14.9
K19 4x4 n
19 6720 4 x 3/8 14.8
K20 n n
10 7500 4 x 3/8 15.2
K21 n n
14 6580 4 x 1/4 19.8
K22 n n
10 6270 4 x 1/4 , 20.1
K23 n n
6 5910 4 x 1/4 19.
Lateral reinforcements in all sections are 1/8 @ 4" . 2
Yield level of the lateral reinforcements is assumed to be = 16.0 Ton/in.
Concrete cover in all sections is 0.75 in. thick, excpet for K11.
Table 4-1 (continued)
6 x 8 x 45 in. columns
No. Axial Load
Tons
Concrete 4 in.cube Strength
psi
Longitudinal Reinforcement Lat. Reinforcement Rate
of Loadino. b
in./min.
Size
in.
Yield Level f sy 2
Ton/in.
Ultimate Strength
f su 2
Ton/in.
e sy e sh e su
Size
in.
Yield Strength r sy 2 Ton/in.
Z1
Z2
Z3
Z4
Z5
50
50
38
30
20
7930
7950
6450
6600
6800
4x1/2
4x1/2
4x3/8
4x3/8
4x3/8
18, 1
18.1
22.0
22.0
22.0
27.5
27.5
34.0
34.0
34.0
1.35x10
1.35
1.6,1
1. 64
1.64
3 6, 75X10
3
6. 75
3.28
3.28
3.28
54x10 3
54
41
41
41
1/4 @ 4
1/4 @4
1/4 @4
1/4 @6
1/4 @4
20.0
20.0
18.8
18.8
18.8
0. 2 If
II
ti
II
Concrete cover in all sections is 0, 75 in. thick.
Table 4-2 Hinge Rotational Properties
4 x 4 in. sections
1 2 3 4 5 6 7 8 9 10
Axial Long. Height Initial K K
No. Load Steel 2 h a Stiffness M--G
with M--0
without K M -p
KM-, A - K S unloading unloading 11/1--9- (Ton) (in. ) Ton/in. Ton/in. Ton-in. Ton-in. Ton-in.
B22 5 4x1/8" 60 -0.21 3.0 -38.0 - -39 -0.55x103 14.1 B30 11 4x1/2 11. 0.24 3.5 64.6 . 66 0.65 9.8 B31 " 4x3/8 11 0.23 3.6 56 0 57 0.57 10.0 K3 6 4x1/4 II 0.26 3.6 53.0 54 0.64 11.8 K23 H IT 48 0.32 6.8 39.6 . 40 0.60 15.0
average -50.2 - -51.2 - 12.1
B14 10 4x1/4" 60 -0.44 3.6 -93 -96 -1.64x103 17.0 B15 4x3/8 0.53 3.2 168 177 1.68 9.5 B17 H 4x1/2 H 0.56 4.0 194 204 1.75 8.6 B24 H 4x1/4 H 0.52 3.2 160 .168 - 1,11 6.6 B26 H H II 0.44 3.2 93 , 96 1.40 14.5 B28 if 2x1/4 H 0.57 3.2 200 '213 1.63 7.7 B35 " 4x1/4 H 0.52 3.3 160 . 168 1.67 9. 9 K1 H If If 0.52 4.2 161 '168 1.68 10. 0 K20 H 4x3/8 48 0.62 7.2 114 117 1.68 14. 4 K22 II 4x1 /4 It 0.68 6.8 147 152 1.71 11.3
average -149 -155.9 - 10.95
The average of "Au without B14, B26 and K20 is 9.1 .
Table 4-2 (continued)
4 x 4 in. sections
1 2 3 4 5 6 7 8 9 10
No.
Axial Load
(Ton)
Long. Steel
Height 2 h
(in. )
a
Ton/in.
Initial Stiffness
S Ton/in.
K l2_.9 with
unloading Ton-in.
KM-43 without unloading Ton-in.
K m_p Ton-in.
K 111-0 A - K 1'I-0
B20 12.5 4x1/4" 60 -0.92 3.2 -398 -453 -3.1x103 6.83 B16 15.0 4x3/8 II 1.36 3..6 640 774 5.3 6.84 B18 II 4x1/2 It 1.28 3.4 585 702 5.1 7.26 B23 " 4x1/8 It 1,15 2.8 489 585 4.5 7.69 B34 " 4X1/4 If 1. 00 3.0 394 450 4.5 10.00 K5 14.0 It It 1.00 4.0 429 480 4.6 9.60 K6 K11
II
" 6x1/4 4x1/4
II it
1.16 0.76
4.0 4.2
540 248
624 264
4.4 1.9
7.05 7.20
K21 It It 48 1.54 6.8 488 551 4.4 8.00
average -495 -577 - 7.83
The average on KM-.g does not include K11 .
B19 B21
19.0 tt
4x1/2" 4x1/4
60 -1.68 1.68
3.6 2.3
-755 694
-942 942
-8.0x103 7.5
8.50 8.00
B32 tt 4x3/8 II 1.32 2.6 510 618 8.1 13.10 B33 " 4x1/4 II 1. 60 2.8_ 679 870 8.2 9.40 K19 " 4x3/8 48 2.34 6.4 734 892. 7.7 8.60
average -674 -853 - 9.52
Table 4-2 (continued)
4 x 6 in. sections
1 2 3 4 5 6 7 8 9 10 11
No.
Axial Load
(Ton)
Long. Steel
Height 2 h
(in. )
a
Ton/ini
Initial Stiffness
S
Ton/in.
K 2-$ with
unloading Ton-in.
M- K .E)
without unloading Ton-in.
K1I-9)
Ton-in.
K111-0 A - Average IT Km-a
K16 6 4x1/4" 48 -0.34 15.0 -51 -52 0.66x103 12.7 12.70
K2 10 n GO 0.51 10.0 156 159 1.42 8.9
K9 11 II n 0.50 9.6 147 150 I, 46 9.7 9.70
K15 n n 48 0.64 20.0 127 ' 129 1.35 10.5
9.55 K8 14 It 60 0.76 10.0 257 264 2.91 11.0
K14 H n 48 1.20 16.0 342 355 2.88 8.1 K7 19 H 60 1.48 8.8 699 762 5.50 7.2
K13 il n 48 1.80 19.2 553 581 5.40 9.3 7.80 K17 n 4x3/8 H 3.24 18.0 1248 1410 CANCELLED
K18 It II II 2.20 20.0 760 811 5.70 7.0
Average 9.38
6 x 8 in. sections
Z5 20 4x3/8" h=45 in -0.58 7.10 -275 -280 -2.8x103 10.0
Z4 30 II II 1.14 7.42 898 950 8.6 9.0
Z3 38 II II 1.72 6.26 1584 1780 10.4 5.8
Z2 50 4x1/2 n 1.93 7.26 1512 1660 18.3 11.0
Z1 50 11 11 2.11 7.26 1809 2025 18.3 9.0
Average 8.96
The overall average of "A" over all the sections is 9.70 . With the assumption of A = 9 the coefficient of variation is 0.30 .
144
Table 4-3 Crushing Strain in Bound Concrete
Equation (3-60)
Cross-section Axial Load T on
Strain Remarks
5-6 O. 0080
10 O. 0060 In B24,e= 0.'010
4x4 in. In B26,ecc= 0. 007
14-15 O. 0055
19 O. 0045
6 0. 0090
10 0. 0075 In K15, ec= 0.008 4x6 in. 14 O. 0070
19 0. 0065
20 O. 0085
30 0. 0065 Spacing = 6 in. 6x8 in . 38 0. 0065
50 0. 0060
145
Table 4-4 Maximum Bending Moment Capacity of Sections
4 x 4 in. sections
1 2 3 4 5 6 1
Axial Experi.- Analytical R=(4)/(3) R with No. Load mental C. T. S.*
considered Ton Ton-in. Ton-in. % %
B22 5 10.5 10.0 95.2 104 B30 11 28.5 24.3 85.3 90 B31 11 20.7 17.9 86.5 .92 K3 6 18.0 16.1 89.5 97 K23 ii 17.0 14.3 84.1 91
Average 88.1 94.8 Standard Deviation (4) (5.2)
Average of R by increase of 20% in fcu is 90. 8% .
B14 B15 B17 B24
10 11 ti it
20.5 25.0 32.4 20.4
20.5 23.5 28.5 18.8
100.0 94.0 88.0 92.0
105 98 91 97
B26 11 21.5 19.0 88.4 93 B28 II 18.0 17.0 94.4 101 B35 II 21.0 18.7 89.0 94 IC1 I, 21.0 19.7 93.8 99 1(20 II 24.0 22.5 93.8 98 K22 It 22.0 19.0 86.0 90
Average 92 96.6 Standard Deviation (3. 9) (4. 4)
Average of R by increase of 20% in fcu is 95. 7% .
* Concrete Tensile Strength
Table 4-4 (continued)
4 x 4 in. sections
1 2 3 4 5 6
Axial Experi- Analytical R...(4)/ (3) R. with No. Load mental C. T. S. *
considered Ton Ton-in. Ton-in. % %
B20 12.5 23.5 21.3 90.6 95 B16 15.0 26.0 26.2 100.7 103 B18 II 32.3 32.3 100.0 102 B23 II 18.2 19.2 105.5 108 B34 il 24.3 23.3 95.9 99 K5 14.0 24.0 23.3 97.0 100 KG 11 28.0 28.4 101.4 104 K11 fl 28.5 25.8 90.5 93 K21 fl 22.1 22.1 100.0 103
Average 98 100.8 Standard Deviation (4. 7) (4.4)
Average of R. by increase of 20% in feu is 104% .
B19 1321 B32 1333 K19
19.0 it it 11 1,
32.0 16.8 27.0 23.3 26.5
35.0 16.6 27.8 22.8 26.2
109.4 98.8
103.0 97.9 98.9
111 100 103
98 99
Average 101.6 102 Standard Deviation (4. 3) (4. 8)
Average of R by increase of 20% in feu is 115. 6%.
Total average of R for 4 x 4 in. sections is 94.8% with S.D. of 6.5%.
With consideration of C. T. S. the total average is 98.6% with S. D. of 3.87%.
G
*Concrete Tensile Strength
147
Table 4-4 (continued)
4 x 6 in. sections
1 2 3 4 5 6
Axial Experi- Analytical n=(4)/ (3) R with No. Load mental C. T. S. *
considered Ton Ton-in. Ton-in.
K16 6.0 29.0 25.3 87.2 98 K2 10.0 48.0 37.8 78.8 86 K9 I! 37.5 34.6 92.2 100 1(15 II 36.0 32.5 90.2 99 K8 14.0 48.5 4?.. 5 89.7 97
. K14 it 44.0 40.2 91.4 97 K7 19.0 55.0 49.6 90.0 " 95 1(13 it 55.2 49.0 88.8 93 K17 il 55.0 51.5 93.6 97 K18 11 67.5 55.1 81.6 85
Average 88.4 94.7 Standard Deviation (4. 4) ( 5 )
Average of R by increase of 20% in fcu is 92% .
6 x 8 in. sections
R with no strain
hardening in steel
Z5 20.0 95.5 103.0 107.8 100.0 Z4 30.0 132.0 120.7 91.4 90.0 Z3 38.0 122.0 130.4 106.9 106.0 Z2 50.0 178.0 169.2 95.0 95.0 Z1 50.0 164.0 169.2 103.0 103.0
Average 100.8 98.8 Standard Deviation (6.5) (5.7)
Total average of R for all sections is 94% with S. D. of 7. 3%.
With consideration of C. T. S. the total average is 98.1% with • S. D. of 3.4%.
*Concrete Tensile Strength
Table 4-5 Comparison of Experimental & Analytical f:esults
"Exact Solution"?
4 x 4 in. sections
1 2 3 4 5 6 1 7 8 1 9 HO 1 n No. Axial
Load
. (Ton)
Max. Lateral Force Deflection at the First Crushing Stage, Di
Test
(Ton)
Analytical
(Ton)
(4) (3) %
Test
(in. )
Analytical
(in. )
(2. ) (6) % 0-
Base Rotation
(in. )
(9) (6) %
. ( 7 H9 ) ( 6 )
of 0
B22 B30 B31 K3 K23
5 IT
tt 6 II
0.30 0.83 0.60 0.50 0.65
0.28 0.75 0.54 0.46 0.54
93 90 90 92 83
0.45 0.70 0.65 0.58 0.30
0.25 0.38 0.35 0.35 0.20
56 54 54 60 67
0.036 0.085 0.063 0.057 0.045
8 12 10 10 15
64 66 64 70 82
Average 89.6 58.2 11.0 69.2
B14 B15
10 IT
0.57 0.66
0.56 0.67
98 102
0.45 0.55
0.35 0.33
73 60
0.080 0.090
18 16
96 76
B17 tt 0.86 0.83 96 0.70 0.35 50 0.120 17 67 B24 it 0.53 0.52 98 0.50 0.31 62 0.077 15 77 B26 it 0.54 0.53 98 0.55 0.31 56 0.077 14 70 B28 tt 0.45 0.47 1u4 0.52 0.30 58 0.066 13 71 B35 II 0.54 0.52 96 0.54 0.32 59 0.076 14 73 K1 ft 0.56 0.55 98 0.50 0.33 66 0.080 17 83 K20 K22
fl
fl 0.84 0.80
0.85 0.70
101 88
0.33 0.33
0.19 0.21
58 64
0.070 0.062
21 19
79 83
Average 97.9 61.1 16.4 77.5
Table 4-5 (continued)
4 x 4 in. sections
1 2 3 4 5 6_J 7 Deaection at
8 the First
9 Crushing
10 i 11 Stage, D/ Max. Lateral Force
No. Axial Test Analytical (4) - Test Analytical 7) (- Base (9) (7 )-L( 9) Load (3) (6) Rotation (6) ( 6 ) (Ton) (Ton) (Ton) % (in) ) (in. ) /0of (in. ) al /0 %
B20 12.5 0.59 0.58 98 0.49 0.31 63 0.083 17 80 B16 15.0 0.65 0.70 108 0.47 0.35 74 0,120 25 99 B18 I, 0.60 0.90 112 0.58 0.36 62 0.140 24 86 B23 Tr •0.43 0.50 116 0.38 0.28 74 0.086 23 97 B34 it 0.59 0.62 105 0.51 0.32 63 0.100 20 83 K5 14.0 0.62 0.61 98 • 0.42 0.35 83 0.100 24 107 K6 1, 0.70 0.75 107 0.57 0.40 70 0.110 19 89 Kll fl 0.76 0.72 95 0.45 0.29 64 0.100 22 86 K21 f1 0.76 0.80 105 0.31 0.19 61 0.080 26 87
Average 104.9 68.2 22.2 90.4
B19 B21 B32 B33 K19
19.0 li t I
rt 11
0.77 0.38 0.58 0.50 0.90
0.93 0.38 0,71 0.56 0,92
121 100 122 112 102
0.50 0.32 0.57 0.49 0.29
0.38 0.26 0.34 0.30 0.21
76 81 60 61 72
0.150 0.100 0.127 0.110 0.100
30 31 22 22 34
106 112
82 83
106
Average 111.4 70.0 27.80 - 97.8
Table 4-5 (continued)
4 x 6 in. sections
1 ., 2 3 4 5 6 7 8 9 1 10 11
No. Axial
• Load
(Ton)
Max. Lateral Force Deflection at the First Crushing Stage, D
Test
(Ton)
Analytical
(Ton)
(-4) (3)
%
Test
(in)
Analytical
(in)
(7) (6)
of /0
Base Rotation
(in)
(9) (6) C
JO
( )±( 9 ) (6)
C'' 10
K16 K2 K9 K15 K8 K14 K7 K13 K17K18•
6 10
ti 11
14 II
19 it ft
"
1.15 1.50 1.15 1,43 1.46 1.67 1.61 2.07 2,08 2.54
1.02 1.18 1.08 1.30 1.35 1.60 1.52 1.94 2.05 2.18
89 79 94 91 92 96 94 94 99 86
0.25 0.40 0.37 0.25 0.40 0.28 0.38 0.30 0.32 0.35
0.12 0.23 0.19 0.12 0.21 0.12 0.19 ' 0.12 0.11 0.12
48 58 51 . 48 53 43 50 40 34 34
0.04 0.06 0.06 0.04 0.07 0.06 0.08 0.07 0.08 0.07
16 15 16 16 18 21 21 23 25 20
64 73 67 64 71 64 71 63 59 54
Average 91.4 45.9 19.1 65.0
Table 4-5 (continued)
6 x 8 in. sections
1 2 3 4 1 5 I 6 7 8 9 10 11 12
No. Axial Load
(Ton)
Max. Lateral Force Deflection at the First Crushing Stage, D1
Test
(Ton)
Analytical
(Ton)
(4) (-3-) %
(4)+* Base
Rotation (Ton)
(6) (3) %
Test
(in. )
Analytical
(in. )
(9) (8) %
(9)4-* Base
Rotation (in. )
(11) (8)
Z5 Z4 Z3 Z2 Z1
20 30 38 50 50
1.83 2.42 2.06 2.76 2.72
2.07 2.44 2.62 3.41 3.41
113 101 127 124 125
1.97 2.36 2.43 3.17 3.17
.108 93
118 115 116
1.00 0.89 0.85 1.15 0.89
0.47 0.35 0.33 0.32 0.32
47 39 39 28 36
0.71 0.48 0.56 0.53 0.53
71 54 66 46 60
Average 118.0 111.0 37.8
59.40
* Base rotations in this group are those measured.
' Table 4-6 Comparison of Column Initial Stiffness
4 x 4 in, section
1 2 3 4 5 6 7 8
No. Axial Test Analytical (4) - Base Analytical (7) Load (3) Rotation
effect + B.R. effect
(3)
Ton Ton/in, Ton/in. Ton/in, Ton/in.
B22 5 3.0 4.60 1.53 23.0 3.8 1.27 B30 ii 3.5 4.70 1.34 23.7 3.9 1.11 P31 It 3.6 4.80 1,33 23,7 4.0 1.11 K3 6 3.6 4.70 1.31 23.6 3.9 1.08 K23 if 6.8 8.30 1.22 32.9 6.6 0. 7
152
Average 1.35
B14 10 3.6 4.6 1.27 21.3 3.8 1.06 B15 II 3.2 4.9 1.53 21.3 4.0 1.25 B17 it 4.0 5.0 1.25 20.4 4.0 1.00 B24 II 3.2 4.4 . 1.38 20.4 3.6 1.13 B26 H 3.2 4.4 1.38 21.3 3.6 1.13 B28 H 3.2 4.4 1.38 21.3 3.6 1.13 B35 it 3,3 4,3 1.30 20.4 3,6 1.09 K1 H 4.2 4.3 1.02 20.4 3.6 0.85 K20 ti 7.2 9.8 1.36 35.0 7.6 1.06 K22 11 6.8 8.2 1.21 32.0 6.5 0.95
Average 1.31 1.06
Average 1.22 1.00
Table 4-6 (continued)
4 x 4 in. sections
1 2 3 4 5 6 7 8
No. Axial Test Analytical (1) Base Analytical (7) Load (3) Rotation
effect +B. R. effect
(3)
Ton Ton/in. Ton/in. Ton/in. Tbn/in.
B20 12.5 3.2 4.5 1.41 21.3 3.7 1.16 B16 15 3.6 4.1 1.14 17.3 3.3 0.91 B18 fl 3.4 4.9 1.44 20.0 3.9 1.15 B23 1, 2.8 3.4 1.21 18.5 2.9 1.04 B34 11 3.0 4.0 1.33 20.0 3.3 1.10 K5 . 14 4.0 4.2 1.05 20.0 3.5 0.88 K6 fl 4.0 4.5 1.13 21.0 3.7 0.93 K11 n 4.2 4.8 1.14 21.0 3.9 0.93 K21 it 6.8 7.8 1.15 29.6 6.2 0.91
153
B19 B21 B32 1333 K19
19 II
II il 11
3.6 2.3 2.6 2.8 6.4
4.8 2.4 4.0 3.3 8.0
1.32 1.04 1.54 1.18 1.25
19.3 13.0 18.3 17.0 26.5
3.8 2.0 3.3 2.8 6.2
1.06 0.87 1.27 1.00 0.97
Average 1.27 1.03
Average 1.77 1.28
Table 4-6 (continued)
4 x 6 in. sections
1 2 3 5 6 7 8
No. Axial Load Test Analytical (4) - (3)
Base Rotation
Analytical +B. R.
(7) (3)
effect effect Ton Ton/in. Ton/in. Ton/in. Ton/in.
K16 6 15.0 28.0 1.87 78.3 20.6 1.37 K2 10 10.0 16,8 1.68 50.4 12.6 1.26 K9 ii 9.6 16.8 1.75 50.4 12.6 1.31 K15 ri 20.0 32.8 1.64 78.8 23.2 1.16 K8 14 10.0 18.2 1.82 52.5 13.5 1.35 K14. II 16.0 30.8 1.93 72.5 21.6 1.35 K7 19 8.8 16.0 1.82 48.5 12.0 1.36 1(13 11 19.2 32.8 1.71 '7 5.8 22.9 1.19 1(17 It 18.0 32.6 1.81 08.3 22.0 1.23 K18 II 20.0 34.0 1.70 78.8 23.8 1.19
154
6 x 8 in. sections
Z5 20 7.10 16.5 2.32 34.9* 11.2 1.57 Z4 30 7.42 16.0 2.16 35.2 11.0 1.48 Z3 38 6.26 15.0 2.40 32.9 10.3 1.65 Z2 50 7.26 18.3 2.52 39.4 12.5 1.72 Z1 50 7.26 18.3 2.52 39.4 12.5 1.72
Average 2.38 1.63
* Base rotations in this group are those measured .
Table 4-7 Comparison of Experimental & Analytical Results
"Simplified Solution"
4 x 4 in. sections
1 2 3 4 5 6 7 8 9 1 10 I 11 1 12
Max. Lateral Force Def. at 1st Crushing, Di. Deflection at F = 0, D2 Axial (4) - (10) R= No. Load Test Analytical -,-)
(o Test Analytical (7) Test Analytical -
II (9) (6) Ton. Ton Ton To in. in. % in. in. /0(7f cr' 10
B22 5 0.30 0.26 87 0.45 0.51 113 1.75 1.56 89 93 B30 11 0.83 0.72 87 0.70 0.52 74 4. '05 3.83 89 104 B31 II 0.60 0.51 85 0.65 0.53 82 3.40 2.60 77 89 K3 6 0.50 0.43 86 0.58 0.57 98 2.40 2.00 83 93 K2Z,' 11 0.65 0.51 78 0.30 0.36 120 2.25 1.66 74 88
Average 84.6 97.4 82.4 93.4 1
Standard deviation of R = 5.7%
Table 4-7 - (continued) "Simplified Solution"
4 x 4 in. sections
1 2 3 1 4 5 6 7 8 9 10 11 1 12
No. Axial Load
Ton
Max. Lateral Force Def. at 1st Crushing, Di. Deflection at F = 0, D9
Test
Ton
Analytical
Ton
(-4) (3) %
Test
in.
Analytical
in.
(-7) (6 )
To
Test
in.
Analytical
in.
7, (10) = rt R - R ce /0
(9) %
B14 B15 B17 B24 B20 B28 B35 K1 K20 K22
10 t! 11 it IT
11 ti It It .II
0.57 0.66 0.86 0.53 0.54 0.45 0.54 0.56 0.84 0.80
0.52 0. (33 0.79 0.47 0.48 0.42 0.47 0.49 0.80 0.65
91 95 92 89 89 93 87 88 95 81
0.45 0.55 0.70 0.50 0.55 0.52 0.54 0.50 0.33 0.33
0.56 0.52 0.52 0.50 0.52 0.51 0.50 0.55 0.35 0.36
124 95 74
100 95 98 93
110 106 109
1.67 1. -/ 6 2.10 1.50 1.67 1.23 1.55 1.60 1.50 1.38
1.45 1.60 2.00 1.50 1.42 1.20 1.31 1.41 1.38 1.17
87 91 95
100 85 98 85 88 92 85
87 97
108 109
96 104
96 94
106 91
Average 90 100.4 90.6 98.8
S.D. of = 7.1%
Table 4-7 (continued)
"Simplified Solution"
4 x 4 in. sections
1 2 3 I 4 I 5 6 7 8 9 10 11 1 12
No. Axial Load Ton
Max. Lateral Force Def. at 1st Crushing, DI Deflection at F = 0, D,
Test
Ton
Analytical
Ton
(4) (-3)
al /0
Test
in.
Analytical
in.
( (6-7) )
%
Test
in.
Analytical
in.
rt = (1 o ) 1i R -1,7
() GI 0
B20 B16 B18 B23 B34 K5 K6 Kll K21
12.5 15.0
it tI
1 I
14.0 I! II 1!
0.59 0.65 0.80 0.43 0.59 0.62 0.70 0.76 0.76
0.52 0.65 0.85 0.43 0.55 0.55 0.69 0.66 0.74
88 100 106 100
93 89 99 87 97
0.49 0.47 0.58 0.28 0.51 0.42 0.57 0.45 0.31
0.51 0.47 0.50 0.47 0.50 0.52 0.56 0.47 0.33
104 100
86 124
98 124
98 104 106
1.13 0.95 1.16 0.74 1.09 1.03 1.15 1.43 0.78
1.13 1.05 1.26 0.84 1.00 1.05 1.25 1.43 0.83
100 110 109 114
92 102 109 100 106
110 109 109 108
96 105 108 110 106
Average 95.4 104.9 104.7 106.7
S. D. = 5.8%
Table 4-7 (continued)
"Simplified Solution"
4 x 4 in. sections
1 2 3 4 5 6 7 8 9 10 11 I 12
Max. Lateral Force Def. at 1st Crushing, D1 Deflection at F = 0, D2
No. Axial Load Test Analytical
(4) - (3)
Test Analytical (-7) (6) Test Analytical R= (- io) R
R () Ton Ton Ton % in. in. % in. in. % - ,0
B19 19.0 0.77 0.85 110 0.50 0.48 96 0.93 1.00 105 96 B21 11 0.38 0.35 92 0.32 0.35 109 0.52 0.53 102 103 B32 11 0.58 0.63 109 0.57 0.45 79 0.97 0.84 87 84 B33 If 0.50 0.49 98 0.49 0.41 84 0.80 0.72 90 92 K19 11 0.90 0.85 94 0.29 0.29 100 0.68 0.67 99 100
Average 100.6 93.6 96.6 95.0
S.D. in 1t/R = 6.6%
Total average of R/R for 4 x 4 in. sections is 99. 7% and S. D. is 7.75% .
Table 4-7 (continued) "Simplified SolutionT1
4 x G in. sections
1 2 3 4 5 6 7 8 9 10 1 11 12
Max. Lateral Force Def. at 1st Crushing, D1 Deflection at F = 0, D2
Axial (4) 7) - (10) R = fi No.
Load Test Analytical
(-3)
Test Analytical (-) (6
Test Analytical (9) -R Ton Ton Ton % in. in. % in. in. & /0 °;70
K16 6 1.15 0.98 85 0.25 0.31 124 3.46 2.70 78 89 K2 10 1.50 1.12 75 0.40 0.45 113 2.70 2.46 91 115 K9 II 1.15 1.03 90 0.37 0.41 111 2.58 2.27 88 95
K15 1, 1.43 1.24 87 0.25 0.29 116 2.44 1.94 80 89
K8 14 1.46 1.26 86 0.40 0 42 105 2.24 1.88 84 94 K14 fl 1.67 1.52 91 0.28 0.29 104 1.70 1.60 94 103 K7 19 1.61 1.43 89 0.38 0.39 103 1.44 1.48 103 114 K13 ?, 2.07 1.84 89 0.30 0.27 . 90 1.40 1.29 92 104 K18 It 2.54 2.10 83 0.35 0.27 77 1.58 1.40 89 109
Average 86.10 104.8 88.8 101.3
S.D. of TYR = 9.9%
Table 4-7 (continued)
"Simplified Solution"
6 x 8 in. sections
1 2 3 4 1 5 6 ' • 8 9 10 11 12 1 " 14
Max. Lateral Force Def. at Ist Crushing, D1 , Def, at F = 0, D9. ,„
No. Axial Load Test Analytical (4)
() (4)+BRE Test Analytical (8)
(7 (7)
(8)+BR Test Analytical - (12) R - R 1), (3 ) (7) (11)
Ton Ton Ton 10 cr' To in. in. % % in. in. /00/ cr:0
Z5 20 1.83 1.98 108 102 1.00 0.70 70 94 4.00 4.00 100 93
Z4 30 2.42 2.27 94 90 0.89 0.63 71 85 2.90 2.80 96 105
Z3 38 2.06 2.44 118 110 0.85 0.59 69 96 1.95 2.44 125 117
Z2 50 2.76 3.20 116 107 1.15 0.57 50 68 2.52 2.18 87 92 Z1 50 2.72 3.20 118 109 0.89 0,57 64 88 2.18 2.18 100 97
Average 110.8 103.6 64.8 86.2 101.6 100.8
S. D. of R = 9.3%
* B. R. E. =Base Rotation Effect
161
base rotation --- . . . / / / /
• ./ ,,,/,.. ---
/1 / / /
• //
/V //'/ /
test
Psi
—Ec=40000Niq —Ec.:600001/C
. fel=5500
I
/ //
/ / //
/*/
i //// II /1 1 . i r //
0.0 01 02 03 0.4 0.5 0.6 0.7 0.8 Deflection, D, in.
Figure 4-1(a) Bending Moment-Deflection Diagram for 'Z3' with Different Ec
1.2
1 0
0.8
0.6
04
0.2
ec
If. = /4
Figure 4-1(b) Stress-Strain Relationship for Concrete in Tension
:I ,= >: P l'> r i III P ni
Exact So1t~Lion
Si rnplifjpd SoluiJon
Cone ret e Tensil e St r e nght
Considered
F'il'St. Crushing Sir1.£;e
Fton.
1.
.c. N
12JJ_ .. { P F~=- ~r2
B22
.1 (;2
P ::: 5 Ton A ::: 4 x 1/8"
s 11 ::: 30 h~ . ::: =--Ju . __ ---!~---. ~---v
0·2 -t.-. .-. -"':~-"':"'---. ___ -. t---r---i 0·' W~- ---- ------4---.---. ~~-:::---. ~ .
L . ---"'2 Din. 0·0 0.5 '·0 1·5 2·0 2·5 3·0
F ton . 0·5 ~----=--r~----.
0·3 ~II~----
0·2
0·1 r
K3
P ::: G Ton A ::: 4 x 1/4"
s h = 30 in.
L' ___ ~r;__--__;;:_L::_--__;~---T7l--=......-~~ 20 in. 0·0 2·0 3·0 4·0 5·0
Figure 4-2 Force-J)of'lecUon Dirt~;l'an1S
123
= 6 Ton A 4 x 1/4"
s
h 24 in.
163
Expo rim ent Exact. Solution Simpliripd Solution.
Bound Concrete Crushing Strain = 0,010 First Crushing Stage
4in.
x
.0
O
9
Figure 4-3 Force-Deflection Diagrams
0.0 0.5
1.0
15
2.0
9
v
-L-A ‘... l.1
P= A= s h
/
/
Fton. 0.5
04
0.3
0.2
01
10 Ton 2 x 1/4" 30 in.
2D in' 25
16.1
Experiment Exact Solution Simplified Solution Steel Strain Hardening considered First Crushing Stage
0 0
B30
P As h
= 5 Ton = 4 x 1/2" = 30 in. N.,
-..........
`-.., N.......,_
----:.:„..--....... '-?!-Jin.
0.0
1.0
2.0
3.0
40
50
6.0
70
80
F ton. 09
08
07
0.6
05
04
03
0.2
0.1
Figure 4-4 Force-Deflection Diagrams
15 10 20 0.0 3.0 05 2.5
10 30 15 25 05 20 0.0
Experiment
Exact Solution
Simplified Solution
First Crushing Stage -C CN
.
1:" -
A = s
h =
11.1 lull
4 x 1/4'i
30 in.
<7 7.
IF
1326
P =
A s =
h =
Spacing
10 Ton
4 x 1/4"
30 in.
= 3 in'.
---- .
• '
/
/•
'--,,..„_
Figure 4-5 Force-Deflection Diagrams
Om. 0.6
Fton. 0.6
165
in '
K1
-
c
0.5
0.4
0.3
0.2
01
05
04
0.3
02
0.1
0
1GG
Experiment Exact Solu ion Simplified Solution Concrete Tensile Strength considered Finst ushing Stage
Fton. 0.6
0.3
0.2
0.1
0.0 0 5 10 15 2 0 2.5 2Di n-
3.0
1324 P = 10 Ton A = 4 x 1/4" h = 30 in. Spacing = 1 in.
0.5
0.4 /v
0.0
Fton. 0.7
02
0.4
0.3
01
06
0.5
..........
l' =-- I
As = 4 h = 30
U .1.. On
x 3/8" in. /.
14-7- .\. 1 .
I . . \ •-• .„_ -*-.-1-' .
'N...
0.5 10 15 2.0
1315
2.5 i 2D n
30
Figure 4-6 Force-Deflection Diagrams
:1()7
4 in. o
0
Experiment I",;xact Sohition Simplified Solution First Crushing Stage
Fton.
P = 10 Ton A = 4 x Sj'8"
S
h = 24 in.
0.5
p.4
0.3
0.2
0.1
0 .0
09
0.8 K20
0.7
06
B20
5
04
0.3
0.2
0.1
0.0 0.5
1. 0
15
20
,-- •/
.
N
.\\NN
P = 1: As = 4 h = 3
/ ' TN. \..
N. N.
7.
I
Fton, 06
I- Ton x1/4" in.
2 Din • 25
Figure 4-7 Force-Deflection Diagrams
4in. Experi ment Exact Solution Simplified Solution First Crushing Stage
10 40 20 30 0.0
/
B17
P = A s = h = / l‘\\
1/ \
/
/
N.
N 2Din• 50
Fton. 09
10 Ton 4 x 1/2" 30 in.
08
07
06
0.5
01.
0.3
0.2
0.1
Fton. 05
0.4
0.3
0.2
7-r B23
P = 15 Ton As = 4 x1/8" h = 30 in.
0.1 7'
2Din• 20 0.0 0.5
1.0 1.5
Figure 4-8 Force-Deflection Diagrams
0 0
0
x
K21
P = 14 Ton A s = 4 x 1/4"
h = 24 in.
0.0 05 10 15 2Din• 20
1.69
Experiment Exact SoI.ili ion Simplified Solution First Crushing Stage
Eton. 07
K5
P = 14 Ton, As = 4 x 1/ 4" h = 30 in.
0.6
0.5
0.1,
0.3
0.2
2Din- 2.5 0.0 0 5
1.0
15
2.0
Figure 4-9 Force-Deflection Diagrams
170
:Experiment P.:xact Solution Simplified Solution First Crushing Stage
4i n-
x -C (NJ
P = 14 Ton A = 4 x 1/4" h = 30 in. Cover = 0 125 in.
0.0 0 5
/"--- B1.6
P = 15 As = 4 ) h = 30 //
\. \
/
/ \
0.0 0 5
1 0
1.5
2.0
Fton. 07
06
0.5
0.4
0.3
0.2
O.
Ton 3/8"
in.
2DirL 2.5
Figure 4-10 Force-Deflection Diagrams
171
4in. 0 0
15 05 20 10 0.0 Fton.
07
0.6
05
0.4
0.3
0.2
0.1
0.0 05 10 15 2Din• 20
Experiment Exact Solution Simplified Solution. First Crushing Stage
Fton. 09
08
07
06
0.5
04
03
02
0.1
15 Ton x
0 in.
2Din• 25
B32
P = 19 Ton As = 4.x 3/8" h = 30 in.
/ .i.
RN
B18 //.
P = As h = //
Figure 4-11 Force-Deflection Diagrams
4in. Experiment Exact Solution Simplified Solution First Crushing Stage
• •
• •
Fton. 08
K6
P = 14 Ton As
= 6 x 1/4" h = 30 in.
0.7
06
0.5
0.4
0.3
02
0.1
2Din.
172
0.0 05 10 15 20 25 3.0
Fton
* 06
05
0.4
03
02
01
0.0 05 1.0 15
B33
P = 19 Ton As
= 4 x 1/4"
h = 30 in.
2Din.
Figure 4-12 Force-Deflection Diagrams
20
K19
P = 19 Ton A = 4 x 3/8"
S
h = 24 in.
10 0.0 0.5 2 Dill'
1.5
09
0.8
03
07
0.2
0.6
05
0.1
F ton.. 10
0.4
1.0 1.5 0.0 0. 5
Fton. 0.4
0.3
0.2
0.1
17:3
Experiment Exact. Solution Simplified Solution First Crushing Stage
Lin.
1321
P = 19 Ton A
s 4 x 1/4"
h 30 in.
Figure 4-13 Force-Deflection Diagrams
174
Experiment :1 .!;xaci Solution Simplified Solution Steel Strain Hardening considered First Crushing Stage
F ton. 1.2 r RiG
P = 6 Ton A
s = 4 x 1/4
h = 24 in.
• ... ••"--.. ?Dill.
•
10 20 30 40 50 b
1.
0
B19
= 19 on A = 4 x 1/2'' h = 30 in,
Figure 4-14 Force-Deflection Diagrams
1.0
0.8
0.6
0.4
0.2
/in. 0
0 0
o 0 Y.
.0 C.1
4in. Experiment Exact So]ution Simplified`volution Concrete Tensile Strength considered First Crushing Stage
Fton: 1.2
1.0
K9
P = 10 Ton A s = 4 x 1/4" h = 30 in.
0.8
0.6
0.4
0.2
0.0 1 0 20 3.0 4.0 5 0
2Din.
60
Fton. 16
0.8
0.6
1.2
1.4
1 .0
K15
P = 10 Ton As = 4 x 1/4" h = 24 in.
0.4 N IN. 0.2
2D'- 50 3.0 4 0 0.0 1.0 2.0
Figure 4-15 Force-Deflection Diagrams
4in. 0
x CD
K8 P = 14 Ton A = 4 x 1/4" S h = 30 in.
5 02Din-
E.-Teri inept Exact Solution Simplified Solution Concrete Tensile Strength considered First Crushing Stage Fton.
K14 P = 14 Ton
/7 •----x, ...,>__.,_
As = 4 x 1/4" h = 24 in.
//
—11
ii
II
1 . "-...„.„....„,,,
I'-....,_,..•--
‘...'N.".....'N''''•-. '..
2Din.
00 05
1.0
15
20
25
30
0.2 ,
0.0
18
16
1.4
1.2
1.0
0.8
06
04
02
1.6
1.4
12
1.0
0.8
0.6
0.4
Figure 4- 16 Force-Deflection Diagrams
177
Experi multi Exact -.;011.ution Simplified .iolution Conereie Tensi]e Strength considered First Crushing Stage
tin.
K13
1? '.'-- 15 Tons As = 4 x 1/4" la = 24 in.
.
// • .
.NN..
/
'N\
\ .
'.
\N\
1 \N
\\N
..\. N
X\
2 Din. 0.0 0 5
10
15
2.0
2.5
3 0
F ton. 2.2
2.0
1.6
16
14
12
1.0
08
06
0.4
02
Figure 4-17 Force-Deflection Diagram
Fton. 1.8
1.6
12
10
08
0.6
04
0.2
0.0 0 5 10 15 20 25
K7
P = 19 Ton A = 4 x 1/4"
h =30 in,
2Din. 30
178
4in. Experiinent Exact Solution Simplified Solution Concrete Tensile Strength considered First Ci-ushing Stage
0 a
x
Figure 4-18 Porce-Deflection Diagram
Gin.
Experiment • Simplified Solution First Crushing Stage
Z5
1) = 20 Ton As = 4 x 3/8" h =45 in.
179
-2.0 -1.0 0.01 1.0 2.0 3.0
-2.0
.... .0
-3.0 1 in.
Fton. 3.0
Z4 P = 30 Ton As = 4 x 3/8" h = 45 in. Spacing = 6 in.
-1.0 -3.0 3.0 Din.
-2.0
Figure 4-19 Force-Deflection Di Luz- ra ms -3.0
in. Fton.
Z3
...) • V
2.0 7" ..---------i. --......, P = 38 Ton
As = 4 x 3/8" h = 45 in.
/ /
1.0
--......, ---.......„
----,......
"--..,......
5 -1.0 0.0i i /
0.5 1.0 1.
"...„ -....,..,,,,,
2.0 "--......, •
-0.5
-1.0
-2.0
-3.0
--
C CO
Experiment Simplified Solution First Crushing Stage
x
2.5Din-
Figure 4-20 Force-Deflection Diagram
.1.0 61n.
/
, N,
--,x
N
• 0
• 0
Z2
P = 50 Ton A = 4 x 12"
s " 2.0 h = 45 in.
i /
/
. N
NN 1.0
5 -1.0 - 0.5 0.0 0.5 1.0
i N N
1.5 2.0 N
-1.0
—2.0 x
-3.0
.
,------ •
-4.0
2.5 Di•n.
Experiment Simplified Solution First Crushing Stage
1
Fton.
Figure 4-21 Force-Deflection Diagram
ton. 3.0
2.0
61n. Z1
P = 50 Ton As 4 x 1/2"
h = 45 in.
-C
N
-1.5 -1.0 0.5 1.5
F,
Experiment Simplified Solution First Crushing Stage -3.0
-4.0
Fig::re 4-22 Force-U.:flec-Lion Dizrani
(a)
x
L A D a
v
I. t
(b)
(e )
V/(qa /7E) 3
2
1
(d)
q/E=6M/Ebt 2
Figure 4-23 Base Ream Deformation
I - x
1_113 A D B
a
Vcr.(201-TE)[(afx)log d/(a4x) -x log cir';;] (qa/n- E)( 1-9 )
(e)
(d)
317437.18
3"i8 374 .1
iici,21ton/in2
1
VD= (2q/rt E)[(a )109 d/(a-ic)+3; log dri,J+(qotrvE)(1-9 )
(a)
/ M=121, 7 , P= 381°n.
r Bin. I • ,
I
L
Z3 Section
Figure 4-24 Base Beam Deformation
( 1) Lin c )' e p l' P S (> n i i 11,(~' l' q II aU () n (-1 - :) ) (2 ) ~\ I C' iL ~.; U l' t~ d }; (1 ::; l' ;:~ () 1 a I j U 11
(~.3) (2) + Exact ~~()Juliol1
( ·1 ) ( 2) + S j 1 n p li r i e cl Sol u L~ (} n ( ;) ) 1;~ x P (~ I • i 111 en t
-;------t------t---
Zl & Z2
a·Gr1-~--~-7----+-----~-----+----~
Z3
~ ____ ~ ______ ~ ______ ~ ______ ~ ____ ~ Din.
a·o 0·2 a·l. a·6 a.8 1·a
Figure 4-25 Bpllcling: 1\'Ton1C~nt-f)cflec1.ion Djagranls for G x Bin. COhtl1111S
(1) Line representing ecit'xii ion (4-)) (2) IVieasured se ilic.Aation (3) (2) + _E.,xact Solution (4) (2) -1- Simplified Solui ion (5) Experiment
1(1) /7- 1 / (2)
1 / (3)
r5)
— (4) -11
I / j i
//r
II ii A
II t Z4
1 0.0 0 2
04
0.6
0- 8
(1) /(2) __– _- (3)
-- -- — — (4)
; / I /
I // ,
,,-- '''.
(5)
1 I /
/ fit I i
/// 1 II /
III/II , Z5
/ V
0.0
02
04
06
0.8
10
Figure 4-26 Bending Moment-Deflection Diagrams for 6 x 8 in. columns
1.;;()
1.0 (test)
0.8
0.6
0.4
0.2
1 M/Mmax.(test) .2
1.0
0.8
0.6
0.4
0.2
Din.
e5y E5= 30x106 P.s.i
Iv
A
vi I
V2I <
, 66in. 12 i n.
Specimens' Size & Loading Arrangement
66'n.
Figure 4-27 Details of Yamashiro's Tests
Experiment
f2C
Beam-Column Deformation (Simplified Solution)
/
i
J24 P = 0.0 fr -=- 5000 psi e f
2 Sy =21.6 Ton in':
As/ht . 1.11%
De ails as fig. 4-27
0 2
L. 10
12
14
.9., 200
0 100
M 300
A 16
Deflection, in. Figure 4-28 Bending -Moment-Deflection Diagram
:7, 300
C)
0
tx 200 • -
100
Experiment Beam-Column Deformation (Simplified Solution) Beam-Column + Stub Deformation
,
.
,1- 25
P = 25 kips . f' = 5050 psi c f sy = 21.9 Ton/in2.
Asi ' b'• = 1.11% '
Details as fig. 4-27
I 1
,
0
2
4
8
10
12
14
16
400M
A
Figure 4-29 Bending Moment-Deflection Diagram
Deflection., in_
400
300
a) 0 0
bf) 2 1:3 a)
10
.
. ---..,.. --,,...
---......„,......
J26• • IP = 50 kips f' = 4600 psi . c . f = 22 Ton/in? sy As ibt = 1,11%
Details as fig. 4-27
I 2
4
6
M 500
A 10
Experiment Beam-Column Defc.»..inal ion (Simplified Solution) Beam-Column + Si ul) Deformation
l!)0
Deflection, i.n.
Figure 4-30 Bending Moment-Deflection Diagram
c .,.....,
Ex peri n1 e nt
Beanl-Colunlll Defornlation (Silnplified Solution)
Bealn-Colunln + Stub Defornlatioll
M 1250~------~--------~--------~-------'
1000~~----+---~~~-------+------~\~
I Ii en (' .~ 750~/~!-------+----------~--------~--~~--~
~, I ~ I (J) r
c:: o '-'
~ 500Hr--------~--------~--------~------~ ~ .J31
P = 75 kips
1~ fl = 42 50 psi c
I f· I_? 1 r: 'f I . ,,", 2 250 HI-----------!...------------l sy -..... ..J 0 n lu.
I A sl bt = 1. 110/0
I' Spach1g = 6 in. Details as fig. 4-2"";"
~--------~--------~--------~------~~. o 2 4 6 8
--' -.,....., I
:rJ.
500M.---~~~--------~--------~------~ r~~
t [,00
.~ 300~--------~--------;---------~---------~
,.---;
g 200 .,....., u ,..... ~
Q) ('(' ~
= 75 Idps = 4500 psi
2'/ "~ ,."
I
I
I j~) c
.T34
= ..... .1 oni J];':-I I l' 100 j------+------I s::·
~ A,jbt = 1. 11% ~ ,c::pacing' = 3 in.
~ ____ ~ ________ ~I_r_)_e_~:a_i_l_s_a_s_,~~_jg_. __ ~f_-_2_; __ ~A L\
o 2 !'t 5 8
DeflecUon. ]11.
2 4 6 10 12
,_ -------
"--........
I
. —____
/-' ,...../....
- - ----__--> / / ----- .. '.-
. • ------------L_____, •-_,,, .
. ••.----.--
• , -
, • v-
. f =7000 P•s•i
— — — — P'=2°10
1 %
0.2%
P 'a= — — • — •
P"–
I 1 0.
1.0
0.8
0.6
0.4
0.2
edel. 0 3 14
(a) Equivalent Concrete 7,ectan;tular Stress-Block
0.3
Figure 4-32 Properties of 'Equivalent. Concrete Stress-Block
t
C '0
0.7 ----- -------,----------,----,---,--------,
0·61-----I-----+------i--- ----j----
0.51-------1
O·t.~- ------l
j ;).)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 0.2 L--_----lL--_---L __ -!. __ --L __ --L... __ --l.-__ ........I e Co xl 0 3 I
o 2 L. 6 8 10 12 11. I
ex: 1·0.----......---.,-----r-·-----r-----,------.-----,
0·81----+---""~/
0·61---1/,
0.4
--- fc=300QP.s.i
---- fc:: 70 OOp·s.i
P"=0.2%
o. 2 6 8 10 12
ji'jglll'P 4-3:i Variation of .oJ.. and 't \Vitl..!.~n
1
1
1
1
1
1
1
1
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1
1
1
1
1
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1
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194
NOTAT I ON
The following symbols are used in Chapters 5 and 6:
a Acceleration equivalent to lateral load yield level, f /M
C Damping coefficient
E, Eo, Eo Earthquake amplification factor
F, F' A constant proportional to earthquake intensity
F(X), f(x) Lateral load
F y Lateral load yield level
g '
Acceleration of gravity
2h Height of a fixed-ended column
K Initial stiffness
M Mass of the system
n Critical damping ratio (nd in Chapter 6 is neutral axis depth)
P Axial load
q. go. 71 qe Ductility factor
R, R Ratios of E/E0 and E/E0 or qo/q and q0/q
Ratio of the gradient of the falling branch of the force-deflection diagram to the initial stiffness in the diagram
S S,, Sd Acceleration, velocity, displacement response spectrum
Sa Acceleration spectrum deduced from the non-linear system analysis
Sad Sam Average, maximum §a SI, SI Earthquake intensity corresponding to Sa and Sa
T Initial natural period of vibration
t Time W, Wo Reserved energy capacity
W Dead load
Z(t) Earthquake acceleration
(,) Circular natural frequency
195
CHAPTER 5
DYNAMIC ANALYSIS OF A COLUMN
The subject of this chapter is the study of the response of
a reinforced concrete column to an earthquake type of loading. The
column is idealized as a single-degree-of-freedom system with
load restoring characteristics similar to those found in Chapter 3.
A model based on the tests by Neal (39) and Koprna (73) is used
for the behaviour of the col umn during the unloading and reloading
stages. A brief summary of these tests and the observations made
on them will be given first. The accelerogram records of six
components from four actual earthquakes are used in the analysis
for the earthquake excitation. In the analysis of the responses,
the emphasis is on isolating the main parameters affecting the
failure of a column during an earthquake. These parameters will
be related to the elastic response of a column having the same
period of vibration as the initial period of the column, and the same
damping coefficient.
Before developing the main subject, a brief review of the
relevant work in this subject is given.
5.1 Review of Relevant Work
Dynamic analysis of structures with regard to earthquake
loading requires two categories of information: ground motion
characteristics for data and the load deformation characteristics
of the structure. With these, the main tasks are the analysis of the
system and the interpretation of the results for design purposes.
Research in the earthquake engineering field is concentrated on
these three aspects.
The study of the ground motion characteristics arises
from the fact that the ground motion acceleration records have a
random nature, and hence the analysis of a structure for them
196
should naturally be based upon a probabilistic approach. The
evaluation of the statistical properties of these motions requires a
great number of accelerograms from earthquakes with different
intensity, duration and distance from the energy released zone.
So far, there are not a lot of recorded earthquake accelerograms,
but the existing records show some statistical similarities. Based
upon these similarities, various statistical models have been
developed to generate an earthquake acceleration record. These
models vary from a simple white noise representation (56, 57) to
a stationary Gaussian process with prescribed spectral density (58),
and to non-stationary processes (59). All show, to a greater or
lesser extent, some similarities to the average properties of the
existing records. The degree of validity of these models, however,
remains to be seen in the future by further records.
A number of experiments have been reported on the load-
deformation characteristics of reinforced concrete members. Some
of these are concerned with the ductility of the members and the
parameters influencing them, and others with the performance of
the member under cyclic and repeated loading. The works related
to the former topics were discussed in Chapter 1. Here some of
the works concerning the behaviour of a member under cyclic
loading are discussed,
The behaviour of concrete material under cyclic loading
has been studied by B. P. Sinha et al (44), and I. D. Karsan et
al (45). The former group found that the stress-strain relation-
ship of concrete under compression is almost unique, in the sense
that at any point, there are only two paths with regard to the un-
loading and reloading stages, no matter what loading history the
specimen has followed. The strength and stiffness of the specimen
gradually drop as the inelastic defoi'mation increases. Unloading
197
paths are governed by a parabolic relationship, and the reloading
paths by an almost linear relationship. In the second group's
experiments, the stress-strain relationship was found to be related
to the loading history, and therefore the idea of uniqueness is contra-
dicted. The envelope curve was found to be the same as the curve
for the specimen under monotonic loading. A series of relation-
ships for the unloading and reloading paths is suggested. They are
parabolic.
A group of tests on beams and columns has been carried
out at the University of Illinois and the results have been summarized
by N. H. Burns et al (9). Based on these tests, H. Aoyama (60)
presented an analytical approach to predict the moment-curvature
relationship in the unloading and reloading stages. These tests
show a gradual deterioration in the stiffness of a member as the
inelastic deformation increases. In cyclic loading in one direction
only, the drop in stiffness is almost linear with an inelastic defor-
mation of up to nearly 50% of the ultimate deflection. It then remains
nearly constant. Deterioration of stiffness is due to the gradual
loss of bond between the concrete and the reinforcement, and the
crushing of the concrete cover. The stabilized stiffness should
belong to the section without cover on the compression side. In
the case of cycling in both directions, drastic deterioration in the
stiffness was found. However, in these tests, and the tests carried
out by B. P. Sinha et al (61) and G. L. Agrawal et al (62), on singly
and doubly reinforced beams, the members showed a considerable
capacity for energy absorption. These results have been generally
confirmed by N. W. Hanson et al (10) in a series of tests on full
size cast-in-place beam-column joints. In these tests, the role
of transverse reinforcements and proper detailing was found to be
very important for the development of ductility.
198
Tests on reinforced concrete frames under static and
dynamic loading have been made by various investigators.
V. V. Bertero et al (11) have carried out a series of tests on a
single storey, one-bay small model frame under proportional and
cyclic vertical and lateral forces. It was concluded that cyclic
loading has little effect on the strength of the frame, as far as
the collapse is concerned, but it gradually reduces its stiffness.
The deterioration in the stiffness is increased by the number of
the cycles, but the rate of deterioration is decreased. The
decrease in stiffness has been attributed mainly to the loss of bond
between the concrete and steel. Losses of 22% in the bond after
10 cycles and 100% after 100 cycles were observed in their tests.
The same trend in the behaviour of frames has been observed by
G. M. Sabnis et al (63).
The relevance of the results of the static tests to dynamic
loading has always been questionable. Considering that under
dynamic loading, such as an earthquake, the load is applied very
rapidly, the answer to this question is very important. Tests (51)
on a concrete cylinder under static and dynamic loading, with a rate
of stressing in the range of 7. 1-17x106 psi/sec., indicate that the
strength is increased with the rate of stressing, but it approaches
a constant level. For a rate between 2000-106 psi / sec.,the
increase is 25% as compared with the static test, and it is 38% for
a rate between 106-107 psi/sec. The strain is also increased with
the rate of loading, but it is in the order of 10%. The increase in
the strength and strain results in an increase in energy absorption
capacity. The maximum value of this increase is of the order of
42%. For steel, the strength is increased by the rate of loading.
When yielding occurs within 0.005 sec. of the application of the
load, there will be nearly 40% increase in the steel strength for
intermediate grade steel (5). However, since earthquake loading
199
is much slower than the above rates, the increase in the strength
of a reinforced concrete member should not be more than 5 - 10%.
T. Shiga et al (64) in a series of tests on single-storey,
one-bay, fixed frames have examined the effect of dynamic loading
on the hysteresis loops, equivalent viscous damping, and equivalent
rigidity. The dynamic load is applied by a vibrating table system.
They found that the hysteresis loops in both static and dynamic tests
were the same for the same amplitude of deflection. The dynamic
equivalent rigidity of the frame decreases with an increase in the
amplitude of cycling and the loss of stiffness appears to be hyper-
bolic. In the tests, the stiffness fell to 1/4 and 1/S of its initial
value, for sways in the column of 0.005 and 0, 025 radian, res-
pectively. The equivalent damping factor increases linearly with
the amplitude of the cycling, but it approaches a constant level.
At a column sway of 0, 04 radian, it has been found to be 0.13,
T. Takeda et al (13) have studied the response of a
reinforced concrete member under simulated earthquake loading.
The specimen was a 6 x 6 x 28.5 in. cantilever column with a mass
of 2025 lb. of steel attached at the head. The specimen was
excited at its base, through a vibrating table system, by feeding a
40 sec. displacement record of the NS El Centro 1940 earthquake
compressed to 5 and 2.5 sec., and also the N21E TAFT 1952 earth-
quake with a time-displacement record compressed by 10 times.
The measured accelerations at the base of the specimen
in these three tests were 1.28, 2.4 and 2.7 times the gravitational
acceleration. The measured acceleration-time histories at the
centre of the steel masses have agreed reasonably well with the
analytically predicted ones. The analytical results were based
upon the response of a single-degree-of-freedom , system with
load restoring characteristics similar to those found in the static
test. For the reversal paths, a series of rules has been introduced.
200
The maximum measured accelerations differ from the calculated
values by less than 20%, which may indicate a rise in the yield
level of steel under dynamic loading. It was observed that the
stiffness and energy absorbing capacity of the specimen changed
considerably and, at certain times, very rapidly during the test.
It was concluded that the static load-deformation characteristic
of a member can be used satisfactorily for dynamic analysis.
Non-linear analysis of structures under earthquake
loading has been the subject of many investigations. In most of
them, a single-degree-of-freedom system with an elasto-plastic
or bi-linear hysteresis characteristic has been subjected to different
records of actual or artificial earthquakes. A number of works on
the response of multi-degree-of-freedom system under these con-
ditions have been reported (70, 71, 72). Most of these works had
a deterministic approach and the results obtained are naturally true
for the earthquakes under consideration. However, an approach
based on statistical properties of earthquake records is required
in order to draw an overall conclusion.
For the case of a single-degree-of-freedom system with
elasto-plastic characteristics, an attempt has been made to relate
its response to the response of a linear system having the same
natural period of vibration and the same damping coefficient.
Extensive work on the earthquake and ground motion shock type
of excitation has been reported by A. S. Veletsos and N. M. Newmark
(65, 66, 67); The results of these works can be summarized briefly,
as follows:
1) For low frequency systems, the total displacement in the
inelastic system is almost the same as for the elastic system.
The ductility factor q, figure 5-2(a), is
201
q Sa/Fy (5-1)
where Sa is the earthquake elastic acceleration spectrum corres-
ponding to the natural frequency of the system, and F is the yield
level of the elasto-plastic system. M is the mass of the system.
2) For intermediate frequency systems, the energies absorbed
in the elastic and elasto-plastic systems are nearly equal.
M. S q = 0.5 [( F a)2 + 1 (5-2)
3) For high frequency systems, a small deviation of Fy with
respect to MSa results in a high inelastic deformation. For this
reason, the yield level in the system is recommended to be
Fy =M. Sa (5-3)
The range of intermediate frequencies is roughly 0.4 < f cps which corresponds to a natural period of roughly 0.3 <T <2.5 sec.
The above rules are true for a damping ratio in the range of 5 - 10%
of critical damping.
On the basis of Hanson tests (10) on beam-column joints,
Clough (12) has introduced a degrading stiffness characteristics
model for reinforced concrete frames, as shown in figure 5-2(b).
The envelope diagram can also be bi-linear. The analysis of the
response of such a system to four different records of actual earth-
quakes, indicates that the ductility requirement in this system, for
structures with the initial period of vibration, T, greater than
0.6 sec, is more or less the same as the ordinary elasto-plastic
system. For structures having T less than 0.6 sec. the system
shows more sensitivity, and the required ductility is sometimes
quite high. Another study by S. C. Liu (68) on a similar system
with T = 0.3 sec. and T = 2.7 sec, but with a statistical approach
based on an ensemble of generated earthquakes, shows that the
system, in both cases,has a lower ductility requirement than the
202
elasto-plastic system. The ratio between the required ductility
in the elasto-plastic system and that of the degrading stiffness
system is nearly 1 for the structure with T = 2.7 sec., and 1.5 for
the case with T = 0.3 sec.
Husid (69) has studied the effect of gravity load on the
response of a single-degree-of-freedom system with an elasto-
plastic or bi-linear hysteresis characteristic to an earthquake
loading. In a statistical analysis based on an ensemble of the
artificially generated accelerogram records (58), he found that
the gravity load increases the drift in the structure considerably.
The drift gradually accumulates and results in the collapse of the
structure. The longest time a structure can withstand an earth-
quake loading was found to be dependent on the height of the
structure, the earthquake intensity relative to the yield level of
the structure, and the upper slope of the bi-linear system. It is
almost independent of the natural period of vibration of the structure.
5.2 Behaviour of a Reinforced Concrete Column under
Cyclic Loading
The following discussion is based on observations made on
the results of tests by Neal (39) and Koprna (73), on a column under
cyclic loading. The specimen and the testing procedure in these
experiments were similar to Bailey's tests, described in Chapter 2.
Neal tested four columns of 4 x 4 x 60 in. under a 10 ton constant
axial load. The programmes of loading in his tests are shown in
figures 5-5 and 5-6. In one programme the cyclic loading was
performed with limited amplitude of deflection and was limited to
one direction only, in order to study the formation of hysteresis
loops and their energy absorption capacity. In the second
programme the cyclic loading was extended to both directions
about the vertical position of the column. Koprna's tests had a
pattern of loading similar to the latter programme, figure 5-7.
203
These tests show that the behaviour of a column at any
stage depends very much on the history of loading it has followed,
In general, a deterioration in stiffness and strength of the column
is seen due to repeated loadings and inelastic deformations. It
seems that the latter effect is more pronounced than the former.
The maximum lateral strength of column N4, figure 5-6, relative
to its strength in the first cycle of loading is shown in figure 5-8(a).
It shows a drop of nearly 12% for each cycle of loading and/or for
each increment in the inelastic range. In a similar test, K4
(4 x 4 x 60 in. under 14 ton axial load), the cycling was performed
with a constant amplitude of 0.9 in. in each direction, and the
reduction in lateral strength was found to be nearly 15% after 8 cycles.
Most of this reduction was produced in the first cycle. This shows
that inelastic deformation is more effective than repeated loading
in reducing the strength.
The same pattern of behaviour is seen in the deterioration
of stiffness. Figure 5-8(b) shows the relative initial stiffness of
the column N4 in different cycles with respect to its initial stiffness
in the first cycle. It is seen that the major part of reduction took
place in the first cycle. It is nearly 50% of the initial value. In
the test K4, the drop in stiffness after 8 cycles is about 40%, most
of which occurred in the first cycle.
The deterioration in strength and stiffness of a reinforced
concrete member under cyclic loading is due to several parameters.
These are: the degree of cracking in the concrete and the inelastic
behaviour of steel, the effectiveness of the bond between the concrete
and steel, the crushing and spalling of the concrete in compression,
the possibility of slip due to lack of effective anchorage, and the
shear deformation and shear cracking at later stages. It can be
seen that as inelastic deformation grows, or repeated loading
continues in a member, on one hand the bond between the concrete
204
and steel is gradually ruined due to high strain or strain reversal
in the steel, and on the other hand the concrete in compression
deteriorates due to propagation of micro-cracking. If the com-
pression strain passes a limit, the concrete is crushed or spalled.
All these phenomena lead to a deterioration in stiffness and strength
of the member in subsequent cycles.
The reversal paths were shown to be highly non-linear in
the load-deformation diagrams of columns. Non-linearity increases
with the extent of damage in the column. A typical reversal path
starts with an initial stiffness slightly less than the corresponding
one in the previous cycle, and as deformation continues in the new
direction, the stiffness gradually drops. After a while, with the
closing of the previous tension cracks at the critical sections, the
stiffness increases again and the column shows resistance in the
new direction. The pattern of behaviour from now on is similar to
the behaviour of a column originally loaded in this direction. The
path eventually joins the previous loading diagram, in this direction,
at a point near to where the last reversal of loading had commenced.
For columns under low axial loads, such as K2, figure 5-7,
when the column is deformed well into the inelastic range, there will
be a sudden drop in stiffness of the column during the process of load
reversal. This occurs because, during the process of unloading,
the previous tension steel which had deformed considerably, now
in compression, may yield before the previous tension cracks are
fully closed. In such a case, the change in the bending moment
resistance at the critical section would be very small with respect
to deformation, and the stiffness would be very low until the tension
cracks are closed and this side of the section starts working in
compression. In some cases where the lateral deflection of the
column is very high, the rate of change of the bending moment
produced by the axial load at the critical section may become equal
205
to, or higher than, the corresponding change in the bending moment
resistance of the critical section. This yields to a null or negative
lateral stiffness for the column. Such a case is seen in K2, figure
5-7, in the reversal paths for the later stages.
A number of conclusions can be derived from these tests
with regard to load-deformation behaviour of a column. These
are summarized as follows:
1) The envelope diagrams in the first and third quadrants are nearly
similar, and are almost the same as the load-deformation of the
column under incremental loading.
2) The reversal paths are non-linear and dependent on the history
of loading. There is, in some cases, a rapid change in the
stiffness of the column at these stages.
3) A reversal path started at any point, joins the envelope diagram
on the opposite side almost at the point where the last reversal
of loading occurred.
4) The initial stiffness of the column in the reversal paths, and
its lateral strength, drop as the inelastic deformation and the number
of cycles increase for the column.
5) The drift in the column increases as the inelastic deformation,
or damage, increases.
6) The energy absorption capacity of the hysteresis loops is
increased with the extent of damage in the column.
5. 3 Idealization of the Force-Deflection Characteristic
In order to study its dynamic behaviour, the column is
idealized as a single-degree-of-freedom system, as shown in
figure 5-1. For the load-deformation characteristic of the column,
an idealized non-linear model is considered. The non-linear model
is to account for the gradual deterioration in stiffness and strength
of the column, and also for its energy absorption capacity. The
basic idea for the model relies on the observations on the tests
206
described earlier. The details of this model are as follows:
5. 3. 1 Envelope Diagram
A bi-linear or a tri-linear characteristic, as shown in
figure 5-3, is chosen for the envelope diagram. Point A on the
tri-linear diagram represents the yielding point of the critical
section, and point B may be considered as the point of maximum
lateral force which, for short columns, is very close to the point
of maximum bending moment capacity of the critical section. The
yield level is assumed at point A and the gradient of line AB is, from
experimental evidence, chosen as 5% of the gradient of OA, the
initial stiffness of the column. The deflection at point B is chosen
to be twice that of point A. The gradient of the falling branch BC is
mainly determined by The axial load and the height of the column. It
can easily be seen that for a column with an elasto-plastic moment-
rotation characteristic for its critical section, the gradient of the
falling branch due to gravity load is -P/hi where P and h are the
axial load and height of the column. For this branch of the diagram,
a negative gradient between 10 and 100% of the initial stiffness of the
column, is considered. This range was observed in the tests des-. cribed ih Chapter 2.
The envelope diagram is assumed to be the same in the first
and third quadrants.
5.3.2 Reversal Path Starting from a Point on the Envelope Diagram
This path is represented by a parabola which is characterized
by three parameters: (1), co-ordinates of the starting point,
(2), co-ordinates of the point where the last reversal of loading took
place on the opposite envelope diagram, and (3), the gradient of the
tangent to the parabola at either the starting or the terminating point.
For most cases, the third parameter is the slope of the
tangent at the starting point. It is taken as the slope of the line
connecting the starting point to the point A or A', depending upon
207
which envelope the starting point is on, figure 5-9. Thus, for a
path starting at point E and travelling towards F, the tangent to the
parabola at this point is EA' and the path is EMF, whereas when
travelling from F towards E, the tangent will be FA and the path
will be FNE. However, the starting slope should not be less than
the gradient of EF itself. Should such a case occur, for example,
travelling from a point very near C' to a point between A and B, the
slope will be the same as the gradient of EF, and therefore the
parabola will become the straight line EF itself.
For a path starting from F' and travelling towards E', it
can be shown that if
SPE' <1 (SPA + SBC) (SBC <0) (5-4)
where S represents the slope of the line, the parabola constructed
according to the above rules will intersect BC at a point E" higher
than E'. If this happens, the parabola is constructed so that BC
will be a tangent at point E', instead of F'A being a tangent at F'.
Therefore, the third rule mentioned above can be summarized as follows: If, SFE (SFA SBC) (5-5)
FA will be the tangent to the parabola; if otherwise, BC will be the
tangent. Should SFA be less than SFE, the path would be the straight line FE.
The choice of the first two rules obviously follows from the
description of the test observations given previously. However, the
third rule is completely arbitrary. Its basis relies on the observed
tendency of the initial slopes of the reversal paths in the tests to con-
verge at a point. The points A and A' are arbitrarily chosen as the
points of convergence. With the above rules, the first reversal path
outside the elastic range is always linear and it passes through either
A or A'. As the column progresses in the inelastic range, the
parabolas deviate more and more from the straight line and,
208
consequently, the hysteresis loop becomes larger with a greater energy dissipation capacity. The experimental and idealized
reversal paths are compared in figure 5-10.
5. 3. 3 Reversal Path Starting in a Loot
There is no experiment which shows the actual behaviour
of a column for reversal paths within a hysteresis loop. However,
by considering the overall behaviour of a loop starting from a point
on the envelope, an idealized pattern can be assumed for the inner
loops. The idealization is here based on the principle that an inner loop joins the original loop on the envelope.
The inner loops are assumed to be parabolic, and are characterized by: (1), the co-ordinates of the starting point, (2), the co-ordinates of the point on the envelope diagram from which
the original loop started, and (3), the gradient of the tangent to the parabola at either the starting point or the terminating point.
For the third rule, consider the reversal path starting at a point L, within the hysteresis loop EMFNE, in figure 5-9(b). The
reversal path passes through L and E and will have a slope at L, parallel to the tangent to the parabola FNE at the starting point F, provided that
Six SFE (5-6)
If otherwise, the reversal path will have the same slope as the parabola
FNE at the terminating point E. It can be shown that, according to this rule, the reversal paths starting at any point below the line FE
and travelling towards E have a tangent at L parallel to the tangent
line at F. Those paths starting at a point above the line FE will
have the same slope as the parabola FNE at E. Those starting on the line FE satisfy both conditions, For this reason, all the reversals
starting at a point on the parabola EMF have the same initial slope,
that is, the same initial stiffness. Physically, this means that for reversals starting on EMF, the initial stiffness does not deteriorate
until the extent of damage is greater than that at F.
209
5.4 Dynamic Analysis
The dynamic response of a single-degree-of-freedom
system under earthquake excitation, as shown in figure 5-1, is
governed by the equation
M (t) + C . + f(x) = - M , E , Z (t) (5-7)
where x(t) is the relative displacement of the mass M with respect
to the groumd, and Z(t) is the ground displacement. 2(t) is there-
fore the earthquake acceleration. E is a constant representing the
amplification of the earthquake acceleration. The damping force
in the system is assumed to be of a viscous type, with C as the
coefficient of damping. The restoring force in the system, f(x), is
the lateral load resistance of the columns supporting the mass.
It includes the effect of gravity load and is characterized by the set
of rules described in the previous section.
Consider the tri-linear or bi-linear characteristic shown
in figure 5-3 for the envelope diagram of the force-displacement
relationship of the system. The natural circular undamped frequency
Go of the system in the elastic range is
G.) = (5-8)
and the natural period of vibration is
T = 2 n (5-9)
where K is the initial stiffness of the system. It includes the effect
of gravity load. It can be shown that to is related to the
ponding natural circular frequency without gravity effect
relationship 2 CU = (.4.) _ g h
where g is the gravity acceleration and h is the height of
It shows that the natural period of vibration is increased
the gravity effect.
corres-
CA) o, by the
(5-10)
the columns.
slightly by
G.)
x X(t ) (t ) x
f F(X) (x) fy (5-12)
X (1 ) + 2 n . X (i) + F(X) - E. Z (t) a y
(5-13)
210
Substituting for C in terms of its critical value in
equation (5-7), results in
(t ) + 2 n . /A) . (t) + . f (x) -E . (t) (5-11)
where n is the critical damping ratio.
In order to consider the parameters affecting the response,
the following transformations are made to the left hand side of this
equation:
. t
where x and f are the deflection and force at the first yield, y y
figure 5-3. With these transformations, the above equation
becomes
where a is the equivalent acceleration of the yield level force,
Y a
Y= f /M. F(X) in the new system of co-ordinates is as shown in
figure 5-4. It is now characterized by the parameters S1 and S in
the case of the tri-linear system, and by S in the bi-linear system.
S1 and S are the relative stiffnesses of the system on the second and
third lines with respect to its initial stiffness. As the reversal paths
are expressed in terms of the co-ordinates of the starting and ter-
minating points as well as the yield point, they are also dependent
on these parameters only.
Equation (5-13) is now non-dimensional and the response
is seen to be dependent on id , n, and E Z (t)/a. as well as the
parameters S1 and S which characterize F(X). For a particular ••
earthquake, with Z (t) as a non-dimensional time varying function
and E as a parameter defining the intensity of the earthquake, it is
seen that the response is influenced by the relative intensity of the
211
earthquake with respect to the yield level of the system, E/a .
Since both the strength and the stiffness in the system
gradually degrade as the relative displacement increases, the
system will fail to resist any further load when the displacement
exceeds a certain level. This level is assumed to be the displace-
ment corresponding to the point on the descending branch of the
envelope diagram where F(X) becomes zero, point C in figure 5-4.
The minimum value of the earthquake amplification factor E, which
causes the relative displacement of the system to exceed Xu, is
referred to as the failure factor.
5.4. 1 Range of Parameters
It was shown that the parameters affecting the response of
a system are T, n, E/ay and, for the tri-linear system, S1 and S,
and for the bi-linear system S. The present study considers the
following ranges of these parameters:
T is varied between 0.1 - 1. 0 sec. and between 1 2 - 2.7 sec.,
with intervals of 0.1 sec., and 0.3 sec. , respectively.
n is taken as zero and 0.02. Considering that the hysteresis loops
are accounted for in the force-deflection characteristic of the column,
no extra dissipation source is in fact necessary, i, e. n = 0.0.
The introduction of the small damping ratio of 0.02 is to account for
any other dissipative force in the system other than the column itself
and to reduce any resonant effect in the initial elastic path of the
system.
ti is taken as 0. lg
S1 is assumed to be 0. 05.
S is varied between -1 and -0.1. This range was observed in the
column tests.
5. 4. 2 Earthquake Ground Motion
Each earthquake ground motion accelerogram has its own
individual characteristics on which several parameters, such as
212
t he distance from the epi centre, the geological properties of the
local ground media, and the mechanism of energy release at the
source, have influence. For this reason, the results obtained
from one earthquake accelerogram record do not represent those
of an average earthquake. Therefore, to have a relatively general
picture of the behaviour of a system under this type of excitation, it
is necessary to analyse the system for several records of earthquakes.
In the analysis in present work, the records of six components
of four strong earthquakes are used.
are summarized below.
The properties of these records
Location & Date Magnitude Duration Component Maximum 2(t), %
EL CENTRO N S 26.3 California 6.5 25-30
Dec. 30, 1934 E W 18.3
EL CENTRO N S 33. 0 California 6. 7 30
May 18, 1940 E W 22. 0
PARKFIELD
California 5. 5 19 N65E Station 2 52. 0
June 27, 1966
KOYNA Longitudinal India 6. 5 11 to axis of 63. 20
Dec. 10, 1967 Koyna Dam
The variation of the ground acceleration and displacement for these
records is shown in figures 5-11 to 5-13. These records have been
digitized in the Seismology Department of Imperial College.
The components of NS EC 40, Parkfield and Koyna have the
greatest accelerations ever recorded during an earthquake, except
for the recent San Fernando earthquake, during which an acceleration
of the order of 1. Og was recorded (77). The record of the Parkfield
213
earthquake was taken at nearly 200 feet away from the slipped
fault (74), and the Koyna one was recorded at a distance of nearly 9 miles from the epicentre (75). These relatively short distances
may be the reason for such high accelerations. The El Centro
records of the 1934 and 1940 earthquakes were taken at distances of 35 and 30 miles from the corresponding epicentres respectively (76).
5.4.3 Method of Solution Equation (5-13) is solved by a step-by-step integration
procedure based on the classical Runge - Kutta method of the fourth-
order (78). The time intervals in the analysis were generally taken
as: T = 0.1 sec. t = 0.005 sec.
T 0.2 - 0.3 sec. t = 0.01 sec.
T 0.4 sec. t = 0.02 sec.
If it was not possible, the time interval between two successive points
in the accelerogram record was used. The accuracy gained by using a time interval shorter than the above values was found to be negligible.
The computation was carried out on a CDC 6600 computer. In the process of integration, the values of the ground motion acceleration
between two successive points in the accelerogram record were found
by a linear interpolation between the values at these points. The main operation in the computer program, other than the
integration, was to select the right path at each step. For this purpose, a check was made on the relative displacement and velocity
at the end of each step. If the displacement exceeded the boundary
of the corresponding path, or the velocity changed its direction, the
exact time of passing the boundary or the zero velocity was found.
The characteristics of the new path were determined and the steps
proceeded along the new path.
214
5.5 Earthquake Spectra
The significant effects of earthquakes, as far as design
engineers are concerned, are the forces and motions they induce
in structures of all types, and the effect of dynamic response on
structural performance. The dynamic response of a structure to earthquake loading depends on the characteristics of both the ground
motion and the structure. From this point of view, an earthquake
ground motion is characterized by the amount of energy it contains
and the way this energy is distributed over the range of frequencies.
This means that the intensity of an earthquake, represented by its
accelerogram record, depends not only on its level of acceleration
but also on its frequency distribution and its duration. An earthquake
of low acceleration and long duration may be more destructive than
an earthquake of high acceleration and short duration.
The dynamic characteristics of a single-degree-of-freedom-
linear system are its natural period of vibration and its amount of
damping. For this reason, the response of such a system to an
earthquake loading depends upon these two parameters, as well as
the properties of the ground motion accelerogram record for this
particular frequency. Thus, a collection of the maximum responses
of systems with different natural periods and different damping ratios
can characterize the intensity of an earthquake over that range of
frequencies and dampings. This is the idea of earthquake response
spectra which was introduced by G. W. Housner (79). A plot of the
maximum value of a function in the response, such as displacement,
velocity, acceleration etc., against the natural period or frequency
of the system for a certain value of damping ratio, is called the
'response spectrum' of that function. The most useful of the
various spectra are the response spectra for relative displacement,
relative velocity and absolute acceleration of the system. The first
indicates the maximum deformation or drift which occurs in the
215
structure, the. second represents the.maxim:urn level of _energy
transmitted to the structure, and finally, the third represents the
maximum force induced into the structure. They are shown by
Sd' Sv' and Sa respectively. It can be shown that these functions
are approximately related by the following relationships (80).
S — 1 S d' v
Sa t:4, . S v (5-14)
where w is the natural circular frequency of the system.
The response spectra for NS El Centro 1940 are shown in
figures 5-14 and 5-17. As an example it can be seen that this
component of earthquake is nearly five times stronger for a structure
with T = 0.5 sec. and n = 0.02, than for a structure with T = L 5 sec.
and the same n, figure 5-17. A comparison of this spectrum with
the corresponding one for N65E Parkfield-2 earthquake, figure 5-20,
shows that the latter earthquake is nearly 1.5 times stronger for a
structure with T = 0.5 sect and n = 0.02 than the former. Housner (76)
defines the area under the velocity spectrum Sv,for the range of
T = 0.1 - 2.5 sec., as a measure of the intensity of the ground
motion, J
0
2.5
SI (n) = .1
Sv (T, n) dT (5-15)
where n is the damping ratio. The SI values for the six records of
earthquakes used in the analysis here, are given for different
damping ratios in Table 5-1. These values will be compared with
the intensity of the records related to the non-linear response of
the structure. The acceleration spectra of these records will be
used later to correlate the non-linear response of the structure with
the corresponding linear response. These spectra are shown in
figures 5-15 to 5-20. They were produced by a step-by-step
integration procedure using the method mentioned previously.
216
5. 6 Analysis of the Response
The main concern in this analysis is to investigate the
parameters affecting the ultimate failure of a system representing
a reinforced concrete column, The system, as described before, has
degrading characteristics with respect to its stiffness and strength,
and naturally cannot withstand an earthquake with an intensity above
a certain level. This level of intensity, in general, depends upon
the reserved energy capacity of the system as well as its dynamic
characteristics. The system, as defined before, is considered to
have failed when its response displacement exceeds the displacement
corresponding to a point on the falling branch of the force displace-
ment envelope diagram, whose force is zero, C in figure 5-3, The
relative earthquake intensity corresponding to this failure limit is
found by increasing the amplification factor, E, in equation (5-13),
incrementally until the prescribed limit is reached. The failure
factors E for systems under different earthquake excitations are
found, and will be related empirically to the elastic spectra of the
corresponding earthquake later in this section.
Another study determines the response of an elastic-perfectly
plastic system with a non-linear reversal path, as used in the present
work, to different earthquake loadings. This may be regarded as
the response of a reinforced concrete frame to an earthquake type of
loading. The ductility requirement of this system is compared with
that of an ordinary elasto-plastic hysteresis system with elastic
reversal paths. Before discussing the results of the above two
topics, some features of the dynamic response of a degrading stiffness system are discussed.
5.6.1 Effect of Stiffness Degradation
The term 'degradation of stiffness' is used in contrast with
systems which preserve their initial stiffness in the reversal
excursions for a relatively long time under dynamic loading, such
as ordinary elasto-plastic hysteresis systems. To illustrate the
217
effect of the degradation of stiffness in a system, its response to
an earthquake type of loading is compared with that of an ordinary
elasto-plastic hysteresis system. The time histories of the dis-
placement response of two systems with initial natural periods of
0.4 and 2.7 sec„ for both types of material, are shown in figures 5-21 and 5-24. The force-displacement history, corresponding to
the degrading stiffness system with T = 0.4 sec., is also shown in figure 5-21.
The most noticeable difference for the systems with T = 0.4 sec., is that the vibration in the degrading stiffness system has
relatively higher amplitudes and lower frequencies. The high
frequency oscillations have been cancelled due to the deterioration
of the system's stiffness which results in a higher natural period of vibration. The yielding in this system is limited to the first
6.0 sec. only, after which the oscillation becomes stabilized within a loop. The elasto-plastic system, however, behaves as a typical harmonic oscillator, in which the centre of oscillation is
gradually shifted and the amplification of amplitudes is limited by
the effect of yielding in the material. In this system, as opposed to the previous one, the yielding occurs in most of the cycles
throughout the time history of the vibration. It seems that the increase in the natural period in the degrading stiffness system,
which reduces the system's sensitivity to earthquake excitation, and a relatively greater energy dissipation capacity in its hysteresis loops, are the main reasons for preventing the system from
yielding further. However, it is interesting to note that the over-
all yielding in both systems is more or less the same.
For systems with T = 2.7 sec., there is not such a distinct
difference between the responses. The degrading stiffness system behaves as a heavily damped oscillator after the first cycle during
which yielding has occurred. The difference between the responses of these systems should be due essentially to their difference in
218
damping, because the change in the natural period in the degrading
stiffness system is not as high as the previous one and, in any case,
it does not significantly affect the sensitivity of the system towards
earthquake excitation for this range of periods. The loss of stiff-
ness in the present case is less than half the initial stiffness,
whereas in the previous case the stiffness dropped to nearly 1/10
of its initial value. The overall yield is the same in both systems.
It will be shown later that the overall yield in both types of system
is appro ximately the same when the initial value of T is above a
certain level.
5.6.2 Failure Analysis
In this analysis, as was described above, a system with
known properties is subjected to an earthquake accelerogram record
amplified by a constant factor. This factor is gradually increased
until the maximum response displacement exceeds the failure limit
of the system. Since, according to equation (5-13), the response
depends upon the relative intensity of the earthquake to the yield
level of the system, this operation is equivalent to reducing the
yield level of the system until it fails under an actual earthquake
accelerogram record, or vice versa. In the analysis here, the
yield level of the system is taken as a = 0. 10g and the variation
of the earthquake amplification factor E is studied with the initial
period of vibration T and the gradient of the falling branch of force-
deflection envelope diagram S for damping ratios of n = 0.0 and
n = 0.02.
To illustrate a typical response of a system with different
S, the displacement-time histories of the response of two bi-linear
systems with T = 0.4 sec. and 2.7 sec., are shown on figures
5-22, 5-24(b) and 5-25. The yield limit in these systems is taken
as twice that recommended by SEAOC (82), equation (5-31). The
system with T = 0.4 sec. is more sensitive to an increase in 1S1
219
than the system with T = 2.7 sec. The former fails with S = -0. 05,
whereas the maximum displacement in the latter for S = -0.5 is
only slightly higher than its response for S = 0. 0. The first few
pulses of the earthquake produced a response in the system's
elastic part which was so severe, and a loss of strength in its
inelastic part which was so rapid, that the system for T = 0.4 sec.
was unable to survive. This pattern of behaviour is seen in most
-cases in the range of low natural period systems.
The earthquake amplification factors corresponding tothe
failure stage for the case of bi-linear and tri-linear systems are
shown in figures 5-26 to 5-37. The accuracy of these factors varies
with T. For T = 0.1 sec., it is generally less than 0.01, for
T 1. 0 sec., it is less than 0.02 and for T > 1.0 sec., it is 0. 05.
As an example, figures 5-28, 5-30 and 5-31 show that a bi-linear
system, with T = 0.6 sec., a = 0.10g, S = -0.2 and n = 0. 02, can
survive earthquakes similar to NS EC 40, Long. Koyna or N65E .
Parkfield-2 with intensities of 0.5, 0.7 and 0.275 times the actual
earthquakes, respectively. Alternatively, in order to have a system
with the above properties which can resist the actual earthquakes 1
mentioned, the yield level in the system. should be raised to 3, and 1 7.7.13 times its gravity load respectively. Figure 5-28 shows that
the system described above with S = -0.1 is 3.2 times stronger than
the same system with S = -0.5, for an earthquake similar to NS EC 40.
The reserved energy capacity of the system with S = -0.1 in the latter
example, defined by the area under its load-deformation envelope
diagram, is nearly 3.6 times that with S = -0.5. This point will be
studied in more detail later.
In most cases, the results corresponding to n = 0. 02 are
not very different from those with n = 0. 0, and generally they are
smoother than the others. As the system under study is damped
by its hysteresis loops, further damping in the form of viscous
220
damping is not necessary. The introduction of a small damping
ratio in the system, such as n = 0.02, is to reduce the effect of any
resonance which may occur in the elastic range of the force-deflection
characteristic. The existence of this damping in the inelastic range,
however, results in overestimation of the resistant capacity of the
system. Comparison of the results for n = 0.0 and n = 0.02 shows
that the overestimation, in general, is not great.
The variation seen in E with respect to S is due to the change
in the ductility and the reserved energy capacity of the system. With
lower I S I , the ductility and the system's reserved energy capacity
are increased, and this naturally results in a greater tolerance for
the system under earthquake loading. The reserved energy capacity
of a system is commonly referred to as the area under its load-
deformation characteristic under monotonic static loading to failure,
which is, in fact, its energy absorption capacity under static loading.
This capacity is greater under dynamic loading due to the dissipation
of energy in the hysteresis loops. According to the above definition
the reserved energy capacity of a bi-linear system, figure 5-3, is
W PI- • q • fy2 (5-16)
and for a tri-linear system, the same figure, is
1 W = K (0.95 + 1.05 q) y22 (5-17) 2
where q is the ductility ratio at the failure stage,
q = xu/xy
To examine the variation of E with the reserved energy
capacity of systems with the same T but different S, the values of
E are reduced proportionally to the square root of the reserved
energy capacity of the corresponding systems. The square root
of this energy is chosen because the maximum level of energy
supplied into a linear system by an earthquake, amplified by E, is
221
a parabolic function of E, equation (5-19). On this basis the values
of E given in figures 5-26 to 5-37 are reduced as follows:
for a bi-linear system, = E /
(5-18)
for a tri-linear system, E = E / / 0.95 + 1.05q
E values are plotted against T for two different earthquake
records in figure 5-38. As seen, the values corresponding to
different S are very close to each other, particularly when T41.0
sec. There is no definite trend in the variation of E with S, but in
general, the systems with lower I S I have greater E values. This
means that these systems have relatively greater energy absorption
capacities than the others. This is because an increase in the
inelastic deformation, which is the case in the systems with lower
I S i , increases the energy dissipation capacity of the hysteresis .
loops quite considerably.
The above figure is typical for all other results. To avoid
repetition, figures 5-39 to 5-41 show the mean value of E and its range
of variation for different S, in the case of bi-linear systems with
n = 0.02. The results corresponding to the same systems with
n = 0.0, and the tri-linear systems with n = 0.02, are also shown
in these figures. The similarity between the results of the bi-linear
and corresponding tri-linear systems is noticeable. The values
belonging to the tri-linear systems, however, are slightly lower
than the bi-linear systems. This is because the averages of E in
the former systems are taken between values corresponding to three
values of S, whereas in the latter systems, the values related to
S = -0.1 are included. This case has relatively greater E values
than the others. These figures show clearly that, in general,
there is a trend of direct proportionality between the earthquake
resistant capacity of a system and its reserved energy capacity.
222
5. 6. 3 Reserved Energy Capacity of a System and its Earthquake
Response
In view of the complexity of the non-linear analysis of
structures for earthquake loading, there has always been a tendency
to correlate the response of a non-linear system to the response of
a relatively similar linear one. The most successful idea which
has been applied, so far, to the response of an ordinary elasto-
plastic hysteresis system, with or without work-hardening effect,
in the range of periods of vibration under consideration here, is
that the maximum level of energy absorbed in an elasto-plastic
system is nearly the same as that absorbed by an elastic system
having the same properties as the elastic part of that system. In
other words, if the maximum displacement responses in elastic and
elasto-plastic systems are represented respectively by points B and
C in figure 5-2(a), then the areas under OB and OAC will be almost
the same. The idea was first introduced by Blume (81) and its
approximate validity has been confirmed by the works of Newmark
and Veletsos (65, 66). This idea will be referred to as the 'energy
rule' hereafter.
The application of the above rule on a degrading stiffness
system will encounter the difficulty of choosing the basic elastic
system on which the comparison is made. An elastic system
similar to the initial elastic part of the degrading stiffness system
is the most straightforward choice, but it may not be the most
representative, since the amount of time the system spends on its
elastic path in the analysis for failure, is relatively short. In
addition, since the system's energy dissipative capacity grows
continuously with the amount of yield it suffers, it is difficult to
assess the correct amount of viscous damping for the basic elastic
system. However, despite all these deficiencies, the basic elastic
system is considered here, in view of its simplicity, to be similar
to the initial elastic part of the degrading stiffness system with the
same damping property.
223
In this section, the earthquake resistant capacity of a
degrading stiffness system, estimated according to the energy rule,
will be compared with its actual resistant capacity found in the non-
linear analysis. The results will show the validity of this rule with
respect to the systems under study and on the basis chosen here.
If Sa is the acceleration response spectrum of the basic
elastic system under a certain earthquake loading, the maximum
level of energy absorbed by this system under the same earthquake,
but amplified by a constant factor Eo' is
Wo = Eo2 . M2 . S: /2K ( 5-19)
where M and K are the mass and stiffness of the system respectively.
If the degrading stiffness system is to withstand the earthquake,
amplified by E0, its reserved energy capacity according to the energy
rule should be
w > w / o (5-20)
Substituting for W from equation (5-16) or (5-17), and for Wo from
equation (5-19), the above equation becomes:
for the bi-linear system Eo ay Sa
for the tri-linear system Eo Ni 0. 95 + 1. 05 q ay Sa
The maximum value of Eo in these relationships is the
maximum relative earthquake intensity that the system can tolerate
before it fails. This value corresponds to the actual value of E
calculated for each system and given in figures 5-26 to 5-37.
If R = E/Eo (5-22)
the energy rule is only valid when R is unity. However, provided
R 1, R may be considered as a factor of safety if the failure limit
is predicted according to the energy rule. Before examining the
actual earthquake spectrum S.., it is interesting to determine under
what acceleration spectrum Sa, R will be unity and the energy rule
(5-21)
applicable.
224
5.6.4 Variation of Acceleration Spectrum, Sa' for R = 1
Using equations (5-21) and (5-22) for R = 1, for the bi-linear
system
Ta = ay / ) (5-.23)
and for the tri-linear system
T = ay /( a f0.95 + 1.05 q
Referring to equation (5-18), Sa will be
= a / a y
)
(5-24)
According to this equation, the diagrams shown in figures
5-39 to 5-41 represent the variation of aY /§
a , or simply the inverse
of §a. The similarity between these diagrams for different earth-
quakes is noticeable. Saa, the variation of a for the mean value of
E. , and §am its variation for the minimum value of E, obtained for
the bi-linear systems with n = 0.02, are plotted by broken lines in
figures 5-15 to 5-20 against T on the actual acceleration spectra for
the earthquakes. Saa represents the spectrum which gives the average
R = 1 in systems with different S. Or in a system, having average S
(between -1.0 and -0.1), the energy rule based on Saa, results in a
response close to the actual response. Tam' on the other hand,
results in R )1 for systems which have S in the range studied here.
It can be considered as an upper bound spectrum for this range of S.
For all earthquakes, Saa follows the same trend as the
actual spectrum, but it is smoother. Its value is generally lower
than the corresponding Sa for n = 0.02, except for values of T (0.4
sec. The close similarity seen, in general, between Sam and Sa
(n = 0. 02), indicates that for values of T > 0.4 sec. , the latter spectrum
can be regarded as a reasonably good upper bound for the prediction
of a system's failure according to the energy rule. However, this
spectrum cannot be used for systems with T 0.4 sec.. Saa is
225
usually closer to Sa corresponding to a higher damping value. This
may be interpreted as the effect of the deterioration in stiffness,
which reduces the sensitivity of the system towards earthquake
loading, and the high energy dissipation capacity in the system.
To examine the similarity between Saa in different records,
the values of Saa are reduced by a factor F proportional to the
intensity of the earthquake. These intensities are calculated on
the basis of the effects of the earthquakes on the non-linear system
under study. The intensity of an earthquake, as defined by Housner
(76), is calculated from equation (5-15). The spectral velocity in
this equation is determined from equation (5-14) and the values of
Saa. The intensities SI, and the corresponding factors F, are
given in columns 6 and 7 of table 5-1, page 233. The components
of El Centro 1934 earthquake show intensities similar to the actual
spectrum for n = 0.05, i.e. SI (0. 05), whereas the intensities in
other earthquakes are closer to their SI (0.20).
The other, more straightforward method, to assess
the relative intensities of these earthquakes is to compare their
corresponding values of amplification factors E, which cause failure
in a system. It is obvious that the more intense an earthquake is,
the less amplification it needs to cause failure in a system. With
NS EC 40 as the basis of the comparison, the values of E belonging
to this earthquake's component have been divided by the corresponding
E in the other earthquakes. The average of these ratios for systems
with the same T and different S are shown in figure 5-42. For
example, according to this figure NS EC 34 is more intense than
NS EC 40 for systems having T) 1.0 sec., and vice versa. The
overall averages of these relative intensities are also given in
table 5-1, column 8, under the heading of F'. Both values of F and
F' are in close agreement.
226
Figure 5-43 shows Saa reduced by factor F for different
earthquakes. Their similarity is noticeable, particularly when
T > 1.0 sec. Most of them show a tendency to vary inversely with
T. The average and maximum values of the reduced Saa are shown in figure 5-44. As shown in the figure, both values fit very well
with a curve representing the ratio of (1/T) in the range T> 0. 3 sec. -
T (2. 7 sec. The values corresponding to T = 0.1 and 0.2 sec.
are not included in the process of selecting the smoothest curves,
because they show a trend dissimilar to the others. The smooth
curve represented by
Ka =0.4 gF/T
0.3 4T < 2.7 sec. (5-25)
is chosen here as the 'smooth acceleration spectrum' to be used in
the prediction of the failure limit of a degrading stiffness system,
according to the energy rule. For values of 0.1 ‘T <0.3 sec., Sa is assumed to have the same value as for T = 0.3 sec. This spectrum
is a reasonably good upper bound as far as these earthquakes are
concerned. The actual smooth elastic acceleration spectrum for
5% critical damping, suggested by Housner (76), has been
represented by Blume (5) as 3
Sa = 0.194 g Fb / T4".
(5-26)
where Fb is a constant, varied for different earthquakes. Its value
for NS EC 40 is 1.83. The above two spectra for NS EC 40 are
shown in figure 5-45. It can be seen that the spectrum represented
by equation (5-25) gives a relatively higher value for systems with
low values of T. Apart from the effect of the damping ratio, this
is due to the fact that degrading stiffness systems show greater
sensitivity to earthquake loading than is expected for this range of T.
The validity of this high value for the spectrum for the above range
can be seen in the values of R which will now be discussed.
227
On the basis of the acceleration spectrum given by equation
(5-25), the values of Eo and R are calculated according to equations
(5-21) and (5-22) for the case of a bi-linear system with n = 0.02.
Figures 5-46 and 5-47 give the average values of R and its range of
variation for each T. With a few exceptions, R is seen to be generally
greater than unity. The range of variation of the average of R is
between 1 and 2 in most cases. It can be observed that in spite of
quite a high value of Sa for the T < 0.3 sec., relative to the actual
smooth spectrum given by equation (5-26), R is quite close to unity
in this range. R is relatively high in the range of T 4 0.5 sec. in
the case of NS EC 34 and, is less than unity for T = 0.1 and 0.2 sec.
in the case of Koyna earthquake. For T in the range of 0.4 - 1.0
sec. , in the NS EC 40 and Parkfield earthquakes, the minimum R
is less than unity, which indicates that Sa is relatively low for this
range. The average values of Ft for the tri-linear system are also
shown in figures 5-46 and 5-47. They are very close to the corres-
ponding values in the bi-linear system. The overall average of
mean value of R throughout the range of 0.1 (T (2.7 sec. is around
1. 6 in different earthquakes.
5.6.5 R Based on the Actual Elastic Spectrum
In the above discussion, it was shown that 7anithe maximum
Sa necessary to give R = 1, was in general less than the actual Sa (n = 0.02), except for T 4 0.4 sec. This shows that the actual Sa
(n = 0. 02) can be regarded as an upper bound for the prediction of
a system's failure limit, according to the energy rule. The values
of Ft found on this basis are discussed here. Figures 5-48 to 5-50
show the average values of R and its range of variation for the bi-
linear system with n = 0.02. The average of R for the same system,
but with n = O. 0 and for the case of a tri-linear system with n = 0.02
are also shown in these figures. The peaks and valleys seen in R
for n = 0.0 are due to the corresponding peaks and valleys seen in
the spectrum for n = 0.0. The R values corresponding to n = 0.02
228
are generally greater than unity, except for T < 0.4 sec. for all
the earthquakes, and on a few other occasions. For the case of
T 4 0 . 4 sec., R varies between 0.5 - 1.
Equations (5-21) and (5-22) can be combined to give
ay E ( R (5-27)
a This shows that, in order to have E = 1 and thereby make the system
safe under the actual earthquake without amplification, the following
relationship should be satisfied:
Sa
P g
where P = M . g is the gravity load, or in the case of a column, its
axial load. For the case of a tri-linear system, CT in this relation-
ship should be replaced by V0.95 + 1.05 q. This relationship shows
clearly the relation between the axial load, yield level, the ductility
in a column and, to some extent, the intensity of the earthquake which
the system should resist. The overall average of the mean value
of R throughout the range of 0.1 ‘T (3.7 sec. is around 1.1 - 1. 8
in different earthquakes.
Finally, the effect of higher dampings on the variation of R
is shown in figure 5-29 by the values of E obtained for a bi-linear
system with n = 0.05 under the excitation of EW EC 40. This com-
ponent of earthquake was chosen arbitrarily. The corresponding
values of R are plotted against T in figure 5-51(a).They are relatively
lower, and also smoother, than those corresponding to the n = 0,02
case. For the lower damping cases, the peaks are cancelled by the
smoothness of the spectrum itself. R is in general greater than
unity for T) 0.4 sec., and the overall average of the mean value of
R is about 1. 4.
(5-28)
229
The conclusion drawn from the above discussion is that
the safety of a degrading stiffness system may, in general, be
predicted by the energy rule, using the actual spectrum of the
earthquake provided that T > 0.4 sec. If the actual elastic spectrum is not available, the smooth spectrum given by equation (5-25) may be used. The safety factor inherited by this prediction will be the value of R, or R, given in figures 5-46 to 5-50.
5.6.6 Ductility Requirement for a Degrading Stiffness System with S = 0.0
In this section a study is made of the response of a degrading
stiffness system with an elastic-perfectly plastic load-deformation envelope diagram (S = 0) to an earthquake loading, figure 5-51(b). As stated previously, this system can represent the behaviour of
a reinforced concrete frame under repeated loading. The main purpose of the study is to investigate the amount of inelastic defor-mation occurring in such a system, and to examine the applicability
of the energy rule in this respect. For this purpose, the required ductility of a system found in the analysis is compared with those
obtained by applying the energy rule, with the smooth spectrum
given in equation (5-25), and with the actual elastic spectrum. In addition, the ductility requirement of this system is compared with
that of an ordinary elasto-plastic hysteresis system.
In order to have a more representative system for actual
structures, the yield level of the system is chosen according to the SEAOC recommendation (82). The total lateral load to be con-sidered for the earthquake design of building, according to this recommendation, is
V c . w (5-29) where K is a constant whose value varies between 0.67 - 1.33,
according to the type of framing of the building. In this analysis
its value is chosen as K = 1.0. W is the total dead load of the
structure and C is a coefficient depending on the natural period of
230
t he structure, T. Its value is given by
0.05 3 / - T in sec. (5-30) v T
Assuming the yield level of the structure to be twice the value of
V given by the above equation, the equivalent acceleration for this
yield level, considered in the present analysis, is
O. lOg a - Y T
Figures 5-52(a) to 5-57(a) show, for different earthquakes,
the required ductility, q, for a system with the above yield level
and a damping ratio of n = 0.02. The corresponding ductility for
a similar ordinary elasto-plastic system is also given in these
figures. The relatively smooth variation of q with T for the case
of the degrading stiffness system, is interesting. The trend of
variation is the same for almost all the earthquakes. For low
period systems, q is relatively high, but it gradually decreases
as the initial period of vibration increases.
Under the components of El Centro 1934 and Koyna earth-
quakes, q is generally less than 5 for systems with T > 0.7 sec.
and it varies between 5 and 10 for systems with T 4 0.7 sec.,
except for systems with T = 0.1 and T = 0.2 sec. under Koyna
earthquake for which q is 29 and 12 respectively (q = 17 for the
elasto-plastic system with T = 0.1 sec.). For the case of the
El Centro 1940 earthquake components, q is relatively high for
the low period system. It is generally between 10 and 20 for
systems with T ( 0.5 sec., except for NS EC 40 when q is about
30 for T = 0.1 and T = 0.2 sec. systems. The Parkfield earth-
quake results show exceptionally high q, particularly in the low
period systems. This record of the earthquake component is
itself rather exceptional, since it was recorded very close to the
slipped fault. The record shows, figure 5-13, that the major
phase of the excitation is a single large displacement pulse of an
(5-31)
231
amplitude of nearly 10.5 in. and duration of 1.5 sec. Such a
pulse type of earthquake has rarely been recorded in the past.
A similar type of excitation was recorded for the Port Hueneme
earthquake 1957 (83). However, the high value of q for this
earthquake indicates that the above yield level is relatively low
for a structure designed for this type of earthquake.
In both degrading stiffness and elasto-plastic systems
q is more or less the same, except for systems with T ( 0.6 sec.,
where the former type of material shows a higher ductility require-
ment. A plot of the ratio of the values in the two systems, q /fie,
is shown in figures 5-52(b) to 5-57(b). q and qe correspond to the
required ductility in the degrading stiffness and the elasto-plastic
systems. This ratio is never less than 0.5 or greater than 2.5,
and in most cases it is very close to unity. The results, in general,
agree with the results from a similar analysis by Clough (12) on a
degrading stiffness system characterized by the force-deflection
diagram shown in figure 5-2(b).
The application of the energy rule to the system under study
gives the ductility requirement as
qo = 0.5 + 1 ay (5 -32) S
In this equation, the value of Sa is found from the elastic spectrum,
figures 5-15 to 5-20, or calculated according to the smooth spectrum
represented by equation (5-25). Figures 5-52(b) to 5-57(b) show
the variation of R and R with T, where R and ft are defined by
R = qoiq and -ft = q0/q (5-33)
for both cases of Sa. When the actual elastic spectrum is considered,
the variation of R is not smooth, but has a pattern similar to the
spectrum itself. Its value is generally greater than unity for the
systems with T 0.4 sec. Et is usually between 1 and 4 for these
systems, but on a few occasions it is greater than 4 and sometimes
232
as high as 9. 0 ( under the Parkfield earthquake, R is 7 and 8.7
for systems with T = 0.6 and 0.7 sec. respectively ). For systems
with T ( 0.4 sec., R is less than unity, and in some cases it becomes
very small.
When the smooth spectrum is used in the calculation of q0,
the variation of 11. is relatively smoother, and R is greater than
unity for all cases. Its value, in general, is between 1 and 3.
However, for the case of El Centro 1934 earthquake components
its value is greater than 3 for low period systems. These results
show that the prediction of q, according to the energy rule and
based on the smooth spectrum, is more realistic than that found
by using the actual elastic spectrum.
R or R, as defined by equation (5-33), has the same
meaning as in the previous section, i. e. it can be regarded as a
safety factor if the ductility requirement is predicted according to
the energy rule. A general conclusion cannot be drawn with regard
to its value. However, these results do show that if the actual
spectrum is used in the calculation, its value is generally greater
than unity for structures with T) 0.4 sec. If the smooth spectrum
is used in the calculation, R is greater than unity, and its overall
average in the range of 0. 1 T 2.7 sec., is around 1.7 - 2.3
in different earthquakes.
233
Table 5-1 Earthquakes Intensities F see. )• sec.
1 2 3 4 5 6 7
Earthquake Components SI(0.0) SI(0. 02) S1(0.05) S1(0.20) SI(0.02) F 11'
NS EC 1934 6.10 4.25 3.95 2.75 4.00 1.27 1.32
EW EC 1934 4.05 2.95 2.00 1.25 2.10 0.67 0.71
NS EC 1940 ' 8.30 5.35 4.60 3.00 3.15 1.00 1.00
EW EC 1940 7.50 4.30 3.60 2.25 2.80 0.89 0.92
Long. Koyna 1967
4.60 3.80 3.40 2.25 2.35 0.75 0.81
N65E Parkfield-2 1966
11.80 8.60 8.00 5.60 5.05 1.60 1.66
f (x)
X
9 3 ,1
0 —777/7?//1/7717777/77777777777/7i z
Figure 5-1. Single-Degree-of-Freedom System.
Figure 5 - 2 Elasto-Plasiie and Degrading Stiffness Systems (1.2)
Figure 5-3 Bi-Linear and Tri-Linear Envelope Diagrains
1F(X) 1.051_- 1.0
1
S =-0.0r■
2 Xur. q
1.0
Figure 5-4 Non-Dimensional 131-Linear and Tri-Linear Envelope Diagrams
1,224 P FTon N2 0.6 4 x 4 x 60 in.
A s = 4 x 1/4 in.
0..4 = 1/8" @ 4 in.
s P = 10 ton
2.5
Figure 5-5 Force-Deflection Diagn-m for Column N2 under Cyclic Loading
N4
4 x 4 x 60 in.
As = 4 x 1/ 4"
A" = 1/8" @ 4 in.
P = 10 ton
Figure 5-6 Force-Deflection Diagram for Column N4 under Cyclic Loading
K2
4 x x GO in.
As
= 4 x 1/ 4"
A" = / 8" @ 4 in.
FTon
2Din.
Figure 5-7 Force-Deflection Diagram for Column K2 under Cyclic Loading
1st half cycle,. (stiffress)1 = 1.65 ton/in. 2nd half cycle, (stiffness)1
1.0 ton/in. 100
80
60 Perc
enta
ge
100 1st half cycle (11max. = 0.54 ton)
2nd half cycle (F' = 0.57 I on) lmax.
80
40
20
2nd .3rd 4th 5th
1.25 1.5 1.75 2.0
(a) Variation of Strength
0 1st
1 nth cycle amplitude, in.
1st 2nd 3rd 4th 5th nth cycle
1 1.25 1.5 1.75 2.0 amplitude, in.
(b) Variation of Initial Stiffness
Figure 5-8 'Variation of Strength and Initial Stiffness in Coiwun N4
0
2:.1!)
Figure 5-9 Reversal Paths in Force-Deflection Diagram
Envelope Diagram
Actual Path
/Idealized Path Details as figure 5-6
Figure 5-10 Idealized Reversal Paths and the Actual Paths
.EW. ELCENTRO 30.12.1934
1— C) CD
L.;
c.) a:
CD a. C)
CD
cr) •
TIME SEC.
241
1? \
1 1 1 , 111111
h 41 1 ,L ,Hi 1Th ) 111 1i, 1 1 .1 .1 111 , ,
(1 • 00 •!.1:-.:5", ' - ' ..1 ' • • '11. • Vik „1, \ r • '1'.'10;I:V7 1 • t. 11 •T!
11 H (i I
10 12 .00 10 .0C . •r
7:1, 0 • 00
217— CD 1-1 CD 4— Ln
_J CL CD cf) •
01
TIME SFr_
NS. ELCENTRO 30.12.1934
c-D
c1)
CD
C)
_ C2) r. CLI1
CD
Cr!.
C) LJ° C.) CD
C)
c7-••J
. 0 .00 5.0) il.O0 15-00 20.00 25.00
SD
LL j
.L.Li IT)
Li
1--.1 •
CD
Firm.° 5-11 Ground Acceleration and Displacement. El Centro, 1934 EQ.
'-"' • - — I-- •
LU C)
CD CL
C C fl .
z CD CD
_ I-i-) :7 7 LL; L.)
_J CL. C.D
CD cr-.,
r 12.00 10.00 24.00 30.00
TIME SEC NS. ELCENTRO 18.5.1940
00
9 42
C)
-.I I '•, , .;;;.1 I ';It r I ,
•/1,.;.!.., 7A I/ I 1 )10/7-CIIA 1 "11
".
' t 11 f ri) 1 ■
1 i \
4__ , , 0 . i'!11 ,Aiki 4 k_,
7.7J-T..4. 1 .
C. CD
C)
CD
LT) C.) ,
F-CE
Lii C)
LL C)
LE
_
C.D CD
ii
7,7 6j.
L
-
)
-J a-
• L".1) • 0 00 12 • 00 lq 0 24.00 730 0 C) L
C) 1
EW. ELCENTRO 18.5.1940 TIME SEC
Figure 5-12 Ground Acceleration and Di splacem-ent, El Centro 1940 EQ.
- 16.00 20.00
CD
CD, E-1(3") • 0 0 \‘
Opi4.11.7)1ilki-,64!
CL-
CD LCD
° C.) LD
TIME ;.:; C
243
C
C7)
"---. CD i -
- f.)
..L.-) .. .; L...., ri n
i i i.., .1. , 1,,1.1 ,;:i i1.,.:i'l .1,;1 .,.!.1 ,i '1 .!' ': ' ryl c.; 1.
;
, • ' ,, ) , .,•• :J.; t.. ; .) l 1: I : t".: 1 1 i „:., ,--)ir. , _ ,:-!...:1 ;4,1 P.. ,..:,,-111: 6 i\ I -.., 1: !.1' -,■ ....
.ri -;.if.•.y -.,,', -.•.,t ,,,i-e....,,11-1,-,k,.,., ';',./11.,..L, .14 .411.1.j1 4; p,1 )5 0 1 A. ' ') .. (in
cl--_ Lu
i,
-̀1- F;-: --
-z
1— C) ,
C
2.00 4.00 6,00 0.00 1.LLOO
TIME GEC-
LONG. KOMP 10,12.67
• 0.00 4.00 8.00 12.00 16.00 20.00 77I j
c.)
CD
j
C)
72.
CI_ CD Li') •
LC
N55E PHRKFIELD-2 276.66
Figure 5-13 Ground Acceleration and Displacement, Long. Koyna and NOSE Parkfield-2 Earthquakes
1.5 2 0 2.5 3.0 Natural period, T, see.
05 10 0
Dis
pla
cem
ent
15
er
10 U
20
0 0.5 Natural period, T, sec.
Figure 5-14 Velocity and Displacement Response Spectra for El Centro 1040 Earthquake (N-S Component)
3.0 2.5 2.0 1.5 1.0
80
n.0.0
0.02
0.05
0.20
.5 100 V
O 4, cC
75 0.)
C.) C.) <'■
50
A V 0 ra g e and Maximum S S S (Section 5. 6. 4) a aa am
175
150
0 05 1.0 15 20 2.5 30 Natural period, T, Sec.
Figure 5-15 Acceleration 'Response Specs.rum for N-S El Centro 1934 EQ.
25
Average and Alaximum Sa: Saa Sam (Section 5. G. 4)
2-1G
1igure 5-16 Acceleration Response Spectrum for E-W El Centro 1934 EQ.
0.5 10 1 5 2 0 2.5 3.0 Natural period, T, sec.
225 A
ccel
era
tio
n S
pec
trum
,
125
100
2.]
A \• e. and Ni ax .
200
175
bL 150
(r)
75
50
25
(Section 5. G. -J) a an*. am
256
0 0.5 1.0 1.5 2.0 2.5 3.0 nturn]. period, T, sec.
Figure 5-17 Acceleration Ilesponse Spectrum for N-S El Centro 1940 EQ.
n=0.0— 0.02 0.05
— 0.20
2.18
Ave rac, and mum S: , (Section 5. G. 4) a isant
0 0.5 10 1.5 20 25 3.0 Natural period, 'I', sec.
Figure 5-18 Acceleration nesponse Spectrum for E-W El Centro lI)4 0 EQ.
423.58 225
200
2-J9
Ave. and Max. Sa: Saa, Sam (Section 5. 6. 4.) ----
175
"150
U)
125
a,
O
c6100 N a) U U
75
50
25
0 0 5 1.0 1.5 . 2.0 2.5 3.0 Natural period, T, sec.
Figure 5-1.9 Acceleration Response Spectrum for Long. lio,yna EQ.
„Average and ..Viaiinum . (Sec 1. ion 5. 4) 11 CI 11)
225
200
175
150 CA
E 125 7_1
100
75
50
25
0 05 10 15 20 25 30 Natural period, '1', see.
Figure 5-.20 Acceleration Response Spectrum for Nii5E Parkfield- 2 Earl hquake.
9 r;
250, 230
n=0.0 0.02 0.05 0.2g.
N
Displacement, in. Yield Displacement
p
0
0
Time, sec.
251
0.0
2.00 3.0
Displacement, in.
Figure 5-21 Response of a Degrading Stiffness System to NS EC 40 Earthquake, T = O. 4 sec. , n = O. 02, S = O. 0
CD
co _CD
-3.00 -2 .00
-12.00 -8.00 -4 .00/, ,7,,,,c101 00 4.00 8.00 12.0
10.00 15.00 20.00 25.00 30.0 _1 I I \_1
Time, sec.
Lat
eral
Fo
rce/
Mas
s,- %
g-
Ni
O CD
Displacement, in.
Figure 5-22 Response of a Bi-Linear System to NS EC 40 Earthquake, T = 2. 7 sec. , n = 0. 02, S = -0.5, E = 1.0
952
Displacement, in. Yield Displacement
253
Time, sec.
Dis
pla
cem
ent,
in
.
o
Yield Displacement
'
- 00 -2.00 -1.00 1.00 2.00.
Displacement, in.
Lat
eral F
orc
e/M
ass,
'70
g
Figure 5-23 Response of a Tri-Linear System to EIV EC 34 Earthquake, T = 0.6 see. , n = 0.0, S -0.2, E =
25.0' 30.0 5.0 10.0 15.0 20.0 0
. Time, sec.
C)
Yield Displacement
00 15.00 20.00 25.00 30.0
c\J 7— (c) Elasto-Plastic System, T = 2. 7 sec., n = O. 02, S = O. 0
2541.
..l!isplacen!cr!l, in. Elastic Range
(a) Elasto-Plastic System, T = 0. 4 sec., n = O. 02, S = O. 0 .
c\J (b) Degrading Stiffness System, T = 2. 7 sec., n = 0.02, S = 0.0
Figure 5-24 Responses of Elasto-Plastic and Degrading Stiffness Systems to NS EC 40 Earthquake.
o
o
~-I
o
(1;_
I
o
CD I
lJ) .
Lf)
.-1
'<;j
I
') ~... r ~ ;.., .) \)
DisplacCllH":nL, in.
L '~{ielcl Displ~cenl Pllt
Tilne, sec.
(a) S = O. 0
(b) S =-0.025
5.00 10.00 15.00 20.00 25.00 30.0 , , I I I
(c) S = -0.05 "
"lyjgHre 5.;..25 Response 01 a Bl-Linl~ar Systenl"to NS EC"40 Earthquake" T = 0.4 sec., n == 0.02, E = 1.0.
J
1-035
2
0
cd c..) 0.50
0.25
o- r.
ct 0.00
1.25
1.00
%
/ /
/
-P-NT
ry
/
/ /
/ //
/
//
/ //
// /
S:-.:--0.1 /
/
. t
- 0 5 V -- .
/
/ /
/ i
N N N
7" ..7
/
l
-- -
Bi-Linear f = 0. 1 W
V n = 0..0 n = 0. 02 •
System, fig. 5-3(a)
----
0.5 1.0 1.5 2.0 2.5 3.0
1.5
2.0
1.0
0.5
2.5
0.0
/ 7S=-0.2 7 ,,,,
,' i
_ -../
/ /
/
/--- / P'''
-:-.-
77 .7 7
, ,-- .-.7 /1 \
\ /
/
/ -----7- z S=.-1.0
2.0
1.5
1.0
0.5
0.0
0.75
0.50
0.2
Natural period, T, see. Figure 5-26 Failure Faci or for a Hi-Linear System under NS EC 34 EQ.
0.00 05 1.0 15
2 0 25
3.0
2.)
4 .5
4.0
3.5 [LI
0
3-0
ce
a)
2.0
lW
;-4
1.
0.
Ili-Linear f = (). 1 \\I Y n = 0.0
n = 0.02
Sysi em. fig. 5-3(a)
/
/ /
/---- /
/ /
/
s=-0.1
1
/S.:-.0.2
-----
/ /
/ /
/ /
/ /
/ . /
\ \
\
--
V Z___
/ /
-----
/ /
/ /
/ /
/
/
/ /
/S=-0.5
, -' \
/ //
// /
/ / \ ______
/
7
/ /
/
/ /
Sr.-1.0
/
/ /
/ ii' r,
I I
/ _ _
---/
/
/ 1
/
7
/ ' i'
,
_I - ....
0.0 0.5 1 0 1.5 2.0 2.5 3.0 Natural period, T, sec.
Figure 5-27 Failure Factor for a 13i- Lihear System under E\V EC 34 :I.:Q.
Bi - linear fig. 5-3(a)
System, II I
I = O. 1 \V Y
n = 0.0 n = 0.02
i I I
1
SL---0.1
/
/ /
I I i
/
) / /
. /
/ z r
/ /
/
/ /
/ S=-0.2
/
/
/ //
/
/ /
/
/ /
/
/ /
1
/ /
...-' —. ..."
/ /
1/ / /
/ /
/
/ /
/ ./
/ /
/ Sr.-0.5
■
•
0
0
.5
.0
.5
.0
.5
2.0
1.5
1.0
0.5
0.0
2.0
1 . 0
0.5
0.2
Failu
re
0.7
Figure 5-.28 Failure Factor for a Bi- Linear System under NS EC 40 EQ.
0.00 05 10 15 2 0 25 30 Natural period, T, sec.
O
•
0-75
c
▪
a
4_, 0.50 r-1
▪ 0.25 /
2.03 n.0.0 n=0.02 n=0.05
1.75
1.50
1.25
S=-01
5
4
3
1.00
—7
Fa
ilu
re
2
/ Bi- Linear System, fig. 5-3(a)
f =0.1W
05
10 15
, , , „ ,
„- ,- r - ....- ...- ...-
, Sr--0.2
r r ,
/
, , , ir --/
•
Iie/
r i
•
...- ....._ / . ..... „ ,...
,..*
......
------ - - - -- -------- - --- -- - " ---::: Sr- -1.0
2›:"/
• •
./ -1
- /-....."
t
.-:-.- -/ -2./3-----
/ ,- .,,../ -
2.0 0
2.5 3.0
1.25
1.0
0.75
0.5
0.2
3
2
1
0
— 7
/
/
/ /
/
./
I' I
/
0.00 05 1.0 1.5 20 2.5 Natural period, T, sec.
Figure 5-29 Failure Factor for a Ili-Linear System under EW EC 40 EQ.
30
Fa
ilu
re
0.75
/". I
J /
r /
I
/
I
}3i-Linear fy = 0.1 W
11 = 0. 0 n = 0. P2
System, fig. 5-3(a)
/ /
/ /
/ /
/
1 /
/
/ /
//
/ / Sr..-0.1
/
/ / S.-----0.2 /
/ I I I i
r i
1 1
7/
i i
/ /
/
/ //
il /
/
/ /
/ / Ii
i i
, /
1
) i
/
/
/ 7 –7—
1 I I I / il
, V ......
--'--St-0.5 S=-1.0
II/ /
7
3
6
5
1
2
4
0
2.00
1.75
0 1.5
1.25
cd
1.0
0.50
0.25
2N0
0.00 05 1.0 1.5 20 25 3.0 Nat Livid period„ sec.
Figure 5-30 Failure Factor for a Hi-Linear System under Long. Koyna Earthquake
? ( ;1
4.0 1.0
131-Linear System, fig. 5-3(a)
f = 0.1 11T
n = 0.0 n = 0.02
/ I
/ 0.2 3.5
/
0.9
0
0 •
•
2 0.7 <11
0.6
*-3
0.5
cd 0.6
0.3
0.2
0.1
3.0 /
25
2.0
1.5
1.0
0.5
/
0.0 2.0 2.5 3.0
Natural period, T, sec. 0.0 0.5 1.0 1.5
Figure 5-31 Failure Factor for a 131- Linear System under Parkfield-2 Earthquake
:2 ;
0.00 05 1.0 15 20 25 30
Ear
thq
uake
Failur
e
0.50
0.75
0.25
0.00
0.75
0.50
1.0
0.25 0.5
1.0
r
--, / Sr:-.0.5 (
7s.
/
/ --- ---
V,"
/
i / Tri-Linear fig. 5-3(b) f = 0.1 NV
n = 0.0 n = 0.02
System,
/
/
/-----v" /
/
_____., ..---------
7 V '' 7
r7
/
7 V
,...--.
,---- ..----."..'. ._._
7 / S=-0.2
/ S:1-1.0
i
i
/ \ /
/ /
/
-....../ /
\i ..-
/
05 1.0 15 20 2.5 30
1.0
2.0
1.5
0.5
0.0
0.0
.5
.0
.0
Natural period, T, sec.
Figure 5-32 I.■'ailure Factor for a Tri-Linear System under NS EC 34 1-,;Q„.
I
T - Linea r ■,- st ern, fir,. 5-3(10
y n O. 0
= 0.02
!:! 1 ;:-1
2.0
1.
1.
3
2
1
1.0 2 0
cd
1 0.5
2.5
0.0
/
0 30
I /
• 1 /
/
-/ / / I
-‘
/ / /
/N\
_ z 7 r/
/
/ /
/
/ /
7y
/ /
/S=.-0.2
7S=-1.0
_I
/ I
I /
/ /
/ —I
/
1
I
/
------/ / /
/ /
/ /- / i
/
I
0.5 1.0 1.5 20 25
5
4
3
2
1
0
2.0
1.5
1.0
0.5
0.0 0.5 10 15 20 25 30 Natural period, 'IT, :,ee.
Figure 5-33 Failure 'F'ael or f0/. a Tri-Linear System under EW .1;',C 34 :EQ.
cr fa
1.2 cj
0.75
0.5
0.25
2 (L)
0
0
r—i
(2.)
. 1.00
0.75
0.50
0.25
0.00
/
/
/
stein,
3
0
Tri-Linear fig. 5-3(b)
/
f = 0.1 W
n = 0.0 n = 0.02
05 10 1.5 2.0 2.5 30
1
5
4
2
3
' /
. /
/ /
/ .
/ /
/ /
/ / _.-
/
/ / z S--:-.0.2
1 1 / /
I / / /
-.,
---s=-1.o
/ s' - -.1 <.--- --‘
K —1
I
/
/ /
---/ /
/
— / 7 / ./
/ /
c•
— Z-
0.00 1.0 15 20 25 3.0 Natural period, T, see.
Figure 5-34 Failure Fad or for a Tri-Ijnear System under NS EC 40 EQ.
0.75
Ear
th
qu
ake
1.2
Fa
ilu
re 1.0
2(L)
2.0
0.5
0.2
,
I i r
( I A
Sr.-, 0.2
1 1 I
r r r
/ /
I I
r r
. / /
/ /
i
/
I /
r / 7
/
...' 7 _.--.-- --
' --
S=-0.5
S=.-1.0
I /
I
-I
r /
I /1 I / .
•
'Fri-Linear System, fig. 5-3(1)) f = 0.1 W y
n = 0.0 n :-- O. 02 -----
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0 0.00 0 5 1.0 1.5 2.0 2.5 3.0
Natural period, '1', sec.
Figure 5-35 Failure Factor for a 'Fri- T.inear System under E\V EC 40 EQ.
2 (i
0.0 05 1.0 15 20 25 30
Natural period, 'I', see. Pip. ure 5-30 Failure Vactor for a 'Fri- Linea v Sysiern
under Lon;,. 1.. I) Earthquake
1 5 10 2.0 2.5 30 05
2.0
1.5
C) 0.0
2.5
1.5
1.0
.
/ /
/ / /
/ /S r. O. 2
/
,
( I I
,..•
/ / i
) 1
/ /
,---.
/ i /
/I
/
/ t- ---
•
/ /
------:77-.--1.0
/ /
,,....------- --- -------- ....-
.
/ ...-
/
5
4
3
2
1
0
It
3
2
1
0
-S=10.5
Tri-Linear System, fig. 5-3(b) f = 0.1 W y n = 0. 0 n = 0. 02
2.5 3.0 7O 1.5 1.0
Failure
0.6
0.5
15 . 0.5 1.0 0.0
'Fri-Linear fig. 5-3(1)) 1 - 0.1.7 .Y n = 0.0 n = 0.02
:-'yslein,
/ /
I
----- /S.-:-.0.5 / i
\-N
I I
1 1 /./
/ /
/
\
..„, ----
,-- .
05
0.5
0.4
0.3
0.2
0.1
0.0
0.4
0.2
0.3
0.1
.
, i
/
//
/ /
/ /
/ /
/
S.-. -0.2
/ 1
/ /
/ ) /
/ / /
,
/
/
/ /
/ /
/ |
.
/ /
/ /
...--
..." "Sr.-1.0
\ / ,./
/ /
/ J
..--.'" ..- ^~~'
--- ' \
0 0 2.0 25
Natural period, '1.', see.
.1.4'iguro 5-37 Failure Fael or for a Tri- Linear System under N-65P, rkfield-2 Earthquake
2.0
1-5
1.0
0.5
0.0
0
2.5
2.0
1.5
1.0
0.5
Natural period, T, see.
Figure 5-38 Variation of P. with 'I' and S in a Bi--Linear System (n = 0. 02)
0.0 05 10 15 2.0 2.5 3.0
1.5 1.0 0.5 0.0 2.0 2.5
3.0
N-S El Centro 1.934
,--
- l'
•;"
, .-.,----
.7 ...."*. ...-- ...,,, ..... ,...-• ."
-.....:-.."--- • ,
/-
--:--1 ______\■,,
. ,-....". ...__ „, , %,
, , _
sr,-0.1
Sr.- 0.2
S.----- 0.5
S.:--- 1.0
— --- --- ----------
, -- ":..1-3->-:.-7 , %, v ,
N-S El Centro 1940
/
......- •
•
•-•- __-
4'. ..../ 4°
••". . / ...,.../ / •-•- /
e'.•• / ....•••
/ , '
..., .....•• /... .r. .....-•""''
/..../. ____ ......rf•••"'
..•
/ .....••
•••;.•• . •••".././- //.../
/ 1 / •
IW
cc ri
I r,'.
0.8
0.6
0.4
0.
2.0
1.5
1.0
0.
1 0 25 05 15 3.0 2.0
Mean value, bi-linear system, n = 0. 02 Max. and Alin, values, hi-linear ern, n = 0. 02 1VJea» value, hi-linear system, n = 0.0 Mean value, tri-linear system, n = 0. 02 0
1.0
co
In
0.5 0
N-S El Centro 1934 .
---- . ...
2...........-77-,..--.
,
, \....,
....- ------ -.:_- ---- ..----- •---1- . .
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Natural period, T, see.
Figure 5-39 Variation of 177, with T in different systems.
2.(i9
• E-W El Centro 1934
,
,
, .
-- ,
, I l. ,
, •
,
, ,, 4-; .
....
-- ---- --- ------- ._+,--
• „,
0.5
IW
1.0
1.5 25 10 2.0 30 05
N-S El. Centro 1.940 , I
,...._
, ; . 1 '/
I
,' .
...- ,-
••• •• • -
.
• , • '
+
..... ....? . ...•" •t----.
• - • '4
E-W El. Centro 1940 ----- .-
--- ---
--;
t g
. , , , - - g __-----
-- - -
. ,.....---- e''
ic ,/'
I / • •
/ ,/......../ _ ■
. ;-"---
2.0
1.5
1.0
0.5
0.5
co 1.0
r.
Syinhol.,i as figure 5-39
270
0.0 05 1.0 15 20 25 30 • . Natural period, T, sec.
• , Figure 5-40 Variation of E T in Different Systems
iN (; 5 . V. I );Irkfield-2
I / / j /
, / / /
V
I
/ I
I
1
.
• •
• .., i
0
?
o
..
- - - - -.. i s 0..
• t
e •
..... '
• '...".
••*-- - ...... - -,---..
•-",------: .......
2.0
1.5
1.0
0.5 0
ct
111
SyJnhols cis figure 5-39 .
2 .1 . 1
0.0
05
1 0
15
20
2.5
30
Long. Koyna
,
/ /
/
•
, ,:
.,' I•
• •
•
• • G
••
•
,
• -•••• • ...+...'......
•
I"' •-• ......-
$.
I'''
/
?.. • 4 .....
d I
.."
.1. „......-
• -- ,
0.0 0 5 1 0 1 5 20 25 30 Natural )eriocl, T, sec:.
Figure 5-41 Variation of E with 'IT in Different Systems
-o C)
0
C) 2.0
1.5
1.0
0.5
0 3
2 '; 2
2.7 F 2.6
01'
NS
ENV
EC 34 F' = 1.32
= 0. 71 JEC 34 17'
—
.. --.. —
.--- ..---
05
10
1.5
ZO
2.5
3.0
ENV EC 40 = 0. 92 F'
-------N.----",. ---
05
10
1.5
2.0
25
3.0
F
N65E Parkfield-2
Long. Koyna
17 ' = 1.66
F' = O. 81 -----
‘
1
..■ 1
\ \ \..
— -- — — — — —
0 05
10
15
2.0
25
3.0 Natural period, T, see.
Fip;ure 5-42 Rein ive ml onsily of Earthquake Response with Respect to NS EC 40 Ea ri hquake
160
140
120
2'i3
NS EC 34 EW EC 34
—.J
NS EC 40 EW EC 40
Long. Koyna NG5E Parkfield-2
\
\ t •
\ 1
\\ 1
• • x ter.
• --- • •
10
co 80
W 60
,a)
40
4.1
4-1
1 v:
05 1.0 1 5 2 0 2.5 3.0
Natural period, T, sec.
li'i■cure :3-43 Reduced .2\verage Acceleration Spectrum, g as
T, 1.. 0, Equal ion (5-25) "
0. 1.94 g b' F = 1.83. Equal ion (5 -9 (. 1
=0.4g /T
0.335g /1.
Max. Sac,
Mean § =0.2559/T
133
120
1 0- 00
ce
80
4.4
O 60 <2,
iv) • 40
20 td
•+'
• •
••••••• •••-•
14 0
tr; 120
100
• 80 c. • 60
• 40
c.) 20
0 05 1.0 1.5 2.0 2.5 30 Natural period, T, se-!.
Figure 5-45 Smooth Acceleration Spectra for NS EC 40 Earthquake
0 0 5 1 0 1.5 2.0 2.5 3.0 Natural period, '1', sec.
.17.igure 5-44 Smooth Acceleration Spectrum, S
lean n 0. 02 Al in. values. Li-linear '-(I11, n = 0.02
7\ic..an value, tri-linear system, a= 0.02
F) 1
. R rn ea.n(n= 0.02 ) 0.1
2.7
44.
...... ....
E /L0
r
'7
+
05
10
1.5
20
2.5
3.0
r
, el' / i i
1,.......- ,i.
+ 0,
• 0
• ,fr---.
..."‘ ■,..,__
• • • • • •
_.■...,....
-.. .......
.... ........ ......
•.... ...... ... ,
....47.:;:s; :-...;,SmadplympiLIZO2r
E1AT EC 34
Rav =
e. 67
L. / ..--...../ f I'. .. . ....,
1/ ../
I ,
0
0.5
10
15
2.0
2.5
30
0.--•••". ..... -- ........ ---" "" ........ ....."
+
...............- \
,.....•-1
■N ti
+
---- ...-
,_...//
A- ..../ .........s.
"........7
....0k. ■ 'Ns ....
.......... ,-
\ /
•
... ....., .,/ ....""" .."
NS EC 40
Rave. = 1.6
0 05 10 1.5 20 2.5 30 Natural period, T, sec.
Figure 5-46 Variation of R = EJED in Different. Systems
3
L
1
2
1
Eiff o 3
Symbols as Figure 5-46
s.. _N.
--
• ,-' ... "*...
• ,„ t .....
..''..--1 t
..
\
e"......,..„
i-
..'' e...,...
---"--• •.
i k e -
•
....
. I
t/
I
4-
'
.....,...--
f-•----.............,...
'-' -•-•-........,.-... -
el. /
# 0
e •.,,,,.. ..... +
`...... + +
.-. ...... ...
EW EC 40 _
= 1.56 ave. 0 5 1D 15 20 2.5 3.0
% % «
4- , I..— ....,
.......e..'.-1.......
Ns, ..,..........t.",,--.......
t I
..— «
+ +
.....— _______
Long. Koyna
Ft = 1.59 ave.
,
+
—
r 1 I
r %
1 r
r
/ e e
l"--, 1 . „..,,
/ /
+ e -~/ I 1I/
0 0.5
l0
15
2 0
2.5
3 0
E ......
X
1 L----,
e /
....o'
o
+
...........,
1 1 1
••• •,,..
%.• ..
...^
... e" /..
V
•-•-••
e e• ___
•..
/ /
/
---.
.. +
. '
+
....."-- ......
+
..
..........."'
.......--
. . . -, *.
\ e #
%,...../
".%--- ................ ,
1V) 5E Pa I'l{ I. i eh] -2
— 11 =l^74 ave.
0 5 10 15 2 0 2.5 Natural period, T, see.
Figure 5-47 Variation of It = E/P3c., in Different Systems
2
1
0
3
2
1
3
2
1
0
3 )
3.0
z
7
• s. ------- „
sutolf-;,is [uo.lopto H Uior'}ef.:.eA 13f,-(.3 o.cukitj•
.Jos `Poi:Io(1 Tu.rtilum
S'Z
O'l 9'0 0
51--
EF, ' I , t f-', a,1
):\F' ',.i
A\<<1:
........--1,04,sts.ulls,-,--. '
--------- • -"" • i
/
i
Y •
...............
I
.1-7-7: - -----
N N
N •,..„ e/
s s s
\
.s/
s...••••••
..•••
--I I I I I ,t■
i /
'" I • / I i V
O'E S'Z O'Z 5'1 01. 9'0 0
1
7
S
.o.A13 E = "1-1
.t' sN
• • s
x
ip•(Z0'0:-.1-1)up;w n 7
Z
. 0 „7. 01. JudiTtir _ t.T1
O •0 `II.101S i. jrs)(01-1171 u(!)[,\?
C.O '(.) • x UR!:
Fs0 .0 7, If .rr!ouTr.-1.(1 unHv
I , 0
Symbols as figure 5-48
-
A
1,,,k tr , si‘.
, , '
I \ A- / \
/ \
/ • s.
\___ --' /- -
,/ .''
/
/
i t
\ k
1
/\\ 1
,./...... '---
\
•• - N.- \ 0
/
..
1
x ,.
-.. ----- -
- .... -..--
x
.... ..... . .... .
'''' ••-•`
NS EC 40
R = 1.75 ave.
0.5
1.0
1.5
2.0
2.5
3.0
R= E/E
11
/\
\ \
-
A /\
----/ \ / / \ \
/
/ / i .
\
1
\
•
i■ / / -4,,
V/4
..-•
..- ..... -
/ J
,'
- - • ,
,I
. .
\•'' . ..77.
F,'11T EC 40
R. = 1.64 ave.
0 05 10 15 20 2.5 30 Natural period, T, see.
Figure 5-49 Variation of R = E/Eo in Different Systems •
I!';8
3
2
1
4 E/E •
A 11
• -------
11;1
, ,
p, , ; , . \
\ , \ „.
•
..,-
___,, .....-- ..../
_....,
x
..... ----------------------
x
.
yrt •
es X t /.....J -,..... .....„
.."--- s .
.....•-•
...—.....:—....... s • -...'.h.
......... ',..
i
, ,/
,
,
X ,`..... ----
,, ........,
Long. Koyna
R = 1.55 -I've.
E/E 4
3
2
1
279
!---;ymbols n.; 5-18
_ ----- '-
,
, 11 ,
,
i-..
A . ‘ / \
I
/
--- ....— ....... ..--- —........... ______ .---
_....-,
%
\
,s. ",
•.'i
IV'
...........
i' _s". ....
X
,....... ..-- --- - --- -
/
,., ......,.....
I
N65E Parkfield-2
R. = i. 81 av e.
0
0.5
1.0
1.5
2.0
2.5
3.0
5
3
2
1
0 05 1.0 1.5 2.0 2.5 3.0 Nal oral period, T, see.
Figure 5-50 Variation of R = E/Eo in Different Systems
Mean value, n 0.05 Min. values, n= 0.05
Mean value, n = 0.02
A /\
/ \ \
7, \ ..
' \
s•
.......
7 rj
--------- __
--------.. ■ - - - -
--... ....... 4 - - ----- .....---- ...- ,
'
--s. ...........,...---__
....-
• ____ ........... , ....
- ____ ,./:-.?-Q,-/
/ ,,--- ./.
.
., E\V EC 40 R ave. (n=0.05) 1.41
05 10 15 2.0 2.5
3.0 Natural period, T, sec.
Figure 5- 51(a) Variation of R. = •ilEo in a Bi.-Ijnear System
2r,0
Figure 5-51.(b) 1 )egracling, Stiffness System (5 = 0.0)
10
Degrading Stiffness System, q Elasto-Plastic System, q
Req
uire
men
t, q
,
2.01 ave. = 0. 4 - 2. 7 see. )
i Z. = q/q 0 ...._
: =-- q/ q o
+ q q ,
\ .--...., ■ \.- -_,
. .
---.. .- - .... 4-
4-
4-•
0 0.5 1.0 1.5 2.0 2.5 3.0 Natural period, '1', sec.
Figure 5-52(b) Variation of R., 1-1, qiq with T
0 05 1 0 1.5 2.0 2.5 3.0 Natural period, T, see.
Figure 5-52(a) Ductility Requirement in Degrading Stiffness and EJasto-P]asi ic Sysi ems, NS EC 34 Earthquake, a .= 0.02
R, 5
4
3
2
1
3
2
1
li = 2.28
R = gig()
4
.., --...-
R. = c1„, q (1,1- ci ____ C-.1
. + +
....... 7
+ \ /
... ....1 -... ........
0
0.5 1.0 1 5 2.0 2.5
3.0 Natural period, T, sec.
Figure 5-53(b) Variation of R, R, (Lige with T
12
Degrading Stiffness System, q Elasto-Plastic System, r - 'e
0 U 5 10 15 20 2.5 3.0 Natural period, T, sea.
Figure 5-53(a) Ductilhy Requirement in Degrading Stiffness and Elasto-Plastic Sy sleins, E\V EC 34 Earthquake, n = 0.02
10
30
Degrading Stiffness System, q Elasto-Plastic System, q
e 2
20 C)
0 0.5
1.0
1.5
2.0
2.5
3.0 Natural period, T, sec.
Figure 5-54(b) Variation of R. 11, q/qc with T
0 05 1.0 1.5 20 2.5 3.0 Natural period, 'r, see.
Figure 5-54(a) Duct 13equiremen! in Dcg ra d i lig Stiffness and Elasto-I'lastie Systems, .NS EC 40 Earth(' llil k e, n = 0. 02
Ii,
= 1. avu. 91
ii -- g/q o
..- , 4- , - _...
.
......- 4...
/ 11 = .
■ v.. .., ‘..-
+
_-__ + q/ qe
______________2-, ,....---____-==---
.1.
4 + .. ........ _ ----
0.5 1.0 15 2.0 25 3.0 Natural period, T, sec.
Figure 5-55(b) Variation of it, R, q/q with T
Degrading Stiffness System, q Elasto-Plastie System, q
0 0.5 1 0 1.5 2.0 2.5 3.0 Natural period, T, see.
Figure 5-55(a) Ductility 13.equirerneni in 1)egrading Stiffness and Ela sto-Plast is ::'.,..si ems, .E\\T EC -1-0 . 1.,:arthquake„ ii = 0. 02
7
6
5
4
3
2
1
0
_±._
_ 1 =l.70 ave.
+ R . q . ! q ______ _ 0 _ ____
. ___ _
R.= gig ()
,—.„.._.... + \ . e A. - +- a
.-- ...._,..,-- -,
. V \ / i ‘... /
-----_______„. .... .-■• + 'V.,----...±..._ _ + i•
-..--vr-- ..... -----,... ,...s.. ....
—
T|. (1'( 7
6
5
4
3
2
1
0 0.5 1.0 1.5 2.0 25 3.0 Natural period, T, see.
Figure 5-560J) Variation of R., 11„ q/qe with T
12.5
Degrading Stiffness System, q Elastb-Plastic System, q
e 10.0
7.5
a) 5.0
c.)
r-4 • r
t 2.5 C.-1)
0.0 0 0 5 10 1.5 20 25 3 Natural period, '1', see.
5-56(a) Duc :Requirement in Degrading Stiffness and
Elasl 0- I 'last ie Sys; his. (mg. Islovna Eart hquake. n 0.02
-
05 10 15 20 2.5 30 Natural period, T, sec.
30 05 1.0 15 20 2.5
V. It, C
. (ii q
4--
___ li = 1.. ave. 77
H.= (A ./ ci 0 (17 q
----- ........ ___
Figure 5-57(h) Variation of 11.1,. R. q/ with T
1
Degrading Stiffness System, q
Elasto-Plastic System, qe • ----
.
I\
\
\
1
\
\ N \
l
\ ...... -.........
Natural period, 'I', see.
.Vip;ure 5-57(a) Ductility Requiremeni in Degrading Stiffness and Elasi.o-.Plas je -;vsiems, .1:avtliquake n 0. n2
0
a)
150
4
3
2
125
100
75
50
25
0
15.0
12.5
10.0
7.5
5.0
2.5
0.0
287
CHAPTER 6
DISCUSSION AND CONCLUSION
6. 1 Summary and Discussion
Two investigations were carried out in this work: the
study of the behaviour of a reinforced concrete column under lateral
load and the study of its behaviour under dynamic loading due to
earthquakes. This section gives a brief summary of each study with
its results and the relevant points are discussed.
6.1.1 Lateral Load-Deflection Behaviour of a Column
In this study, the results of 39 tests on 4 x 4 in. and 4 x 6 in.
reinforced concrete columns carried out by the investigators mentioned
in Chapter 2, and 5 tests on 6 x 8 in. columns performed by the author,
were analysed and a method was developed for determining the lateral
force-deflection of a column under monotonic loading to failure. The
purpose of the latter tests was primarily to examine the effect of
size by comparing the results with those of the former series of
tests. In addition, they were to examine the effect of a relatively
thick concrete cover of the section in the former tests, and the effect
of welding the longitudinal reinforcement in the column to that in the
connected end beams. The results of tests on the 6 x 8 in. columns
showed, in general, no sign of any effect due to size nor any effect
with regard to the other two points. The maximum bending moment
capacity of the non-welded-reinforcement column, Z2, was nearly
8% greater than the corresponding welded-reinforcement column, Zl.
However, since such a discrepancy is within the range of scattering
seen in the results of other similar tests, it may have nothing to do
with the effect of welding.
The solution for determining the load-deflection characteristic
of a column was developed in two stages: the stage of loading before
the first crushing of concrete represented by the rising branch of
288
the M-P-P diagram, and the stage shown by the falling branch of
the same diagram. The M9-P relationship for a section was
derived according to the assumptions described in section 3.3,
and the material stress-strain laws given in section 3.4. In the
falling branch of this diagram, it was assumed that the concrete
cover on the compression side of the section is gradually crushed,
and the crushing strain in the concrete layers varies linearly
between 0.0035 and the strain corresponding to the ultimate
crushing strain of the bound concrete in the core, given by
equation (3-60). The maximum bending moment capacity of the
sections found in the analysis were, on average, 94% of the
corresponding experimental results with a standard deviation of
7. 3%. The discrepancy was higher for the range of low axial
loads, and it was shown that it could be attributed partly to
ignoring the steel strain-hardening effect and partly to ignoring
the stiffening effect of cracked concrete on the reinforcement, as
was discussed in section 4.2.
Two solutions were examined for obtaining the load-
deflection diagram of a column in the rising branch of the M-P-P
diagram. The first one was based on the orthodox method of
integrating the curvaturesalong the column's length. The results
showed a large underestimation in the deflections at the higher stage
of loading, which caused an overestimation in the lateral load of the
column. This underestimation was expected, because in this
solution, as was discussed in section 1.5.1, the inelastic rotation
which occurs in the cracked zone is grossly underestimated.
However, the results showed that for 4 x 4 in. columns, the
deflections at the first crushing stage calculated by this method
were, on average, 68% of those seen in the tests. The remainder
was due to inelastic deformation in the hinge zone and deformation
in the concrete base-beam. The contribution due to inelastic
deformation in the hinge varies from approximately 22% for low
289
axial loads to 3% for high axial loads. The underestimation in
deflections was greater for columns with a larger section, and
it was shown that the deformation of the test apparatus could be
a part of this discrepancy.
The above solution was simplified in the second approach
by assuming that the entire length of the column had a flexural
rigidity of EI = M/F, where M and T- were the values at the
critical section. It was also assumed that the inelastic defor-
mation at the hinge zone was accounted for by the relationship
(3-46) given by Soliman (35). The results of this solution showed
better agreement with the experimental results, and in some cases
they even showed an overestimation of deflections at the first crushing
stage. As the point corresponding to the first crushing stage is not
quite clear on the experimental force-deflection diagrams, particu-
larly for the range of low axial loads, no definite conclusion can be
drawn in this respect. However, for reasons given in section 4.5,
it was suggested that in equation (3-47) a coefficient of 2 instead of
2.5 would be preferable in this context.
The solution for the load-deformation characteristic of the
column in the falling branch of M-p-P diagram was based on the
assumption that during this stage of loading, the hinge zone follows
the falling branch of M-P-P diagram, while the rest of the column's
length preserves the deformation corresponding to the first crushing
stage, the'-eby ignoring the unloading of the column. It was shown
that this assumption does not have a significant effect on the result
for the normal range of loading. From the experimental results
concerning the hinge rotational property during this stage, an
empirical relationship was derived for the variation of the hinge
length during the post-crushing stage, equation (3-76). The hinge
length varies linearly with the neutral axis depth ratio, and is a
function of the concrete strain level. Its range of variation for
the columns analysed in this work was between 2. 5nd and 6. 75nd,
290
which gives the hinge length as 1.3d - 3. 9d for the high axial load
case (50 ton on the 6 x 8 in. section) and 0. 65d - 2. 2d for the low
axial load case (20 ton on the same section). The comparison of
the analytical and experimental results is shown in figures 4-2 to
4-22.
Apart from the axial load, the parameters studied in these
series of tests were longitudinal steel ratio, height of the column
and the rate of loading. The amount of shear reinforcement was
varied in three tests. The analysis of the test results showed no
sign of any significant effect on the hinge behaviour due to the steel
ratio or the rate of loading. In most of the tests, the more slender
columns showed a smaller rotational capacity for the hinge. However,
due to an insufficient number of tests, no conclusion could be drawn
in this respect. In the case of columns with different amounts of
shear reinforcement, no significant change was observed in the
behaviour of the hinge which could be attributed to the effect of
this parameter. The effect of this parameter, however, should be
studied in future tests.
With the above information concerning the behaviour of the
hinge, the qualitative discussion made in Chapter 1 on the behaviour
of a column under non-axial load can be further clarified. As
stated above, the hinge rotation is directly related to the level of
strain which the concrete can tolerate, equation (3-73). This
level of strain in unbound concrete is limited to between 0.003 and
0.004, and in bound concrete it depends upon the neutral axis depth
ratio and the amount of confinement provided at the critical zone,
figure 3-21. As the neutral axis depth for columns is directly
related to the axial load, the hinge rotational capacity is mainly
a function of the axial load and the shear reinforcement ratio.
As an example, figures 6-1 and 6-2 show the interaction
diagrams of M-P and P-P for the cross-section of a typical column
291
and its bending moment-deflection diagram. Two different values
for the lateral reinforcement are considered in the example. The
curves on the interaction diagrams representing the first crushing
and the ultimate stages correspond to a concrete strain of 0.0035
in the outermost fibre of the cover, and the strain given by
equation (3-60) in the outermost fibre of the concrete core. It is
seen that an increase in the shear reinforcement does not affect the
behaviour of the column up to the first crushing stage, during which
the strain in the core is relatively low, but beyond this stage it
increases the ductility of the member considerably. In both cases,
the bending moments corresponding to the ultimate stage are very
close to each other.
The effect of the axial load on the ductility of a member is
seen clearly in these two figures, and will not be discussed again
here. As an example, when a column with p" = 0.41% under 10 ton
axial load is considered, the ductility factor at the first crushing
and the ultimate stages (p c /py and p u /0y ) are 8 and 34 respectively,
while the drop in the bending moment capacity between the two stages
is only 7%. The corresponding ductility factors in terms of deflec-
tions (D c /Dy and D u IDy ) are 1.5 and 5.5 respectively, figure 6-2.
In the case of the same column under 60 ton axial load, the ductility
factor at the first crushing is negligible and at the ultimate stage is
ip <2 and D <3. In contrast, the drop in the bending u y u y
moment is as much as 20%. An increase in shear reinforcement
from p" = 0.41% to p" = 1.25% raises the ductility factors in terms
of deflection at the ultimate stage to 8.5 and 4.5 respectively, for
the above two cases. It is seen that in the cases where high
ductility is required in a column, special attention should be paid
to the amount of axial load and/or the amount of shear reinforcement.
The role of lateral reinforcement in increasing the rotational
capacity of a hinge, becomes more apparent by considering the modes
of failure of a hinge in a column. Observations made in these tests
292
showed that a column fails in one of the following three modes:
an excess of crushing of the concrete core under compression,
the buckling of the compression reinforcement, and shear failure.
All these can be prevented or delayed by providing enough lateral
reinforcement at the critical zone. The role of this reinforce-
ment is three-fold.
(a) By preventing the lateral expansion of the concrete under
compression, it exerts lateral pressure on the concrete in the
core and increases its strength and ductility. Its effect is more
pronounced in the post-crushing stage, as shown above. By
increasing the crushing strain of the concrete, the section is able
to undergo a greater deformation before failure.
(b) By limiting the unrestrained length of the longitudinal reinforce-
ment, which is exposed at the failure stage, the buckling load of the
reinforcement is increased and its lateral deformation is reduced.
The latter effect confines the spalling of the concrete cover to a
smaller area.
(c) By increasing the shear strength of the member at the critical
zone, shear failure is prevented.
In view of the above discussion, the role of the shear
reinforcement in the post-crushing behaviour of a member should
receive more attention in future tests. Its effect on the behaviour
of concrete under high strain, i.e. the falling branch of stress-
strain relationship, and on the ultimate crushing strain of concrete,
should be studied in more detail.
With regard to the hinge rotational capacity, its ultimate
value could not be estimated in these series of tests, since in cyclic
loading the direction of loading was reversed on completion of the
first quadrant of the F-D diagram. The completion point changes
in columns under the same axial load but of different height or
different bending moment capacity of the section (compare B28 with
B17 in figures 4-4 and 4-8), and by no means represents the ultimate
293
rotational capacity of the hinge. However, it can at least be said
that for these columns, the ultimate rotational capacity of the hinge
was such that the corresponding F-D diagram could be completed
in its first quadrant. In the case of column K13, figure 4-17, the
failure occurred before the completion of the F-D diagram, which
may be due to shear failure.
In the analysis of the columns, it was found that the
calculations of ultimate hinge rotation based upon the concrete
crushing strain given by equation (3-60), underestimated the
rotation, especially in the cases of low axial loads. In these
cases the F-D diagram was completed by extending the M-P-P
diagram of the cross-section, without concrete cover on the com-
pression side, to a higher strain, which in some cases was nearly
30 - 40% higher than the prescribed value. Such a case is seen
in the F-D diagramsionhe column discussed in the above example,
figure 6-3. Point A or A' in these diagrams corresponds to the
concrete strain given by the equation (3-60). If the underestimation
in the strain were taken into account, these diagrams would probably
be completed before failure occurred. This point should be clarified
in future tests and the above strain should be modified.
6.1.2 Response to Earthquake Loading
In this study a column was idealized as a single-degree-
of-freedom system with a bi-linear or a tri-linear load-deflection
characteristic as shown in figure 5-3. A parabolic variation for
the reversal paths was assumed to simulate a pattern similar to
the actual behaviour of the column observed in the tests by Neal (39)
and Koprna (73). These tests showed that the behaviour of a column
at any stage depends upon the history of loading it has undergone.
The stiffness and strength of a column deteriorate with the repetition
of loading and the amount of its inelastic deformation. The latter
contributes more to the deterioration of these parameters. The
hysteresis loops grow considerably as the inelastic deformation
294
increases in the column.
The idealized system was subjected to six components of
four real earthquake accelerogram records, page 212, and its
displacement-time history was studied. However, the main con-
cern in the analysis was to determine the maximum earthquake
resistant capacity of a column. For this purpose, the earthquake
ground acceleration was incrementally amplified by a constant E,
in the analysis, until the displacement response exceeded the failure
limit. This limit was assumed to be the displacement level at which
the lateral force on the falling branch of the F-D envelope diagrams
became null, when the strength of the column was just equal to the
effect of gravity load alone. The response was shown to be depen-
dent on the ratio of the relative intensity of earthquake to the yield
level of the system, Eia . The variation of E, for a constant ay,
for different systems is shown in figures 5-26 to 5-37.
The analysis of the limiting values of E for systems with
the same properties, except the gradient of the falling branch of
their F-D envelope diagram S, showed that there is a tendency
towards a linear variation between these values of E and the square-
root of the reserved energy capacity of the system, defined by the
area under its F-D envelope diagram. The variation of E, reduced
in proportion to this parameter, is shown in figures 5-38 to 5-41.
These reduced values, E, show a tendency to vary linearly with T,
the initial natural period of the system, for different earthquakes.
Considering that the square-root of the reserved energy capacity of
the system is itself proportional to T, it is concluded that E is nearly
independent of T and varies according to the reserved energy capacity
of the system. This point will be discussed later.
The "energy rule'', based on the idea that the maximum level
of energy absorbed during an earthquake by the system under study
is the same as that absorbed by the corresponding similar elastic
system, was applied. A comparison was made between the predicted
295
maximum earthquake intensity that a system could withstand, E0,
given by this rule, and the actual value of E given by the analysis.
The comparison showed that the value predicted by this rule was
usually on the safe side for systems with T > 0.4 sec. On this
basis, a bi-linear system will be safe during an earthquake if its
yield level and ductility, a and q, satisfy equation (5-28), in which
Sa is the response acceleration spectrum of an elastic system with
the same T and n, and R is the safety factor whose range of values
is given in figures 5-48 to 5-50 for different earthquakes. For
systems with T 4 0.4 sec., R is less than unity and varies between
0.5 and 1.0. The overall average of the mean values of R for
n = 0.02 varies between 1.1 and 1.8 for the earthquakes considered.
In another similar analysis, it was found that for the energy
rule to be applicable with R = 1 for all values of T, the average and
maximum values of the acceleration spectra of the earthquake
records for n = 0.02 should be similar to those shown by the broken
lines in figures 5-15 to 5-20, Saa and Sam. These spectra showed
a general tendency to be inversely proportional to T. The earth-
quakes intensities, SI, calculated according to equation (5-15) and
based on the average spectrum Saa, were found to be, in most cases,
similar to the intensities calculated on the basis of the actual spectra
for n = 0.20, table 5-1, page 233. These intensities were proportional
to the factors F which are also given in this table. On the basis of
this analysis, a smooth acceleration spectrum was suggested for
predicting the safety of a system according to the energy rule.
The spectrum is:
Sa 0.4 F
T 0.3 sec. (T in sec.)
§a =T 0.3 sec. (6-1) a(T = 0.3 sec.)
The application of the energy rule on the basis of this spectrum,
resulted in values of the safety factor R givenin figures 5-46 and 5-47.
296
They were generally greater than unity and in most cases vary
between 1 and 2. The overall average of the mean of R for n= 0.02
is about 1.6, for all the earthquakes.
In a study of a system with an elastic-perfectly plastic load-
deformation envelope diagram and non-linear reversal paths similar
to those used in the previous study, figure 5-51(b), the ductility
requirement of the system, q, was determined, and it was com-
pared with that of an ordinary elasto-plastic hysteresis system.
The results showed that when T 0.6 sec., the ductility require-
ments in both systems are virtually the same, and in the case of
T 0.6 sec., the degrading stiffness system in the majority of
cases requires greater ductility, figures 5-52 to 5-57. The
application of the energy rule in this case gave similar results
to the above case: using the actual spectrum, the rule gave a
safety factor R greater than unity when T 0.4 sec. and a very
low value of R when T ( 0.4 sec. Using the smooth spectrum,
equation (6-1), the rule gave more uniform results of R, which
are generally greater than unity. The overall average of R in
this case is about 1.7 - 2.3 for the various earthquakes.
It will be noticed that in both the above analyses, the
degradation of stiffness in systems with T 0.4 sec. made the
system more sensitive to earthquake loading. In both cases,
the application of the energy rule based on the actual spectrum
resulted in R < 1, and in the case of the ductility analysis, the
required ductility for the system was usually higher than that of
the corresponding elasto-plastic system. This sensitivity may
be due to the fact that in these systems the deterioration of stiff-
ness leads the system towards higher periods of vibration which
include the region of T = 0.4 - 0.6 sec. In most earthquake
records, this region corresponds to the frequencies with the
highest energy content, as seen in the spectra in figures 5-15 to 5-20.
This sensitivity is not so severe in the case of systems under the
297
Koyna earthquake, in which the high energy content region is around
a frequency corresponding to T = 0.1 sec., figure 5-19. In this
case, the values of R are closer to unity than for the other earth-
quakes, in the region of T 4. 0.4 sec., figures 5-50 and 5-56.
However, the sensitivity of the degrading stiffness material in low
period systems may be detrimental in tall building frames, whose
natural period of vibration in the higher modes is relatively low.
In both the above analyses, the application of the energy
rule based on the smooth spectrum, equation (6-1), resulted in a
safe answer, R > 1. The use of this spectrum enables a simpler
relationship to be derived for determining the earthquake resistant
capacity of a system. Equation (5-19) shows the maximum level
of energy transmitted into an elastic system during an earthquake
with a relative intensity of Eo. Substituting in this equation for
Sa = a from equation (6-1) and for
2
Wo will be
K - 4 112
T2
0 02 Wo = ' • Eo2 . F2 . M . g2 n2
M
(6-2) for a system with T 0.3 sec.
According to the analysis discussed above, a system under such an
earthquake is safe provided its reserved energy capacity, W, is
W woiR2
(6-3)
where R is the constant discussed previously. Substituting for W0
from equation (6-2), the maximum earthquake intensity that the system
can withstand will be
(6-4)
According to figures 5-46 and 5-47, R varies slightly for systems
with different T or gradients in the falling branch of their F-D
diagrams, S. The variation of R in these figures for systems
Eo< 5111 x/ 2W F.g M
298
with the same T is due to the variation of S between -1 and -0.1.
In an ordinary reinforced concrete column, S is generally greater
than -0.5. For this reason, the mean value of R in these figures
can be considered as the lower bound for R. The overall average
of R for 0.1 4 T 4 2.7 sec. was found to be about 1.6 for all the
earthquakes. The corresponding average in the case of the ductility
analysis, where S = 0.0, was about 2. It can therefore be said that
in the above relationship,Tivaries on average between 1.6 and 2.0,
for systems with -0.5 S 0. 0.
The above relationship shows clearly that the earthquake
resistant capacity of a system, depends on its reserved energy
capacity per unit mass. Ignoring the slight variation of R with T,
Eo is independent of the initial natural period of the system. For
a degrading stiffness system, this conclusion is to be expected
because the system holds its initial stiffness for a relatively short
time under a severe earthquake. The same trend, however, has
been observed in a non-degrading stiffness system by Husid (69).
In a similar analysis on a bi-linear hysteresis system, as in
figure 5-3, but with elastic reversal paths, under an ensemble of
artificially generated earthquake records by Jennings (58), Husid
shows that the statistical average time that a column can withstand
an earthquake with the relative intensity of E, is a
t = 2000 h (Eg)2 (6-5)
where h is the height of the column (in feet). In other words, if
a column is to survive an earthquake with a duration of t and relative
intensity of E, the above relationship between its yield level and its
height should be satisfied. The relationship is independent of T,
which shows that the independence of Eo from T in equation (6-4)
is not an exclusive characteristic of the degrading stiffness system.
However, it should be mentioned that equation (6-5) has been derived
by analysing systems with T = 0.5, 1.0, 1.5 and 2 sec., and its
validity does not necessarily hold for systems with other values of T.
299
The results of equation (6-4) are not very different from
equation (6-5), if the properties of the system that Husid analysed
are also considered. His system has an elastic-perfectly plastic
hysteresis characteristic, in the absence of the gravity load effect.
With this effect, the system will be similar to the bi-linear system
shown in figure 5-3(a), with the gradient of the falling branch as
a = - P/h. The reserved energy capacity of such a system is
1 2 1 P) W = M• ay2 (k- +
and substituting for W in equation (6-4)
(6-6)
5 nik E o F LIX 1 g Y g (6-7)
The term 1 —2 is small in comparison with — and it can be ignored,
(for example, a column with T = 1.0 sec. and h = 10 ft. has
= 0.025 and 12 = 0. 3). Ignoring this term and expressing h
in feet,
E < 2.77 — o N F
which, for NS EC 40 component, F = 1, will be
Eo < 2.77 (6-9)
According to Jennings (58), NS EC 40 is equivalent to a 25 sec. duration
earthquake of the above-mentioned ensemble amplified by 3.17,
(Jennings actually gives the amplification as 2.9 for the average of
the NS and EW components of this earthquake. The value of 3.17
is derived on the basis that, according to Jennings the NS component
is 2.2/2.01 = 1.094 stronger than the average.). Substituting for
t = 25 sec. and E = 3.17Eo in equation (6-5), Eo corresponding to
a Eo = 2.82 --Z • (6-10)
This shows that equations (6-4) and (6-5) are similar.
a
g (6-8)
NS EC 40 is
300
The similarity between the above two expressions leads to
two conclusions:
(a) Since the ensemble of the artificial earthquake records used in
the derivation of equation (6-5) is based on the statistical properties
of a relatively large number of ordinary real earthquake records,
the results obtained in the present work for only six earthquake
records are probably valid for the case of any ordinary earthquake.
However, this should be verified in future studies.
(b) Considering that the overall average of R in equation (6-9) is
about 1.6, it is concluded that, at least in the case of NS EC 40,
the degrading stiffness system shows more earthquake resistant
capacity than an ordinary elasto-plastic system. To verify this
conclusion, the E values calculated for a bi-linear system with
elastic reversal paths are compared with those of the degrading
stiffness system in figure 6-4. As can be seen in practically all
cases, the latter system tolerates a more intense earthquake than
the former. The average ratio between the two sets of E values,
calculated on the basis described in section 5.6.4, is 1.64. This
confirms the above result. Figure 6-5 gives the R and R values
for the elastic-reversal system, calculated on the basis of the
energy rule, using the actual and smooth spectrum, equation (6-1).
As an example of the application of the above results,
consider the column examined in the previous section, figures 6-1
to 6-3, Under a 40 ton axial load, the column may be considered
to have a bi-linear characteristic, as shown in the figure, with
properties of K = 2.5 ton/in., Fy = 3.45 ton and S = -0.14. These
parameters give T = 1.28 sec. and a = 0. 086g. According to
figure 5-28 this column can tolerate an earthquake similar to
NS EC 40 with a relative intensity, Eo, of nearly 0.86 x 1.8 = 1.54,
for damping ratio of n = 0.02. Alternatively, from equation (6-4)
with W = 19.5 ton-in., the value of Eo = 0.785 R. The average
value of R for this column, figure 5-46, is R = 1. 7, but since
301
S = -0.14, R is closer to the upper bound and is probably about
R = 1.9. On this basis, Eo = 1.5. Under a 60 ton axial load,
the column may be considered to have a tri-linear characteristic
similar to the one studied in this work. Its properties are:
T = 1.5 sec., S = -0.22 and a = 0.052g. From figure 5-34,
for n = 0.02, Eo = 0.52 x 2.6 = 1.35. Alternatively, from
equation (6-4) with W = 13.8 ton-in. and Tt 02.2, Eo = 1.2.
6.2 Conclusions
The conclusions drawn from the results of the present
work can be summarized as follows:
1. The results of tests on 6 x 8 in. columns did not show
any size effect when compared with the results of 4 x 4 in. and
4 x 6 in. columns.
2. The force-deflection analysis for columns based on the
orthodox method of integration of curvatures along the member,
underestimates the deflections. The deflections found at the
first crushing stage were nearly 68% of the actual deflections.
The remainder could be attributed to rotations occurring at the
hinge zone and the deformation of the concrete base-beam.
3. The hinge length varies with the neutral axis depth ratio
and the level of concrete strain, during the post-crushing stage.
Its value is governed by equation (3-76), and in the columns
analysed in this work, it varied between 2. 5nd and 6.75nd. The
hinge rotational capacity depends directly on the ultimate crushing
strain of the bound concrete.
4. The columns showed deterioration in stiffness under
repeated loading and increase in their inelastic deformations.
The latter effect was found to cause more severe deterioration.
5. The earthquake resistant capacity of a column, predicted
according to the energy rule and based on the elastic response
spectrum of the earthquake record, is usually safe provided
T) 0.4 sec. The overall average ratio of the actual values to the
302
values predicted according to this rule, varies between 1.1 and
1.8, for the case of damping ratio of 0.02, for the various earth-
quakes. In the majority of cases, use of the smooth spectrum,
equation (6-1), gives predictions which are safe. The overall
average ratio of the actual to predicted values is about 1.6 for
the various earthquakes.
6. The ductility requirement for a degrading stiffness system
with an elastic-perfectly plastic envelope diagram, was found to be
almost the same as for the corresponding similar ordinary elasto-
plastic hysteresis system, when T ) 0.6 sec. For systems with
T ( 0.6 sec., the former generally requires a greater ductility.
The ductility requirement, as predicted by the energy rule and
based on the smooth spectrum, is safe, and is, on average, about
twice the actual requirement.
7. Based on the results concerning the earthquake resistant
capacity of a column, the relative intensity of the earthquake records
used in this analysis were found to be proportional to the factor F
given in table 5-1, page 233. With the exception of the 1934 El Centro
earthquake, these intensities are very close to those calculated on
the basis of the elastic response spectrum with a damping ratio of
n = 0.20,
8. The earthquake resistant capacity of a column depends upon
its reserved energy capacity and is almost independent of its initial
natural period of vibration.
9. Finally, the results confirm the concept that the design of
earthquake resistant structures should be based on their reserved
energy capacity.
6.3 Recommendations for Future Work 1. In view of the importance of the shear reinforcement in
increasing the ductility of concrete, its effect on the crushing strain
of concrete and on the ultimate hinge rotational capacity should be
studied in more detail. The study of the behaviour of bound and
303
unbound concrete is relatively new for the range of high strains,
i.e. the falling branch of stress-strain diagram, Full information
about concrete behaviour in this range of strain is required for a
thorough investigation of the hinge performance.
2. The method developed in this work for determining the
load-deformation of a column is limited to its performance under
monotonic loading to failure. To assess the dynamic behaviour of
a column more realistically, a study of its behaviour under repeated
loading is necessary. The unloading and reloading paths should be
investigated in greater detail. Such a study requires full information
about the behaviour of the concrete under repeated loading, and a
full study of the mechanism of deterioration of the bond between steel
and concrete.
3. The effect of the vertical component of the ground motion
acceleration was not considered in the present analysis. This
component of an earthquake has a spectral intensity of about 20-30%
of its horizontal components (84). Considering the effect of axial
load on the hinge rotational capacity, this component of an earthquake
may be detrimental. The degree of its influence on the performance
of a column should be studied.
4. The approach followed in the dynamic analysis of the column
was a deterministic one, and naturally, the results are valid only
for the earthquake recoirds studied. The validity of the results should
be examined by a statistical solution of the problem.
5. Finally, the results obtained in this study concerning the
dynamic analysis of the column need to be verified by some experi-
mental evidence. An experimental investigation into the problem
would be most valuable.
10"
1/4 6 or 2"
P / Pu 1.0
0
kk - 1"
0.
0.
N .N
N .N
/ P.6010
./7 P. ioln
0.0 50 100 150 200 250 300 350 Bending Moment, Ton-in.
0.8
0.6
= 3000 psi 2 A= 2 in.
s
9 f = f" = 20 Ton./ in. sy sy
P =171 Ton
First Crusi,ir Stage Yield Stage Ultimate Stage (P" = —• Ultimate Stage (P" = 1.23"0)
0..0 2 4
------------------ ------
60 10 12 0
Curvature. x
Figure Cj-1 BEmcling 7joment - Axial Load. Axial Load-Cut•vatnre eraction DiasTrams
2 0 6 8 10 12 14 16 18 Deflection. 2D, in. Fig-zrce 6-2 Bending Moment-T,eflecion Diagram
P= 60Ton. .., P" = O.
P" = 1.25%
as Figure Cross-section
41(7/0
6-1 properties ------
p ,... 40Tori"..
-......... ..., ....„ -.., ......,
------- ---z.l.„------____
I
i ,
P=10Ton. __ ____, ......_
,---, eefl
{Y
ii I i
350
300
250
200
150
100
'4' 50
P = 40 Ton
o
r
.
P=
a . • .... •
.... . Ns, . \ . •
i
---.. ‘
. \
, \\B
• \B'
0 4 4
La
ter
al
3
2
1
60 Ton
20 IF ?'Y
])" = 0. 4 ] c,,i )
1.25u(. ) lii-1,incar Tri-I..inear Idealization
Cross-Section Properties as Figure 6-1
2G 5.0 7 5 10.0 12.5
Deflection, 2D„ in.
o uz
(NJ
0 2 5 5.0 7.5 10.0 Deflection, 21), in.
Figure 6-3 I..,ateral J,oad-Deflection Diagrams
5
4
3
2
15 10 2.5 20 3.0 0.5
1.25
1.00
0.75
0.50
0.25 a)
FL-1
/
/ i
/
/ i ____
/ /
lli- Linear._ f '0. 1. kV, Y
IA a sue 1 (,,,,ersa.i. :\:on-Linc.ar
. -
/
/ /
7
System. Fig. 5-3(a) n := 0. 02
/
/ /
/
_
3=-0.1
_____________ 11.eversal--- _______________________
//' ---- —7', ■
/
/
1/
/ /
I
/ /
/ 1
---r- /i
I I
/ ..)
/ I
Li V ,,.. -_,.
.
/ //
/-- 5=-0.5
"
0.00 0
1.00
cd
0.75
0.25
0.00 0.5 1 0 1 5 2 0 2 5
0.5
7 / / / / v .
/
// 1 /
..- /
,--- S=-0.2 0.2
1
,-'
..
/
..---,
___.------ ../-
... --- ..-- ---,
0 3.0
3
4
2
Natural period, '.I', sec. Figure 6-4 Failure Factor for a RI-Linear System under NS EC 'if) EQ.
Mean Value, Elastic Reveral System
max. & values, Elastic ;y stem
Mean Vall,e, Non-.I...inezit.• ersal System
R=E- 1E0
ll, _ a V e.
1.07 A
/ \ /
,_... J
N —N
'■ ......, 7 N.
/ .."...."s .....".
i / / /
(7,
.1
i'," /
/
' • . , ',..
/ /
.../--..
0.5 1.0 1.5 2.0 2.5 3.0 Natural Period, T,
R :: 1.0 ave.
„.... ..--
-..., ..-..
....-- ...„
„,
..- ---
" tk % % , ....,.. ...../. ............• ...........-- .....
..„^ .. ...r ...
...,
% , -,—..—■ r ...... " -„...... — ----- - - .......s.,m,-- ----,7:: I 11
Iv................. .....-"--'''...
• ..... ......
05 10 15 20 2.5 3.0 Natural Period, T, see.
1■Igu1.•e 6-5 Variation of It and R in a Bi-Linear. System
Under NS EC 40 Earthquake, n = 0.02
3
2
0
2
1
0
3 R= E/ Eo
309
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