HINGE ROTATION CAPABILITY OF PRESTRESSED ...digitool.library.mcgill.ca/thesisfile47696.pdfHINGE ROTATION CAPABILITY OF PRESTRESSED CONCRETE BEAMS by NEVILLE CHAI A thesis submitted

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HINGE ROTATION CAPABILITY OF

PRESTRESSED CONCRETE BEAMS

by

NEVILLE CHAI

A thesis submitted to the Faculty of Graduate

Studies and Research in partial fulfillment

of the requirements for the degree of

Master of Engineering

Department of Civil Engineering

and Applied Mechanics

McGill University

Montreal, Canada

® 1:evi11e Chai 1971

August, 1970

To

Christina

- i -

Hinge Rotation Capabi1ity of

Prestressed Concrete Beams

Neville Chai

Department of Civil Engineering

and App1ied Mechanics

ABSTRAGr

M. Eng.

August 1970

The present study was conducted to investigate the magnitude of the

hinge rotation and the extent of the plastic region of rectangu1ar prestressed

concrete beams under a concentrated load at midspan. The effects of the mag

nitude of the prestressing force and the spacings of the web reinforcement

were studied.

Extensive ~urements of rotations were ca1cu1ated by severa1 methods.

Hinge rotation was ca1cu1ated from the curvature distribution measured from

concrete strains.

A theoretica1 moment-curvature re1ationship using equi1ibrium of

forces, compatibi1ity of deformations, the stress-strain curves of concrete

and tension steel and the physica1 beam properties is presented. The results

thus ca1cu1ated were found to compare we11 with the experimenta1 resu1ts.

The experimenta1 resu1ts provide data of 1imiting values for concrete

deformations in ine1astic design theories.

- ii -

ACKNOWLEDGEMENTS

The author is deeply indebted for the advice and guidance provided

by Professor J.O. McCutcheon in directing this program. The helpful sug

gestions by Drs. M.A. Sheikh and M. Celebi in the initial phase of the

project is appreciated. The author wishes to extend his acknowledgements

to Drs. M.S. Mirza and J. Nemec who willingly offered their assistance and

helpful criticism during informal discussions.

The ideas provided by his fellow graduate students and the tech

nical assistances given by Mr. B. Cockayne and his staff are deeply appre

ciated.

The author acknowledges financial assistance from the Emergency

Measures Organization of Canada under Grant No. 275-22. Support from the

Steel Company of Canada and the Canada Cement Co. Ltd., who provided the

prestressing wires and the cement respectively, is wholeheartedly

appreciated.

ABSTRACT .•••

ACKNOWLEDGEMENTS

TABLE OF CONTENTS.

LIST OF FIGURES. •

LIST OF TABLES •

l - INTRODUCT ION

- iii -

TABLE OF CONTENT S

1.1 Object and Scope •••

1.2 Nature of the prob1em .

II - SURVEY OF LITERATURE • • •

III - TEST PROCEDURES AND INSTRUMENTATION

3.1 Strain Measurements on Prestressing Strands •

3.2 Strain Gauges on Concrete •.

3.3

3.4

Lever Arms at End Supports ••

Def1ections at Midspan. . • .

IV - EXPERIMENrAL RESULTS AND ANALYSIS

4.1 CurvatureS and Curvature Distributions .•

4.2

4.3

4.4

Strain Distributions.

Rotations • . • • •.• •

Hinge Rotation measured from Curvature Distribution .

4.5 Average Curvature over the Hinge Length .••

4.6 Rotation measured from Midspan Def1ections.

4.7 Rotation measured from Lever Arms • . • • •

Page

i

ii

. • iii

v

vi

1

2

4

6

7

7

7

8

9

10

11

12

· 12

· 12

4.8 Total Rotation measured from Curvature Distribution . 13

V - THEORETICAL ANALYSIS

5.1 Assumptions ..

5.2 General Formulas. .

5.3 Hinge Rotation ..

. • 14

· 18

21

- iv -

VI - CONCLUSIONS

6.1 Comparison of Experimental and Theoretical Results • 23

. . 24 Hinge Rotation and Total Rotations 6.2

6.3 Differences between Reinforced and Prestressed Members 25

6.4 Recommendations for Future Research.

FIGURES.

TABLES .•.

APPENDIX A.

APPENDIX B.

BIBLIOGRAPHY.

. . . 26

• • • 27

• 75

82

. • 85

93

Figure No.

1

2

3

4

5-13

14

15-21

22

23-35

36-42

43

44-45

46-48

BI

B2

B3

- v -

LIST OF FIGURES

Title Page

Beam Dimensions 27

Load-Strain Curve of Steel 28

Schematic Representation of Instrumentation 29

Concrete Stress-Strain Distribution 30

Strain Distribution 31

Histogram of Ultimate Strain Values in Hinge Length 40

Curvature Distribution 41

Typical Envelope of Moment-Curvature Relationship 48

Moment vs Hinge Rotation 49

Moment vs Total Rotations (Lever Arm) 62

Typical Midspan Load vs Midspan Deflection 69

Moment vs Hinge Rotation/Effect of Web Reinforcement 70

Moment vs Hinge Rotation/Effect of Prestressing Force 72

Apparatus for Prestressing 90

Apparatus for Release of Prestress 91

Prestress Bed 92

Table

1

2

3

4

5

6

7

No.

- vi -

LIST OF TABLES

Tit1e

Beam Designation

Theoretica1 Moment-Curvature

Theoretica1 Ana1ysis of Hinge Rotation

U1ttmate Strains, Curvatures, Def1ections

Experimental Rotations at U1timate

Effect of Web Reinforcing on Rotations

Effect of Prestressing Force on Rotations

Page

75

76

77

78

79

80

81

1.1 Object and Scope

- 1 -

CHAPTER l

INTRODUCTION

A logical and realistic method of design and analysis applicable to rein

forced or prestressed concrete structures must be based on the actual conditions

which exist on the structure when that structure is said to have failed. The

criteria for failure must be clearly defined and the prescribed limits must not

. be exceeded under the worst condition of loading.

Limit Design or Ultimate Load theories recognize the inelastic behaviour

of concrete and steel. However, one of the aspects of structural behaviour in

which there is very limited information is that of the plastic behaviour of con

crete structures. Of fundamental interest to any inelastic theory is the rota

tion capability of concrete hinges in complex indeterminate structures.

The present study was made to investigate the hinge rotation of pre

t~nsioned prestressed rectangular concrete beams, simply supported and under a

concentrated point load at midspan. The investigation could further be applied

to the conditions at the support of a continuous beam wherein the midspan con

centrated load represents the support and the simple beam supports represent

the points of contraflexures.

Tests were performed on rectangular, pretensioned prestressed beams of

dimensions as shown in Fig. 1. The variables investigated included the effects

of the level of prestressing and the percentage of web reinforcement. Except

for the initial two tests which were performed under two point loading, a single

load was applied at midspan.

Rotations were calculated from the actual curvature distribution over

the span of the beam, from measurements of deflections of lever arms at the end

- 2 -

supports and from the load-deflection curve at midspan.

A conventional theoretical analysis to de termine the Moment-Curvature

relationship of a section of the beam using equilibrium of forces, compatibility

of deformations and the physical properties of the beam section is presented.

Various assumptions were made, the most important being the stress-distribu-

tian in the compression zone and the value of the ultimate strain in concrete.

1.2 Nature of the problem

13 14 24 30 Recent developments in limit design theories ' , , or inelastic

theories of indeterminate concrete structures, in which redistribution of inter-

nal moments and forces at various cross-sections are recognized, have indicated

the necessity for obtaining realistic Moment-Rotation relationships for a given

section. The need for simplified Moment-Curvature or Moment-Rotation relation··

ships is evident. The danger of theorists indulging in complicated mathematical

theories for complex structures based on the simple plastic theory, without

experimental evidence of the limitations of reinforced or prestressed concrete

structures, is real.

Definitions and assumptions are required to develop a practical design

theory. The designer must however be constantly aware of the many inaccuracies

and limitations inherent in the analysis and must intuitively keep in clear

perspective the actual behaviour of the real structure in its entirety.

Keeping this in mind, a review of the most frequently used definitions

relevant to ultimate load analyses for concrete structures will he given.

Plastic Hinge is a section in a structural member at which the inelas··

tic rotations are assumed to be concentrated.

Hinge Rotation is the total change of slope at a hinge or critical

section due to inelastic or plastic behaviour •

. ' "' ... ,

- 3 -

Equivalent plastic length is an assumed length of the member, over

which the curvature is assumed to be constant, such that the calculated rota

tion at the hinge equals the actual rotation at the hinge.

A structural member behaves elastically according to the initial

linear relationship of the curve. The remaining contribution is due to

plastic action.

- 4 -

CHAPTER II

S URVEY OF LITERATURE

There is ample evidence IO ,12,20,2l to show that for under-reinforced

concrete indeterminate structures, sufficient plastic curvature is developed

to produce substantial redistribution of moments at ultimate load.

Limit Design theories 13 ,14,24 have been proposed which predict in-

elastic hinge rotations necessary to develop full redistribution of moments.

14 32 33 The work undertaken by Professor A.L.L. Baker ' , at Imperial

College is of conside~able significance to the development of limit design

theories. Baker reported a simplified method of calculating rotations using

safe limit design values. These limiting values were based on experiments on

statically de termina te members and were found to be very conservative.

10 12 13 25 1 28 2 37 Chan , Ernst ' , , Mattock' , Corley , Obeid have made con-

siderable contributions to the limited experimental knowledge of rotation capa-

cities of reinforced concrete sections.

Chan has developed an expression for the rotation capacity using

flexural rigidities and a curvature distribution factor. Ernst has adopted

revised moment-area theorems for computations of elastic slopes and deflections

that occur during the yielding of the plastic hinge. Mattock, Corley developed

expressions using an average curvature measured over the hinge length.

Attempts to evaluate magnitudes of hinge rotations have in Many cases

led to proposed determinations of the MOst fundamental relationship, the Moment-

Curvature diagram.

16 38 Unfortunately, experimental results ' do not confirm the existence

of a moment-curvature relationship for a given beam cross-section. It is vir-

tually impossible to predict accurately the value of the curvature at or very

- 5 -

near the critical section after the reinforcing steel has yielded.

A more realistic representation of the behaviour for a length of the

beam under consideration would be the moment-average curvature relationship

over the particular length.

The load and loading distribution, the rate of loading, the stress-

strain relationships of concrete and steel, the variation in the structural

system, i.e. relative stiffnesses, local behaviour e.g. bond slip and cracking,

the span to depth ratio all have an appreciable effect on the rotation capa-

bility of the hinging regions.

18 26 Roy and Sozen ,Nawy in their experiments to determine the effects

of rectangular ties and rectangular spirals have shown that the ductility of

concrete governs the rotation capacity of the section. 21 .

Macchi has observed that the rotation capability is limited if the

percentage of reinforcing increases, or if axial load is acting at the same

time as bending. Obeid37 has also confirmed the effect of axial load on limit-·

int the rotation capacity of the section.

- 6 -

CHAPTER III

TEST PROCEDURES AND INSTRUMENTATION

The rectangular beam section with an effective depth of 4.5 inches

is shown in Fig. 1. The strength of concrete was determined from axially

loaded compression tests on standard 12" x 6" diameter concrete cylinders. The

concrete strength was not a parame ter to be studiéd. Because of the difficulty

of measuring the average water content of the sand used in the concrete batch,

the strength of concrete displayed some scatter.

Stress-relieved high strength 1/4" diameter 7-wires prestress ing

strands manufactured by Stelco Company were used as the prestressing reinforc

ing. The stress-strain curve is shown in Fig. 2.

Deformed bars designated D-2 (area = .02 in2) with a yield stress of

80 ksi was used as web reinforcing. The dimensions are shown in Fig. 1.

The load was applied by a 400 ton capacity Baldwin-Lima hydraulic

testing machine. This load was applied through a steel roller 4 inches long and

1 3/4 inches in diameter. The roller was welded to a plate 4" x 2 1/4" x 1/2"

thick. A small groove was cut from the bot tom of the plate so that the con

crete strain gages in the vicinity of the midspan would be operat.ive.

3.1 Strain Measurements on Prestressing Strands

Polyester gauges, PS-20, manufactured hy Tokyo Sokki Kenkyujo Co.,

Ltd., were applied on the prestressing wires. Eastman 910 adhesive cement

was used. GW-5 waterproofing manufactured by Automation Industries Inc. was

applied after the lead wires were connected.

The gauges were located at the midspan of the beam to measure the pre

stressing force (by using the stress-strain curve of the steel). An electrical

SR-4 strain indicator box was used for measuring strains before the test and a

- 7 -

Budd automatic strain indicator was used during the test.

3.2 Strain Gauges on Concrete

Polyester gauges, PL-2~ were applied on the concrete surface at the

top of the beam along the longitudinal center-line and ahalf inch from the

top along one side of the beam, Fig. 3. The gauges on the concrete gave the

strain and curvature distributions along the span of the beam at any parti

cular loading stage.

3.3 Lever Arms at End Supports

Two small angles 3" x 1 1/2" x 1 1/2" were cemented at the mid-height

of the beam at the end-supports. At the time of test, rigid steel bars were

clamped to the angles in a horizontal position. Direct Voltage Differential

Transducers (DVDT) were placed over the bars at a distance of 20 inches from

the end-support. Continuous deflections measured at these points, by means

of a Sanborn automatic recorder, gave values for the end-rotations at any

particular load.

3.4 Deflections at Midspan

A small angle was also cemented at the midspan and another DVDT was

placed at this point. Continuous deflections were recorded to ultimate.

- 8 -

CHAPTER IV

EXPERIMENTAL RESULTS AND ANALYSIS

1 10 16 In experimenta1 programs' , concerned with the measurement of

hinge rotations, the most frequent method of ca1cu1ation is based on the

integration of the curvature distribution. 16 26 38 The use of c1inometers ' ,

4 16 38 to measure rotation direct1y and the use of def1ection gauges' , to in-

direct1y ca1cu1ate relative rotations are a1so common.

4.1 Curvatures and Curvature Distributions

1 Some investigators who use the curvature distribution method ini-

tia11y assume the 1ength of the hinge and th en proceed to measure the strain,

or more precise1y, the average e10ngation within that predetermined 1ength and

thence ca1cu1ate the hinge rotation. Whereas this is a very convenient method

(and especia11y so in theoretica1 analyses) this procedure does not i11ustrate

the true extent of the p1asticity region.

In this experimenta1 program, curvatures were measured at c10se1y

spaced interva1s near the critica1 section and more wide1y spaced a10ng the

remaining 1ength of the beam.

The resu1ts indicate that there is no true symmetry of curvatures at

symmetrica1 sections of the beam (Fig. 15-21). A1so, there is a wide scatter

of Moment-Curvature re1ationships for different sections a10ng the beam (Fig.

22). As a matter of fact, there is great doubt as to whether there is a defin-

able Moment-Curvature re1ationship for a given beam cross-section.

The curvature distributions i11ustrate definite maximum peaks even

at c10se1y spaced sections (Fig. 15-21). This i8 due to the concentrated strain

distribution in the vicinity of widening cracks. In between cracks, the

- 9 -

"

curvature is less. For load stages approaching the ultimate, the magnification

of the 'peak' curvatures are more pronounced in the vicinity of the critical

section (Fig. 15-21). In regions outside of the critical sections, the peaks

are not as pronounced. Here, there is a semblance of the curvature distribu-

tion 'following' the shape of the moment diagram, especially at the lower loads.

The values of the ultimate curvatures measured display a great scatter

(Table 4). These values were mainly dependent on the time at which the last

reading at the section with the greatest curvature was taken. The automatic

recorder for the Budd indicator was available only for half of the tests per-

formed and the other results had to be taken manually.

Experimental results in the later experiments however indicate maxi

-1 mum curvature readings in the range of .006 to .008 in with an exceptionally

-1 high value of .026 in in beam lPW6-A. An analysis of the sections at which

it was still possible to measure curvature readings after considerable yield-

ing shows that the maximum curvature rose sharply at the most critical section

and decreased or remained constant at immediate neighbouring sections. This

demonstrates the hinge mechanism in concrete elements and supports partially,

theoretical assumptions of hinges assumed to be located at the critical sec-

tion at ultimate.

4.2 Strain Distributions

The strain distributions on the outermost fiber in compression are

illustrated in Fig. 5 to Fig. 13. Strain concentrations observed in the vicinity

of cracks result in peaks in the strain distributions. Outside of the critical

region the strain distribution closely followed the shape of the Bending Moment

diagram.

- 10 -

The ultimate strains measured display a wide scatter. Eighty-five

percent of absolute ultimate concrete strains measured within the hinge

length varied from .002 to .0052. A histogram of the ultimate strains mea

sured in the critical hinging region is shown in Fig. 14. Since most of the

rotation is assumed to occur within this region, it is felt that the value

of the ultimate strain to be used in a theoretical analysis must lie within

the range indicated in the histogram. More results of this nature are re

quired to evaluate statistically the much sought after "true" value of ulti

mate or limiting strain in flexure of concrete to be used for rotation cal

culations.

4.3 Rotations

Measurements for rotations were calculated directly by integrating

under the curvature distribution along the span. MOment·Rotation curves for

the beams tested are shown in Fig. 23 to Fig. 35. The rotations indicate

total rotation for the hinging length.

In inelastic theory analyses, the plastic rotation capability of

hinging regions is of prime importance. The moment-rotation curves enable

the designer to de termine the moment distribution at any section. From the

moment-rotation curves as shown, the elastic contribution of rotation must

be subtracted.

A real problem is to de termine the contribution due to elastic action

and that due to plastic action in reinforced or prestressed concrete elements.

In other words, how much of the rotation is available for redistribution of

moments?

In prestressed concrete, there are 3 distinct stages of the Moment

Rotation characteristic. The first stage relates to the stage before cracking.

- Il -

It can safely be assumed that this stage is elastic. The second stage relates

to the region after cracking has taken place to the time of yield (normally

defined as when yielding of steel has begun); and the third stage shows the

portion of the curve between yielding of the steel and the ultimate moment.

For convenience of calculations and analyses, it is normal1y assumed

that the first linear stage can be extended and the value of the e1astic

rotation at any particular moment can be determined. The remainder of the

rotation is then assumed to be plastic. Imp1icit in this calculation is the

assumption that plasticity of concrete occurs as soon as cracking begins.

True plasticity in concrete occurs only to a limited degree. The extent of

the deformations is due to the opening of cracks. It must be kept in mind

that the value of rotation calculated as above is a maximum value.

4.4 Hinge Rotation measured from Curvature Distribution

The hinge 1ength is vaguely defined as the length in which a large

portion of the inelastic rotation is concentrated. An examination of the

strain and curvature distributions indicate that within a length of five

inches (l.ld) on either side of the midspan, the strains and curvatures re

corded were distinctly magnified as compared to the readings outside of this

region. The magnitudes of the concrete strains on the surface of the beam

in this hinging region were in the inelastic range.

From the curvature distribution, it was evident that there were

portions outside of the hinging region which were not elastic. However, for

the purposes of calculations in inelastic analyses it was felt that the plastic

hinge rotation was more realistical1y estimated by measuring the rotations con

tributed from the ten inch hinge length.

- 12 -

Graphs of moment versus total rotation over the hinge 1ength are

shown in Fig. 23 to Fig. 35. The e1astic contribution was determined by

extending the initia11y 1inear portion of the graphe The plastic rotation

avai1ab1e in the hinge is the difference between the total rotation and the

e1astic rotation. These values at u1timate are shown in Table 5.

4.5 Average Curvature over the hinge length

The average curvature over the hinge length is defined as that cur-

vature which is assumed to be constant over the hinging region and which will

produce the required rotation over the hinge length. The average constant

curvature value is a convenient value to be used in ca1cu1ations but is not

rea1istic even in constant moment regions. Average curvature values at the

u1timate are shown in Table 4.

4.6 Rotation measured from Midspan Def1ections

The deflection at midspan is assumed to comprise of an elastic de-

f1ection together with a plastic deflection. The elastic deflection is that

represented by the continuation of the initial linear uncracked re1ationship.

The plastic deflection is the difference of the total def1ections and the ela-

stic deflection.

-lle. The plastic rotation, 9 is equal to 2 tan • The elastic p z

contribution to rotation is determined using the second Moment-Area theolem.

The values of these rotations are shown in Table 5.

4.7 Rotation measured from Lever Arms

The angulat" end rotations are determined by dividing the measured

- 13 -

deflection of the lever by the length of the arm from the support. These

values are shown in Table 5.

4.8 Total Rotation measured from Curvature Distribution

The total rotation over the span of the beam at ultimate obtained

by integrating under the curvature distribution is also shown in "Table 5.

A comparison of the results at ultimate of the total rotations

as obtained [rom the midspan deflections, the curvature distribution and the

lever arms show good agreement considering the difficulties encountered in

measuring deformations at ultimate.

- 14 -

CHAPTER V

THEORETICAL ANALYSIS

The ana1ysis as presented is one which is based on the equi1ibrium

of forces and the compatibi1ity of deformations. The genera1 ana1ysis for

ca1cu1ating the MOment-Curvature re1ationship from the point of cracking to

u1timate is presented. It is assumed that the re1ationship for the uncracked

section is 1inear. An equation for the moment and curvature at cracking is

given.

5.1 Assumptions

The fo11owing assumptions are made in the ana1ysis.

1) The concrete stress distribution in flexural compression is of the form

shown in Fig. 4, i.e. parabolic to the maximum stress and horizontal to

the u1timate strain.

2) The strength of concrete in tension is neg1ected.

3) Concrete strain distribution is 1inear across the section.

4) The stress-strain re1ationship of the prestressing steel is as shown in

Fig. 2.

5) The bond between concrete and steel is perfecto

6) In under-reinforced sections, the section is said to have failed when the

ultimate concrete strain is reached.

7) Yie1ding is defined as the point when the strain in the steel reaches the

yie ld va lue E: • sy

Each of these assumptions requires careful consideration, and a

discussion follows.

- 15 -

Assumption 1

27 31 33 The stress distribution in concrete in f1exural compression ' ,

has been the subject of much controversial discussion. Most investigators

have assumed that the stress-strain curve of concrete in uniaxial compression

is a good approximation of the stress distribution in concrete under bending

with or without axial load. The result is that attention has been focussed

15 18 on obtaining the complete stress-strain curve under uniaxial compression ' .

It is now well-established that the form of the stress-strain curve and also

the ultimate strain observed is a function of many parameters. These para-

meters include

i) The type of loading machine used, i.e. whether the loading head is con-

tro1led by deformation or by applied load.

ii) The rate of straining.

iii) The size of the gauge length used.

iv) The manner of measuring strains.

v) The type of specimen used.

vi) The length of time the load is sustained near ultimate.

Beyond the ultimate load, the specimen is generally badly cracked and the re-

sponse to load is highly sensitive to time.

Added to all the above factors involved in determining the stress-

strain curve in uniaxial compression is perhaps the most important considera-

tion which is whether it truly represents the stress response to strain where

the stress is dependent on the space gradient of strain. Also, one must con-

sider whether the state of stress is one of simple bending or whether the

section is in fact subject to a complex distribution of localized stresses.

After taking all the above into consideration, the form of the

stress-strain curve is determined and the value of the absolute ultimate strain

- 16 -

in the concrete must be found. In ine1astic theories where u1timate values

are of major importance, perhaps the most important parame ter which affects

the values obtained in theoretica1 analyses is the value of the u1timate

strain in concrete. 15 18 27 31 Here again many investigators ' , , have attempted

to solve this prob1em. The u1timate strain de pends 1arge1y on whether the

concrete is we11-bounded or not. That is to say, it depends on the degree

of confinement of the concrete. To what extent confinement is obtained will

de pend not on1y on the spacing or volumetrie percentage of the binding rein-

forcement but a1so on the manner in which the binding reinforcement is ap-

p1ied. The effect of rectangu1ar, spiral, vertical, inc1ined binders p1aced

inside or outside of the concrete section will certain1y affect the value

of the u1timate strain in concrete.

On the basis of tests performed by Hognestad on reinforced concrete

members subject to combined bending and axial 10ad, the author has adopted

the form of the Ritter's parabo1a for the stress-strain curve up to the

u1timate strength of concrete as obtained by test cy1inders. The value of

~ = .002 is obtained as the 10wer 1imit of the strains obtained in the exper~ o

mental resu1ts in the hinging region of the beams tested. The equation as

3 derived by Cor1ey was adopted for the u1timate strains.

~ u

b P"f" ) 2 = . 003 + . 02 + (----Y

z 20

The values th us obtained lie within the range of the experimenta1

values obtained during the test.

Assumption 2

A1though the tensi1e strength of concrete i8 approximate1y 10 per-

- 17 -

cent of the compressive strength in uniaxial compression, the effect of in

cluding the tensile strength is negligible near ultimate conditions.

Assumption 3

Results obtained from tests to de termine concrete strains over

the depth of the section show that the strain distribution is linear across

the depth of the compression block and fairly linear across the tensile

region near ultimate. Because the curvature was measured within the compres

sion region, this assumption is valid for the curvature. However, in the

analysis of ultimate moment, a small error may occur depending on the ultimate

strain in the steel. In the analyses performed, this error is negligible

because the stress in the steel at ultimate invariably occurred in the almost

horizontal region of the stress-strain curve for the steel.

Assumption 4

The stress-strain curve of the prestressing strand used was ob

tained using an extensometer with a gauge length of twenty-four inches. The

dotted line represents the region where the extensometer was removed. The

ultimate load and the total elongation was used to define the ultimate co

ordinates.

Assumption 5

For the convenience of analysis, bond between concrete and steel

was assumed perfecto Although perfect bond is impossible to obtain, the bond

obtained as measured from the observations of the crack patterns was suffi

ciently good to justify the assumption.

- 18 -

5.2 General Formu1ae

Forces in Concrete

Case (i) € c ~ ~o

C ab r fI [2 .f. (..f.) 2] d€ =-

c Cc c €. é 0 0

0

ab f' ~ 3~ - € c c ( 0 c) (1) = éo 3 ~

0

Case (ii) E: c '> é o

Ccl = (~) f' b (lo) a (2) 3 c E'c

f - € Cc2 = ( c 0) a f' b (3)

€ c c

[~ fo ~ - <; C = + ( c 0) fI b a (4) c 3 € ~c c c

Moments in Concrete (about Neutra1 Axis)

Case (i) / c:c

M = (!!....)2 b fI J [2 ~ -c € c é c 0

o

8 e - 3 ~ (

0 c) 12 E: o

(5)

Case (ii)

5 Mc1 - 12

Compatibi1ity of Strains

~c €. sa - = -..,....:==---,-kd d (1 - k)

- 19 -

+

é. = (1 - k) E + é s k c se

Equi1ibrium of Forces

Case (i)

Le.

bkd fI ~ c C

€ o

3€ - G ( 0 c) = A f = pbd f

3 é s s s o

(6)

(7)

(8)

(9)

(l0)

(11)

From equations (9), (11) and the stress-strain curve of steel, the unknown

parameters € , é , k can be found if one of € , , , k is assumed. c s c s

- 20 -

Moment

Taking moments about the neuta1 axis

Curvature

Case (ii)

a 2 b fI If. c c M = ---.,;~.;;.

~o

8~ - 3~ ( 0

12 e c)

o

Equi1ibrium of forces gives

f s

Similar1y

fI ·C

=-p

+ A f s s (d - a) (12)

(13)

Using (9), (13) and stress-strain curve ~ , é , k can be determined. c s

"1 = ~t:. ., kd

Moment at Yie1d

The moment at yie1d i8 defined as the moment when the strain in the

reinforcing reaches the yie1d va1ue,( • From eqn. (8) sy

say

- 21 -

By equating the force in the steel with the force in concrete as before

e ,k can be found. c y

Also, M is found by taking moments about the neutral axis. y

Moment at Cracking

The moment at cracking was found by equating the sum of the compres-

sive stresses produced by the prestressing and the tension stress produced by

the flexural moment at cracking to the tensile stress of concrete in bending.

midspan.

i.e.

or

The

p P ey M y -E + --E-:. + cr = A l l

! !P [f + (1 + M = cr y r A

\fi M = cr cr E l c

as used expression for f r

f r = 3000

3 + 12,000

fI c

f r

A e l!y) ]

by Warwaruk 4 was adopted

The value of E l was determined from the load-deflection curve at c

Tri-linear moment-curvature relationships were calculated using the

above analysis for each beam tested and the results are presented in Table 2.

5.3 Hinge Rotation

The moment-curvature relationships as derived from the above analysis

were used to calculate the hinge rotations of each beam.

- 22 -

l 10 The plastic region at ultimate is defined by some authors' for

reinforced concrete as the region where the moment exceeds the yield moment.

This definition is convenient but it hardly represents the true length of the

plasticity zone if it infers that outside of this zone the beam is truly

elastic, that is uncracked. It is well-known that sections are cracked

long before the steel reinforcing begins to yield.

In inelastic analysis, it is the plastic rotation within the hinge

length which is of real significance and not the contribution due to plastic

rotation over the entire beam. The hinge length represents the length over

which a major portion of the plastic rotation is assumed to be concentrated.

However, it must be remembered that in the analysis, the rotation is assumed

to occur at a particular section and hence the contribution to the hinge

rotation must necessarily be derived from that in the immediate vicinity of

the hinge.

1 14 21 Many authors' , have for convenience assumed the hinge length

23 for reinfcrced concrete to be equal to the effective depth. In the Report

of the Institution of Civil Engineering Commit tee it is reported that the

hinge length may vary from O.4d to 2.4d.

In this analysis, the hinge length, based on experimental evidence,

is taken to be approximately 2.2d or 10 inches.

The total rotation within the hinge length was calculated by

integrating under the curvature distribution obtained from a simplified tri-

linear moment-curvature relationship.

The total elastic rotation was calculated by extending the moment-

curvature portion for the uncracked section and deriving the curvature dis-

tribution using the uncracked section relationship.

The total plastic rotation was taken as the difference between the

total rotation and the elastic rotation.

- 23 -

." CttAPTER VI

CONCLUS IONS

6.1 Comparison of Experimental and Theoretical Results

For each beam, total rotations were calculated by integrating under

the curvature distribution, by deflections measured at midspan and by rota

tions of the lever arms. The results obtained were found to be comparable

with deviations of approximately 10-15 percent in most cases (Table 5).

The values of ultimate hinge rotations obtained experimentally

were compared with those obtained theoretically. Here again there was good

correlation with deviations of the same magnitude (Tables 3 and 5).

The average ultimate curvatures as measured within the hinging

region (Table 4) showed deviations of 10-20 percent from the values calculated

theoretically. This is thought to be within acceptable limits of deviation

for deformations at ultimate.

An attempt was made to evaluate experimental moment-curvature rela

tionships for various sections within the hinge length of each beam. The

results obtained were very scattered, thus confirming that there is no unique

moment-curvature relationship for a given section (Fig. 22). As a last re

sort, a tri-linear curve was fitled using a least squares law. The rotations

obtained in the hinging region were compared with the rotations obtained from

the curvature distribution (Fig. 23 to Fig. 34). Because of the wide scatter

of the curvatures at different sections, it is not recommended that evalua

tion of a moment-curvature relationship for a given beam section be attempted

experimentally.

- 24 -

6.2 Hinge Rotation and Total Rotations

Effect of Web Reinforcing (Fig. 44,45,46 and Table 6). At both

leve1s of prestress of 147 ksi and 109 ksi, the effect of reducing web

reinforcing spacing is to increase the amount of rotation.

At the higher 1eve1 of prestress, the resu1ts do not indicate

the trend very strong1y because of the relative ineffectiveness of the web

reinforcing in confining the compression b10ck of the concrete. The depth

of the compression b10ck at u1timate was approximate1y 1.25 inches whi1e

the concrete cov~ at the top was 0.5 inch.

At the 10wer 1eve1 of prestress, the effect is distinct1y shown

in both the hinge rotation and the total rotation.

If confinement was more efficient a greater variation of rotation

capabi1ity cou1d be expected using the same percentages of reinforcing.

Better confinement cou1d be obtained by using rectangu1ar spirals or by

placing the stirrups so as to enclose the compression b10ck more effectively.

Effect of Prestressing Force (Fig. 47,48 and Table 7).

The effect of increasing the prestressing force is to decrease the rotation

capacity. This is shown for the different spacings of web reinforcing.

Again, the effect is recognized more distinct1y by comparing the total rota

tions. The effect is magnified by considering the rotations for the beams

with no prestressing force.

The mode of fai1ure at high prestressing forces tends to be of a

more sudden britt1e nature than that at lower prestressing forces. The

higher the prestressing force, the higher is the cracking load, the shorter

is the region from cracking to yield, and the smal1er the ultimate rotation.

- 25 -

6.3 Differences between Reinforced and Prestressed Members

The differencesbetween reinforced and prestressed members in hinge

formation and rotationcapBbility are due to the effects of the axial load

and also to the different types of reinforcing normally used in reinforced

and prestressed concrete members.

Axial load with an eccentricity produces a longitudinal compression

and a uniform bending moment in the opposite direction to the load bending

moment. The effect of the longitudinal compression is ta produce a stiffer

member and the effect of the moment is to produce a camber in the member.

An increase in the axial load increases the stiffness of the member

with the result that the elastic portion of the rotation is also increased.

The mode of failure is of a more brittle nature and the total ultimate

rotation is lower.

By keeping the prestressing force constant and increasing the eccen

tricity of the force, the prestressing moment can be increased. If the camber

is sufficiently large and the load moment is applied in the opposite direc

tion, the curvatures along the member may or may not be reversed, dependent

on the magnitude and the type of loading conditions. That is to say, there

may be points along the beam with increasing negative curvatures. This be

haviour was observed experimentally in regions away from the hinge. This

effect would tend to reduce the total rotations of the beam.

Tension reinforcement for reinforced concrete members normally

has a stress-strain relationship which exhibits a definite yield point stress

and plateau. The yield stress i6 much lower than that for reinforcement

used in prestressed concrete members.

Provided that the section is under-reinforced and the same area

- 26 -

of steel is used for both types of reinforcing, th en at ultllnate the force in

the mild steel is much lower than that in the prestressing steel. Rence,

the force in concrete compression is smaller for the reinforced concrete sec

tion and the depth to the neutral axis is less. The ultllnate curvature and

ultllnate rotation is therefore greater than for the prestressed concrete

section (Tables 2,3,4,5).

The above discussion also applies when the yield strain is reached

in the prestressing reinforcement.

Up to the point of yield of mild steel, the behaviour of the sec

tions using both types of reinforcement will be the same.

6.4 Recommendations for Future Research

The number of variables that affect the rotation capacity of

prestressed concrete members is so great that information must be gathered

from separate sources.

Parameters such as percentage of reinforcing, amount of compression

steel, effects of different types of binders, span/depth ratio and varying

moment gradients need to be studied before an understanding of hinge rota

tion can be achieved in indeterminate structures.

Limit Design theories can only be used as a practical design method

when sufficient confidence can be given to the values of the hinge rotation

capability. Because of the number of parameters that influence the hinge

length and the hinge rotation capability, it is evident that a statistical

analysis using more results would be the logical solution to this problem.

It is recommended that smaller sized specimens be used in further investiga

tions to decrease the manual labor involved and to increase the efficiency

of producing more data. Further research must include studies on simple in

determinate structures.

- 27 -

: =

V k i'-'t.

( 1-1 .u CIl

CIl CIl Q) 1-1 .u CIl Q) 1-1

Il-<

~rI "1 0

\0 N 0\ .-f

W ['-1

Beam Dimensions

Fig. 1

. ... . ,;., .. ,~.; ~.,:,.' .. -':..: : ... L!, .. ' "

10

8

"'"' CIl 0. 1

":1 or! .... ~ 6 ()Q '-"

"0 N cu

0 ...:1

4

2

. 01 ~~!i

- ---

.02 .03

Load - Strain Curve

1/4 inch diameter Prestressing Wire

-- - - ---

6 E = 30.2 x 10 psi s

.04 .05

Strain

.06

-?C

.07 in in

N (Xl

CI) (')

=-~ " .... (')

~ en "CI t;

'Tj en ID .... en

OQ ::s " III w " .... -0 ::s 0 HI

H ::s CIl

" t;

m ::s " III

" .... 0 ::s

DVDT Lever

~ f..

33"

21"

9"

ATm \ \ \ \ \ \ 1 \ 1 ,.concrete strain ------ / gauge 1/2" - -- - - - r~ ------- -

IQ H--.........--. 1/4" l'

I-.J \0

Steel Strain Prestress Gauge Wire

Def1ection Gauge

~

20" 20"

96"

120" _~ J ____________ _---"='-- ____________________ _ ,

1

d

b

f

fi C

\

c

LA

Section

1

s

- 30 -

f f i [2 é: (;0)2] C C_=r--C _é_O ----~ --,

€ ( o u c

Concrete Stress-Strain Distribution

f

H Cc2 Cc

Ccl

T "" T

l t s s

Case (i) Case (ii) é é

sa se

Strain Forces

Fig. 4

,< - '.\ :1..

- 31 -

~ 6

48

= 5.55k

Strain Distribution

Bearn 1 PW3 - A 3

1

-5 CIL 8 16 24 32 48

Distance fram Center-Line (inches)

Fig. 5

64

48

32

16

-5 CIL

k = 5.5

8

- 32 -

Strain Distribution

Beam 1 PW3 - B

16 24 32 40 48

Distance from Center-Line (inches)

Fig. 6

- 33 -

48

Strain Distribution

Bearn 2 PW3 - A 32

16

-5 CIL 8 16 24 32 40 48


Fig. 7

..;t 1

o !=lI ....... .-j.

~

1 1

1 1

Max é = .016 u

64 'I

48

~P

32

16

-5 CIL 8

- 34 -

Strain Distribution

Bearn 1 PW6 - A

16 24 32 40 48


Fig. 8

64

48

32

16

-5 CiL

k ~P = 5.4

8

- 35 -

Strain Distribution

Beam 2 PW6 - A

16 24 32 40 48


Fig. 9

- 36 -

48

• ...--p = 4.8k

32 Strain Distribution

Bearn P4

16

-5 CIL 8 16 24 32 40 48

Distance from ··Clmter-Line (inches)

Fig. 10

"" 1

o C::I .-1 .......

>:

32

16

-5 CiL 8

- 37 -

Strain Distribution

Beam PS

16 24 32 40 48


Fig. 11

48

32

16

-5 CIL

k ,r-P = 4.8

8

- 38 -

Strain Distribution

Bearn P8

16 24 32 40 48


Fig. 12

- 39 -

64

48

Strain Distribution

Bearn RlA

32

16

-5 CIL 8 16 24 32 40 48


Fig. 13

4

al oU ~

"" 0 o 1 Po<

~ 0

t'Zj ~ .... CIl OQ ~ t-' Z 12 ~

8

-

4

o .0024

Histogram of Ultimate Strain Values in Hinge Length

Web Reinforcement at 6 inches

1 1

1

No Web Reinforcement or

Web Reinforcement at 9 inches

l

1

.0032 .0040 .0048 .0056

-

1

1

1

.0064

.

.0072

1 in in

~ o

• 1

Il

64

48

32

16

-5 CIL

- 41 -

\11 = .00796 in-1 u

P = 5.55k

8 16

Curvature Distribution

Bearn 1 PW3 - A

24 32 40

Distance fram Center-Line (inches)

Fig. 15

48

64

48

32

16

-5 CIL

- 42 -

Ip = .008 in- 1 u

8 16 24


Beam 1 PW3 - B

32 40


Fig. 16

48

16

-5

" ,1

CIL

'-Pu

- 43 -

. -1 = .008 ~n

8 16


Bearn 2 PW3 - A

24 32


Fig. 17

48

- 44 -

\ilu = .02634 in- 1

64

48

~P


Beam 1 PW6 - A 32

16

-5 CIL 8 16 24 32 40 48


Fig. 18

"" r-I 1 1 - 45 -o r-I J:l

• .-1 li' u = .00976

-1 in

64

48


Beam 2 PW6 - A

32

16

-5 CIL 8 16 24 32 40 48


Fig. 19

- 46 -

64

48


Beam P4

32

16

-5 CIL 8 16 24 32 48


Fig. 20

1 1

64 1 1

'fi u = .0099 -1

in

k ~p = 4.9

48 /"

32

16

-5 CiL 8

- 47 -


Beam RIA

16 24 32


Fig. 21

40

,.... s::

..-1 >Jj 1-"

OQ P-. ..-1

N ~ N '-'

~ .S::

Q)

l· 1 .

12

80

1 ( ./

40

.002

Typica1 Enve10pe of Moment-Curvature

Re1ationship

.004

Beam 1 PW3 - B

-1 .006 in

.po oo

--... ------120

Moment vs Hinge Rotation

,,-.... r::

..-l r 0 Bearn 1 PW3 - A

t'%j 80 (M = 133.0 kip-in)

1""- 0-u

O'Q ..-l . ~ X Bearn 1 PW3 - B +='

~ (M = 132.0 kip-in) \0

N U W ~

r:: Least Squares Q)

a 1 0 ~

1

1

1 40 ~I

1

1

~ 11, ':\? L..R v ln ""!:Irl; !::IInc:

t'!j ~.

OQ

N ~

120

,.... = ~

~ 80 ~ '-"

.u = g ::l!:

40

1 1

Il

8

j/

16

M = 127.2 kip-in --. u

/ Moment vs Hinge Rotation

Bearn 2 PW3 - A

- - - Least Squares

24 32 40 48 56 -3 x 10 radians

VI o

,.... = ..-4

02:1 1-'-

OQ p.. . ..-4 ~

N -VI ~ = ~ ~

120

80

40

1

1

p

8 16 24

,.. .Ir

o o


0 Bearn 1 PW6 - A (M = 129.5 kip-in)

u )( Bearn 1 PW6 - B VI

1-' (M = 134.2 kip-in) u

- - - Least Squares

32 40 48 56 x 10 radians

-120

/


;-.. / ~

/ 0 Bearn 2 PW6 - A

'o-l (M = 129.5 kip-in) t'%j 80 u ...,.

0. / Bearn 2 PW6 - B OQ x . oo-l ..!I\ (M = 125.0 kip-in) lJ1

N '-" Il U N 0\

/ oU - - - Least Squares ~

~ 0 / ~

f 1 l,

40

8 16 24 32 40 48 56 -3 x 10 radians

120

,.... s::

·rI

~ 80 t-'o 0.

1 JQ . 'rI

~ N '-' -...J .w

s:: Q)

~ // ~

40

//

//

8 16 24 32

M = 113.0 kip-in u


Bearn P2

- - - Leas t Squares

40 48 x 10

·1

VI w

radians

>'Jj 1-'.

OQ

N 00

120

3 80 t / / p.. / ~ ~ ...... .u / ~ QI

~

'" 40 ~ ,r Il 1

1

8

/ /

/

16 24 32

----M 124.8 kip-in

u


Bearn P3

- - -Least Squares

40 48 -3 x 10 radians

\.J1 ~

120

,...., 80 c:: .,-1

'::1 J? ..... OQ Po. .

"'"' ~ N .......

// \0 oU c:: al

~ ~

40

8

/ /

/

,/

16 24 32

M = 120.0 kip-in u


Beam P4

Least Squares

40 -3 x 10 radians

\JI VI

120

,.... 80 r // ~ ~

t'%j t-"o

OQ P-.,,4

U) ~ 0 -

.j.J

~

~ 0

::t::

40

'/

8

/ /

/

16 24

M = 113.5 kip-in u


Bearn PS

- - - Least Squares

32 40 -3 x 10 radians

U1 (J'I

I%j t-'.

OQ . VJ 1-'

120

,..... ~ 80

0.. -..1 ~ '-'

.u

J

40

8 16

7

/

24 32

M = 124.8 kip-in u


Bearn P6

- - - Leas t Squares

40 -3 x 10 radians

lJ1 -...J

120

'2 80 '.-1

'>:j 0. .....

'.-1 OQ ~ . '-"

w ~ N t::: Q)

g ~

1 Il

~ 40~ fI

~ ~

8 16 24 32

M = 126.5 kip-in u


Beam P7

- - -Least Squares

40 -3 x 10 radians

~

~.

U1 00

,-..

= ..-1

t'%j ... ()Q p.. . ..-1

W ~ '-'

W oU

~ ~ ~

120

80

40 t-, 1

(

1

1 1 1

8

/'

/' /'

16

/

./

/'

24 32

M = 117.5 kip-in u


Beam P8

- - - Least Squares

40 -3 x 10 radians

Ut \0

>zj .... OQ

w .p-

12

'2 8 ·roI

P. 'roI ~ '-"

.u

~ o ~

8 16 24 32

.. -- -K .-.4' ;> --

M = 125.0-kip-in u


Bearn 9 PW9 - B

- -- -- Least squares

40 48 -3 x 10 radians

0'\ o

I"%j .... OQ . \.Al \J1

" .E

P. '.-1 ~ '-'

.u c::

1

120

80

40

r 8 16 24 32 40

o


x

o

Bearn. R1 - A (M = 120.0 kip-in)

u

Bearn. R1 (M = 117.5 kip-in)

u

-3 48 x 10 radians

0' t-'

y. o

o

120

Moment vs Total Rotations (Lever Arm)

:§' 80 1 lr

"':1 / x Bearn 1 PW3 - A

.... OQ . p. Je/ 0 • .-1 0 Bearn 1 PW3 - B (,..) ~

IJ'

IJ' '-' ~

oU ~ al a 0 ~

40

8 16 24 32 40 48 56 -3

x 10 radians

120

"""' "%j

= 80 -,-l

1-'-OQ

0-or-l

W ...... ~ ..., .u = J

40

8 16

o

o

o

24 32 40

o

~


o Bearn 2 PW6 - A

X Bearn 2 PW6 - B

48 56 -3 x 10 . radians

0\ W

o

o o

-"t

120


-s:: 80 "%j .,.,l

.....

/ x Bearn 1 PW6 - A OQ .

P- a-w o,.l .p-oo ..!o:

Bearn 1 PW6 - B '-'" 0 .u s:: Q)

El 0 ~

40

8 16 24 32 40 48 56 -3 x 10 radians

120

-r:: t'%j ~ .... OQ 80

0.. W ." \0 ~ -

.j.J

s:: al S 0 ~

40

8 16 24 32 40

Moment vs Total Roations (Lever Arro)

Bearn P2

48 56 -3 x 10 radians

0'\ \..TI

120

,..... c:: 80 'rl

0. orl ~ '-"

oU c:: al g l::

40

8 16 24 32 40


Bearn P3

48 56 64 x 10

0' 0'

radians

,,-..

"!j ~ 1""

OQ . p..

.p-t-'

.~

~ '-"

+J I=l QI

~

120

80

40

8 16 24 32


Bearn p4

40 48 56 x 10-3 radians

0\ -....J

120

""' r::: "'1

'rl

1-'- 80 (JQ . 0. .j:'-

.~

N ~ ........ ~ r::: al

5 ::E:

40

8 16 24 32

Moment vs Total Roations (Lever Arm)

Beam 9 PW9 - A

40 48 56 -3 x 10 radians

0\ (Xl

6

5

4

'"' tf.I t>j P. 1"" ..-l ()Q ~ . '-'

~ '"é lN tU 0 3 ~

s:: tU p. tf.I

'"é .,.c ::E:

2

1

.2 .6

,c 9 Q X

Typical Midspan Load vs Midspan Deflection

x Beam 1 PW3 - A

o Beam 1 PW3 - B

.8 1.0 1.2

0'\ 1..0

inches

120

-~ s:: .,-l

t-'. 80 ()Q .

al ~ p..

o.-l ~ ~ -

~

s:: <Il !3 0

1 q. 11 ~

40

8 16 24 32 40


Effect of Web Reinforcement

f = 147 ksi pe

A 3 inches spacings

,.;. 6 inches spacings

0 9 inches spacings

48 -3

x 10 radians

1 PW3 - B

1 PW6 - B

9 PW9 - A

-..J 0

120


Effect of Web Reinforcement

f = 109 ksi pe

/// ,.... 80

'%j .E 1-'-

OQ . c.

~ ..-1 Ut ~

Il / A 3 inches spacings 2 PW3 A '-1 '-' t-'

oU s:: al

il/' ~ ~ 6 inches spacings 2 PW6 - B ~

/// 0 Infinite spacings p4

40

8 16 24 32 40 48 -3 x 10 radians

120


Effect of Prestressing Force

Web Spacing = 3 inches

80 " s::

~ .,-l

..... 1 1 / Il f = 147 ksi 1 PW3 A OQ -...J .

p.. pe N

~ .,-l

0- ~

1 1 / f = 109 ksi 2 PW3 - A '-' 0 ~

pe s:: CIl a 0 ~

40

l 2 32 4 48 -3 x 10 radians

120



Web Spacing = 6 inches

"'1 - 80 .... r:: OQ ~ . 1 1 f 147 ksi 1 PW6 B A ...... ~ 0-

pe w ......

.~

~

/ / ..., ·f = 109 ksi 2 PW6 - A 0 "-1

pe r:: Q)

!3 ~

40

8 16 24 32 40 48 56 x 10-~ radians

120



'2 80 ~ // / No Web Reinforcement ..-l

l'Tj ra-

OQ 0. ...... .

/ / • ,.l / .po. .po. ~ 00 '-'

oU

f = 147 ksi P8 s:: f r / A CIl pe a 0 ~

0 f pe = 109 ksi P4

40 l f / / )C. f = o ksi R1 pe

8 16 24 32 40 48 -3 x 10 radians

- 75 -

BEAM DESIGNATION

Designation fI é se Web

c Spacing p.s.i. x 10-6 in (inches)

in

1 PW3 - A 6350 4870 3

1 PW3 - B 6350 4870 3

2 PW3 - A 6250 3600 3

1 PW6 - A 6000 4720 6

1 PW6 - B 6000 4720 6

2 PW6 - A 5840 3320 6

2 PW6 - B 6250 3600 6

pl 6100 4785 None

P2 6500 3040 Il

P3 5840 3320 Il

p4 5830 3710 Il

p5 5800 3730 Il

p6 6550 4010 Il

P7 7000 4400 Il

p8 6200 4870 Il

9 PW9 - A 6430 4575 9

9 PW9 - B 6430 4575 9

R1 6550 0 9

RlA 6550 0 9

Table 1

M Bearn u

kip - in

1 PW3 - A 109.9

1 PW3 - B 109.9

2 PW3 - A 109.53

1 PW6 - A _108.46

1 PW6 - B 108.46

2 PW6 - A 107. 72

2 PW6 - B 105.36

Pl 114.91

P2 109.51

P3 107.84

P4 107.8

P5 102.67

p6 110.33

P7 109.66

P8 108.94

9 PW9 - A 113.14

9 PW9 - B 113.14

RI 109.08

RlA 109.08

THEORET ICAL MOMENT -CURVATURE

'fJ u M 'II y y

10-6 . -1 x ~n kip - in 10-6 . -1 x ~n

4980 97.68 2430

4980 97.68 2430

4900 97.32 2820

4120 96.21 2520

4120 96.21 2520

4000 96.54 2958

4280 97.90 2800

3310 97.30 3260

3800 97.45 3070

3370 96.09 2970

3370 96.09 2860

3160 95.61 2850

3800 98.45 2660

4180 99.93 3070

3600 97.33 2480

3600 97.92 2850

3600 97.92 2850

3870 99.97 3810

3870 99.97 3810

Table 2

M cr kip - in .

52.8

52.8

43.1

52.1

52.1

40.4

43.4

49.0

46.0

40.4

44.2

43.4

46.3

49.7

52.6

52.3

52.3

14.9

14.9

\41 cr

x 10-6 in- 1

162

162

132

160

160

124

133

150

141

124

136

134

142

152

162

161

161

46

46

'-1 0'\

THEORETICAL ANALYSIS OF HINGE ROTATION

U1tirnate (rad) Yie1d (rad) Cracking 1

Bearn geh 9ph 9 th 9 th

geh

1

1 PW3 - A .0032 .03574 .03894 .0217 .00154

1 PW3 - B .0032 .03574 .03894 .0217 .00154

2 PW3 - A .00318 .03625 .03943 .0252 .00125 1

1

1 PW6 - A .00315 .03063 .03378 .0225 .00128

1 PW6 - B .00315 .03063 .03378 .0225 .00128

2 PW6 - A .00314 .03165 .03479 .02695 .00164

2 PW6 - B .00307 .0317 2 .03479 .02547 .00126 -...J -...J

Pl .00322 .02963 .03285 .0293 .00138

P2 .00318 .03313 .03631 .02777 .00133 i

P3 .00314 .02926 .03240 .02715 .00117

P4 .00315 .02889 .03204 .02604 .0013 1

P5 .00315 .02889 .03204 .02604 .0013 P6 .00319 .02871 .0319 .02418 .00133

P7 .00318 .03202 .0352 .02767 .00144

P8 .00318 .02730 .03048 .02212 .00153 1

1

9 PW9 - A .0331 .02976 .03307 .0255 .00153

9 PW9 - B .0331 .03976 .03307 .0255 .00153 1

1

RI .00316 .03440 .03756 .03576 .00043

RlA .00316 .03440 .03756 .03576 .00043 -- 1

Table 3

- -ULTIMATE STRAINS. CURVATURES. DEFLECTIONS Expt. Expt. Theoretica1 Expt. Avg.

Beam Eu

"'u 'Vu \&1 in Hinge in u

x 10-6 in- 1 x 10-6 in- 1 x 10-6 in- 1 in

1 PW3 - A .00604 7960 4980 3540 1 PW3 - B .00665 7980 4980 3860

2 PW3 - A .00544 7980 4900 3880

1 PW6 - A .01592 26340 4120 5500 1 PW6 - B .00484 5560 4120 3340 2 PW6 - A .00704 9760 4000 5660 2 PW6 - B .00624 6780 4280 3240

Pl .00460 4190 3310 2680 P2 .00480 5850 3800 3040 P3 .00630 7280 3370 3960 P4 .0054 6200 3370 3180 P5 .0033 3750 3160 2080 P6 .00589 5790 3800 3200 P7 .00460 4500 4180 3160 P8 .0043 4600 3600 2480

9 PW9 - A .00672 7440 3600 3220 9 PW9 - B .00660 9180 3600 4540

R1 .0062 8280 3870 4200 RlA .00708 9900 3870 3920

Table 4

Theoretica1 Avg. 'IIu

10-6. -1 x l.n

3894

3894

3943

3378

3378

3479

3479

3285

3631

3240

3204

3204

3190

3520

3048

3307

3307

3756

3756

Expt. U1timate Def 1ec t ( in)

1. 79

1.57

1. 71

1.95

2.00

1. 70

1.56

1. 97

1.23

1.59

1.10

1.00

1.88

1.80

1.12

1.36

1.09

2.10

2.20

~ ~ '1

EXPERIMENTAL ROTATIONS AT ULTIMATE

Hinge Rotation Total Rotations Over Span Bearn (From Curv. Distr.) I(From Curv. Distr.) (From Midspan Def1ect.)

geh 9ph 9 th 9 t 9 9 9 t e p

1 PW3 - A .0028 .0326 .0354 .0697 .02 .0592 .0792

1 PW3 - B .0028 .0358 .0386 .0790 .02 .0531 .0731

2 PW3 - A .0046 .0342 .0388 .1251 .0194 .0912 .1106

1 PW6 - A .0032 .0518 .0550 .0881 .0194 .0684 .0878 1 pw6 - B .0034 .0300 .0334 .0681 .0194 .0704 .0898 2 PW6 - A .0050 .0516 .0566 .1129 .0294 .0512 .0806 2 pw6 - B .0048 .0276 .0324 .0967 .0206 .0696 .0902

Pl .0041 .0227 .0268 .0694 .028 .0690 .0970 P2 .0034 .0270 .0304 .0820 .0162 .0404 .0566 P3 .0038 .0358 .0396 .0860 .028 .0478 .0758 P4 .0084 .0234 .0318 .0577 .0172 .0344 .0516 P5 .0034 .0174 .0208 .0398 .0156 .0312 .0468 p6 .0066 .0254 .032 .0888 .0131 .0696 .0827 P7 .00398 .0276 .0316 .0692 .0181 .0630 .0811 p8 .0036 .0212 .0248 .0634 .0231 .0312 .0543

9 PW9 - A .0042 .0280 .0322 .0704 .02 .0434 .0634 9 Pl-l9 - B .0066 .0388 .0454 .0799 .0187 ,0329 .0517

R1 .0061 .0359 .042 .1309 ,015 .0775 .0925 RIA .0064 .0328 .0392 .0941 ,OIS .0816 .0966

Table 5

(From Lever Arms) et

.070

.0772

.0987

.1070

.0852

.0838

.0920

.0625

.0671

.0868

.056

.040

-.01022

.0550

.0807

.0690

.1645

-- - .

--.J \0

'-",:'

EFFECT OF WEB REINFORCING ON ROTATIONS

-6 in Spacing Hinge Total ~ x 10 in Web Reinf. Rot. (rad) Rot. (rad) se

3" .0354 .0697

3" .0386 .0790

4870 6" .0550 .0881

(147 ksi) 6" .0334 .0681

None .0248 .0634

3" .0388 .1251

3600 6" .0324 .0967

(l09 ksi) None .0318 .0577

None .0208 .0398 -

Table 6

Bearn

1 PW3 - A

1 PW3 - B

1 PW6 - A

1 PW6 - B

P8

2 PW3 - A

2 PW6 - B p4

P5

1

(Xl o

EFFECT OF PRESTRESSING FORCE ON ROTATIONS

€. se

-6 in Hinge Total Web Reinf. x 10 in Rot. (rad) Rot. (rad)

3" 4870 .0354 .0697

4870 .0386 .0790

3600 .0388 .1251

6" 4720 .0550 .0881

4720 .0334 .0681

3600 .0566 .1129

3320 .0322 .0967

No web 4870 .0248 .0634

Reinforcement 4785 .0268 .0694

4400 .0316 .0692

3710 .0318 .0577

3320 .0396 .0860

3040 .0304 .0820

0 .042 .1309

0 .0892 .0941 -

Table 7

Bearn

1 PW3 - A

1 PW3 - B

2PW3 - A

1 PW6 - A

1 PW6 - B

2 PW6 - A

2 PW6 - B

P8

Pl

P7

p4

P3

P2

R1

RlA

1

i

1

1

00 t-'

- 82 -,

APPENDIX A

NOTATIONS

a = depth of neutral axis

A = cross-sectional area of concrete

A = area of prestressing steel s

b = breadth of beam

C = compressive force in concrete in flexure c

Ccl = compressi.ve force in concrete corresponding to the parabolic

portion of the stress block

Cc2 = compressive force in concrete corresponding to the rectangular

d

D

e

E c

~s

E say

Ese

f c

fi C

f r

f s

fil Y

portion of the stress block

= effective depth of beam

= depth of beam

= eccentricity of prestressing force

= strain in concrete at extreme fiber

= concrete strain corresponding to maximum concrete strength

= strain in steel

= additional strain in steel due to loading

= additional strain in steel dce to loading at yield

= effective prestrain in steel

= yield strain in steel

= ultimate strain in concrete at extreme fiber

= stress in concrete

= maximum strength of concrete cylinders

= modulus of rupture of concrete

= stress in tension steel

= yield point stress of web reinforcement

l

k

k Y

- 83 -

= moment of inertia of beam section

= ratio of depth of neutral axis ta effective depth

= ratio of depth of neutral axis ta effective depth at yield of

prestressing reinforcement

M = midspan moment

M = resisting moment due ta concrete about the neutral axis c

Mcl = resisting moment corresponding to the parabolic portion of the

stress black about the neutral axis

Mc2 = resisting moment correspondi.ng to the rectangular portion of

the stress block about the neutral axis

M = midspan cracking moment cr

M = yield midspan moment y

M = Ultimate midspan moment u

p = ratio of area of tensile reinforcement to the product of the

pli

p

p P

T s

y

z

\Vavg

'lIy

'l'u

9

breadth and effective depth of the section

= ratio of volume of binding steel ta volume of concrete bond

= midspan load

= prestressing force

= tensile force in steel

= distance of neutral axis to top fiber

= distance along span from section of maximum moment to adjacent

section of zero moment

= curvature at a section

= average curvature over hinge length

= curvature at first yield of tension reinforcement

= ultimate curvature at ultimate moment

= rotation

- 84 -

9 = elastic rotation e

geh = elastic rotation over hinge length

9 = plastic rotation over span p

9ph = plastic rotation over hinge length

9 = total rotation r

9t

= total rotation

9 th = total rotation over hinge length

9 ty = total rotation at yield

d = midspan deflection

- 85 -

APPENDIX B

In this appendix, the experimenta1 techniques found to be successfu1 are set out

for' the benefit of other researchers.

Application of s~rain gauges on prestressing strand.

1. Locate position of strain gauge.

2. C1ean are a initia11y with emery paper. Use acetone to remove partic1es of

dust, oi1, etc.

3. Position strain gauge on the strand by means of tape (Scotch tape). Keep

in this position for a few minutes so that the gauge and tape will keep

the shape a10ng the strand.

4. Peel back the tape with the gauge unti1 the gauge is off the strand whi1e

keeping the remainder of the tape on the strand and still maintaining the

initial shape.

5. App1y Eastman-910 cata1yst on the strand and 1eave to dry for a few minutes.

6. App1y a thin coating of Eastman-910 adhesive on the gauge.

7. Quick1y and accurate1y place the gauge on the strand. Using a sharp instru

ment, cut the tape a10ng the sides of the strand on the outside of the

gauge. This is to ensure that the adhesive spreads over the entire gauge.

8. App1y firm pressure by p1acing a thumb over the gauge and ho1d for a few

more minutes.

9. Peel the Scotch tape off, thus 1eaving the gage in contact with the strand.

The technique as described was found to be quite efficient. However,

other variations may equal1y we1l be app1ied as long as it is efficient1y and

cornfortab1y performed.

- 86 -

Application of Strain Gauges on Concrete Su~face

1. Locate areas wherestrain gauges are Lo be applied.

2. Clean surface by using an electric sander. Make surface smooth and fIat

if necessary.

3. Finally, clean surface with emery paper and remove particles uSlng a

cotton swab and acetone.

4. Apply a thin layer of PS-adhesive on concrete surface and leave to dry

for at least four hours.

5. Locate the position for the gauge.

6. Apply a thin layer of PS - cement and position the gauge. Place a piece

of polyethylene over the gauge.

7. Applya small pressure on the gauge with the thumb.

8. Place a plece of foam over the gauge and then place a small weight on

the gauge for a few more hours.

9. Remove weight, foam, and polyethylene.

Waterproofing of Gages

After soldering the lead wires to the strain gauges, the gauges

and a portion of the lead wires must be waterproofed.

Gauges on the strands.

GW-5 Waterproofing was used. The setting time of this mixture is

about 1 hour. In order to get a good consistency so that the water-proofing

does not flow too easily, the waterproofing is left to stand for about half

an hour before applying to the strain gauges and about four inches of lead

Wlre extending from the gauge.

Black electric tape is then wrapped around the waterproofing to

strengthen and lessen the chances of the wires ripping away from the water

proofing and the gauge.

- 87 -

Gauges on the Concrete

GW-l Waterproofing is applied over the strain gauge and a short

length of the lead wires. The application of this waterproofing is a

precaution for the gauges from being moisturized with water from the

atmosphere.

Prestressing the Wires

After the strain gauges were installed, the wires were placed

through the jacking apparatus as shown (Fig. Bl). A small load was applied

to initially remove the slack from the wires and the zero strains were

taken.

The wires were then tensioned until.the run of the jack was fully

extended. The strain in the wires were occasionally checked. If the

required tension was not reached, bolts (A) and the grips as in the dia

gram were moved forward. The load was then released and plate (C) and the

grips were moved forward.

More load was then applied until the required tension was approxi

mately acquired. Final adjustments to each wire were made by moving the

nuts on bolts A.

Releasing the Prestress (Fig. B2)

The middle wire was cut so that the equipment (D) could he placed

within the jacking apparatus. The jack was then placed as shown and just

enough load was applied so that the blocks (E) could be removed. The load

on the jack was released slowly thus releasing the prestress in the wires.

- 88 -

Fabrication of the Concrete Beam

Mix design batches were made before the start of the program.

Portland Cement Association mix design tables were used with slight varia

tion to obtain the required concrete strength and workability.

The water content of the sand was taken before each mix was

made. This water content which was meant to be representative of the

water content of the sand to be used in the mix was sometimes incorrect

because of the variation in water content inside the bin.

Half-inch and quarter inch size coarse aggregate were used.

High early strength Portland Cement was used in order to obtain the re

quired strength in approximately ten days.

Two batches of concrete were necessary for casting two specimens.

The first bat ch was used for the bottom half layer of each beam. The

second bat ch was used for the top half layer of each beam. A 'pencil'

vibrator was used for compacting the concrete.

Four standard 6" x 12" concrete cylinder specimens were pre

pared from each batch. These cylinders were compacted according to ASTM

specifications.

The beam specimen was trowelled as nearly horizontal as possible.

The concrete was left to harden for about an hour before it was covered

with polyethylene sheets to keep in most of the moisture whilst the con

crete was setting.

The steel forms were removed after about eighteen hours and the

specimens were then covered with 2 layers of wet burlap over which the

polyethylene sheets were then placed.

- 89 -

The specimens were kept wet continuous1y for about five days

before removing the bur1ap. The specimens were then a110wed to air

cure and dry. The strain gauges were app1ied and the beam was ready

for testing about 10 days after casting.

The concrete cy1inders were a110wed to cure in a simi1ar manner

as the beam specimens.

> "tl "tl ~ 11 ~

" c: al t%j ....

OQ HI . 0 11

~ .... '"tI 11 CD al

" t1 CD al al .... ::1

OQ

~---End of Prestressing Bed

,,- -(C) , Elevation

,

---6 ------------------------jI.:Jl:=:J-;lJ;I.H+ Il - -- - - - 1 Y

) Steel Plates --/ we1ded ta Plate

8" x 6 " x 1" Removab1e Steel Blacks

Supreme Grips

l nO

/- uD ...... }

Prestressing Wires .< We1ded Frame

Bo1t and Nut (A) , (for fine adjustment of Prestress)

\0 o

- 91 -

r~ ---

-...:::

~ 1 1 ........ .-1

,-. IJ 0 ri! '-' / 1

.-1 1 1

-~ 1/

0 .-1 al

tIl m "CI 0 p::

.-1

,-. ,,1 ~

j::l ~ '-' ~ 1 / "CI

al al al .u Cf.l

V'

~ l r

u

Apparatus for Re1ease of Prestress

Fig. B2

~ ~

CIl

al

8 CIl

...c:: u (

N ...... r-I

~ N ...... r-I

C"'l -~ 1t"I r-I

r-I al s::

V

1

- 92 -

1

1

1 1

1

1

1

1

1 ~

J N r-I

1 al \0 P-

I .,-l

~ r-I ~ N

1 r-I al

1 al +J CIl

1 \j 1 1 1

p (

Prestress Bed

Fig. B3

- 93 -

BIBLIOGRAPHY

The following abbreviations have been used:

ACI

ASCE

ICE

PCA

PCI

MCR

1. Mattock, A.H.

2 • Khan, A. Q •

American Concrete Institute

American Society of Civil Engineers

Institute of Civil Engineers (London)

Portland Cement Association

Prestressed Concrete Institute

Magazine of Concrete Research

Rotational Capacity of Hinging Regions

in Reinforced Concrete Beams.

An Investigation of the Behaviour of a

3D Reinforced Concrete Connection.

3. Corley, W. Gene Rotational Capacity of Reinforced

Concrete Beams.

4. Warwaruk, J.,

Sozen, M.A. &

Siess, C.P.

Strength and Behaviour in Flexure of

Prestressed Concrete Beams.

5. Ingerle, K. The Flexural Rigidity of Reinforced

Concrete Beams.

6. Popovics, Sandor A review of the Stress-Strain

Relationships for Concrete.

7. Sturman, Shah,

Winter

Effects of Flexural Gradient on

Stress-Strain Characteristics.

PCA Bulletin

DIOL

M. Eng. Thesis

McGill Univ.

Ju1y 1969.

PCA Bulletin

Dl08.

Univ. Of Illinois

Bull. No. 464.

International

Civil Engineering

March 1969.

J.A.C.I.

March 1970.

J.A.C.I.

Ju1y 1965.

8. Ma11ick, S.K.

9. Soliman, M.T.

10. Chan, W.W.L.

Il. Ma11ick, S.K.

12. Ernst, G.C. &

13. Ernst, G.C.

14. Baker, A.L.L.

15. Barnard, P.R.

16. Burnett, B.

- 94 -

Redistribution of Moments in Prestressed

Concrete Continuous Beams

The F1exura1 Stress-Strain Re1ationship

of Concrete Confined by Rectangu1ar

Transverse Reinforcement.

The Rotation of Reinforced Concrete

Plastic Hinges at U1timate Load.

Redistribution of Moments in Two-Span

Prestressed Continuous Beams

U1timate Loads and Def1ections from

Limit Design of Continuous Structural

Concrete.

A Brief for Limit Design.

U1timate Load Theory for Concrete Frame

Ana1ysis.

Researches into the Complete Stress

Strain Curve·for Concrete.

F1exura1 Rigidity, Curvature and Rota-

M.C.R.

Dec. 1966.

M.C.R.

Dec. 1967.

M.C.R.

Ju1y 1962.

M.C.R.

Nov. 1962.

J.A.C.!.

Oct. 1959.

Trans. ASCE

Paper No. 2812

Vol. 121, 1956.

Trans. ASCE

Paper No. 3386

Vol. 127, 1962.

M.C.R.

Dec. 1964.

M. C. R.

tion and their Significance in Reinforced June 1964.

Concrete Design.

17. Aoyama, Hiroyuki Moment-Curvature Characteristics of

Reinforced Concrete Members Subjected

to Axial Load and ReversaI of Bending.

ACI SP-12.

18. Roy, H.E. and

Sozen, M.A.

19. Everard, K.A.

20. Ernst, G.P.

- 95 -

Ducti1ity of Concrete.

The F1exura1 Rigidity of Reinforced

Concrete.

Plastic Hinging at the Intersection

of Beams and Co1umns.

21. Macchi, Georgio E1astic Distribution of Moments on Con-

22. Wright, D.T. &

Berwanger, C.

23. Institution

Research Connu.

24. Comité European

du Béton

25. Ernst, G. C.

26. Nawy, E.G.,

27. Rusch, H.

28. Mattock, A.H.

29. Yu, C.W., &

Hognestad, E.

tinuous Beams.

Limit Design of Reinforced Concrete

Beams.

U1timate Load Design of Concrete

Structures.

Note on the U1timate Load Design of

Statica11y Indeterminate Structures.

Moment and Shear Redistribution in 2

Span Reinforced Concrete Beams.

Rectangu1ar Spiral Binders Effect on

Plastic Hinge Rotation Capacity in

Reinforced Concrete Beams.

Researches Toward a General F1exura1

Theory for Structural Concrete

Redistribution of Design Bending Moments

in Reinforced Concrete Continuous Beams.

Review of Limit Design for Structural

Concrete.

ACI SP-12.

M.C.R.

Nov. 1962.

A.C.I.

June 1957.

ACI SP-12

A.S.C.E.

Ju1y 1960.

Proc. I.C.E.

Feb. 1962.

Bull. No. 21

J.A.C.I.

Nov. 1958.

J.A.C.I.

Dec. 1968.

J.A.C.I.

Ju1y 1960.

Paper 63/4

Proc. I.C.E.

May 1959.

Proc. ASCE

Dec. 1958.

30. Cohn, M.Z.

31. Rognestad,

Eivind

32. Baker, A.L.L.

33. Baker, A.L.L.

34. Bate, S.C.C.

35. Anchor, R.D.

36. Smyth, W.J.R. &

Feneron, S.G.S.

37. Obeid, E.R.

- 96 -

Limit Design of Reinforced Concrete

Frames.

A Study of Combined Bending and Axial

Load in Reinforced Concrete Members.

Further Research in Reinforced Concrete

and its Application to U1timate Load

Design.

Present Research in Reinforced Concrete

and its Application to Design.

Why Limit State Design?

The Application of Limit State Design.

Limit State Design: Its Practica1

Effects.

Proc. ASCE

Oct. 1968.

U. of Illinois

Bull No. 399.

Proc. T.C.E.

V. 2, Part 3,

No. 2, Aug. 1953.

J.T.C.E., V. 35

No. 4, Feb. 1951.

J. of the

Concrete Society

V. 2, No. 3

March 1968.

J. of the

Concrete Society

V. 2, No. 4

April 1968 ..

J. of the

Concr~te Institutl

Vol. 2, No. 4

May 1968.

Compression Hinges in Reinforced Concrete Dept. Civil

Elements. Engineering &

App1ied Mech.,

McGill Univ.

M. Eng. Thesis

Sept. 1969.

38. Burnett, E.F.P.

& Yu, C.W.

39. Burns, N.H.

- 97 -

Reinforced Concrete Linear Structures at

Ultimate Load.

Moment-Curvature Relationships for

Partially Prestressed Concrete Beams

A.C.1. SP 12

J.P.C.I.

Feb. 1964.

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