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

1

1

1

1

1

1

1

1

1

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

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./7 P. ioln

0.0 50 100 150 200 250 300 350 Bending Moment, Ton-in.

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= 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

------------------ ------

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as Figure Cross-section

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P=10Ton. __ ____, ......_

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350

300

250

200

150

100

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Cross-Section Properties as Figure 6-1

2G 5.0 7 5 10.0 12.5

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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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