The Structural Behaviorof Higher Strength Concrete

by

Narayan Babu D. Hiremagalur

B. E. , Bangalore University, 19 82

A MASTER'S THESIS

submitted in partial fulfillment of the

requirements for the degree

MASTER OF SCIENCE

Department of Civil Engineering

KANSAS STATE UNIVERSITYManhattan, Kansas

1985

'

Mi TABLE OF CONTENTS.TV

Iff5

M?7A115D5 btllOM

iAGE

List of Tables III

List of Fi gures V

Chapter 1 - Introduction 1

Present Usage 1

Objectives 3

Chapter 2 - Selection of Materials 4

Introduction 4

Cement 4

Coarse Aggregate 6

Strength 7

Maximum Size and Graduation 7

Particle Shape and Surface Texture 8

Cleanliness 8

Mineralogy and Formation 8

Aggregate- Paste Bond 9

Fine Aggregate 9

Water 10

Admixtures 10

Chapter 3 - Literature Survey For Compressive Stress Block 12

Chapter 4 - Test Specimen and Method of Testing 15

Method of Testing 15

Strain Gages 17

Chapter 5 - Experimental Work and Results 18

Casting 18

l&BLfi 3E CONTENTS (Continued)

PAGE

Curing 18

Instrumentation and Apparatus 18

Test Procedure 19

Results and Discussion of the Results 20

Strain Behavior 23

Chapter 6 - Summary and Conclusions 25

Summary 25

Conclusions 2 5

Recommendations For Future Work 26

Appendix I References 27

Appendix II a) Mix Proporations 30b) Derivation of Ultimate Strength Factors 31

kik3, k2c) Steel Frame dm Q df 33d) Computer Program for Calculating 5— & 3— 34

e) Calibration of Minor Load P2 36

Appendix III - Tables and Figures 37

Appendix IV - Notation

Acknowledgements

ii

LIST QE TABLES

TABLE PAGE

5.0 Information about the test specimens 3 8

5.1 Cylinder data corresponding to Beam I 39

5.2 Cylinder data corresponding to Beam II 40

5.3 Cylinder data corresponding to Beam III 41

5.4 Cylinder data corresponding to Beam IV 42

5.5 Stress-strain relation for cylinder #1, 43corresponding to Beam I

5.6 Stress-strain relation for cylinder #2, 44corresponding to Beam I

5.7 Stress-strain relation for cylinder #2 45corresponding to Beam II

5.8 Stress-strain relation for cylinder #1 46corresponding to Beam II

5.9 Stress-strain relation for cylinder #1 47corresponding to Beam III

5.10 Stress-strain relation for cylinder #2 48corresponding to Beam III

5.11 Stress-strain relation for cylinder #1 50corresponding to Beam IV

5.12 Load-Strain data for flexural test of Beam I 51

5.13 Load-Strain data for flexural test of Beam II 53

5.14 Load-Strain data for flexural test of Beam III 55

5.15 Load-Strain data for flexural test of Beam IV 57

5.16 Load and average strain at each level for flexural 59test of Beam I

5.17 Load and average strain at each level for flexural 61test of Beam II

5.18 Load and average strain at each level for flexural 63test of Beam III

5.19 Load and average strain at each level for flexural 65test of Beam IV

JLIST .QE TABLES (Continued)

TABLE PAGE

5.20 Load and ultimate strength factors for Beam I 67

5.21 Load and ultimate strength factors for Beam II 69

5.22 Load and ultimate strength factors for Beam III 71

5.23 Load and ultimate strength factors for Beam IV 73

5.24 Load and stress data for flexural test of Beam I 75using equations (4.1) and (4.2)

5.25 Load and stress data for flexural test of Beam II 78using equations (4.1) and (4.2)

5.26 Load and stress data for flexural test of Beam III 80using equations (4.1) and (4.2)

5.27 Load and stress data for flexural test of Beam IV 82using equations (4.1) and (4.2)

5.2 8 Load and stress data for flexural test of Beam I 84using equations (4.1) and (4.2) and cylinderstress-strain curve

5.2 9 Load and stress data for flexural test of Beam II 86using equations (4.1) and (4.2) and cylinderstress-strain curve

5.30 Load and stress data for flexural test of Beam III 87using equations (4.1) and (4.2) and cylinderstress-strain curve

5.31 Load and stress data for flexural test of Beam IV 88using equations (4.1) and (4.2) and cylinderstress-strain curve

iv

LIST OF FIGURES

FIGURE PAGE

2.1 Effect of various brands of type I cement on 89concrete compressive strength (20)

2.2 Effect of various cements on concrete 90

compressive strength (2)

2.3 Effect of size of coarse aggregate on 91

compressive strength in different types ofconcrete (adapted from Ref . 5)

2.4 Maximum size aggregate for strength efficiency 92

envelope (15)

2.5 Compressive strength of concrete using two 93

sizes and types of coarse aggregate for 7500psi concrete (1) .

2.6 Compressive strength of high-strength concrete 94

using three sizes and types of coarseaggregate.

2.7 Slump vs. time when using superplasticizer 95

(11)

4.1 C-Shape structural element for flexure test. 96

4.2 Concrete stress-strain relations (7) 97

4.3 Reinforcement layout 98

4.4 An arbitrary section A-A with rebar arrangement 99

at each leg (11)

4.5 An arbitrary section of column part with rebar 100arrangement (11)

5.1 Cylinder compressive stress-strain curve for 101Beam I

5.2 Cylinder compressive stress-strain curve for 102Beam II

5.3 Cylinder compressive stress-strain curve for 103Beam III

5.4 Cylinder compressive stress-strain curve for 104Beam IV

5.5 Location of strain gages vs. strain for Beam I 105

LIST OF FIGURES (continued)

FIGURE PAGE

5.6 Location of strain gages vs. strain for Beam II 106

5.7 Location of strain gages vs. strain for Beam 107III

5.8 Location of strain gages vs. strain for Beam IV 10 8

5.9 Stress block of Beam I 109

5.10 Stress block of Beam II 110

5.11 Stress block of Beam III 111

5.12 Stress block of Beam IV 112

5.13 Axial load vs. strain for gages 4 and 5 for 113Beam I

5.14 Axial load vs. strain for gages 4 and 5 for 114Beam II

5.15 Axial load vs. strain for gages 4 and 5 for 115Beam III

5.16 Axial load vs. strain for gages 4 and 5 for 116Beam IV

5.17 Stress factors k_, k,k, vs. concrete strain for 117Beam I

z j. j

5.18 Stress factors k, , k, k, vs. concrete strain for 118Beam II « J. J

5.19 Stress factors k, , k.k, vs. concrete strain for 119Beam III x J

5.20 Stress factors k_ , k.k, vs. concrete strain for 120Beam IV * ± J

5.21 Condition at ultimate load in test specimen 121

5.22 f vs concrete strain for Beam I 122

5.23 fQ

vs concrete strain for Beam II 123

5.24 fQvs concrete strain for Beam III 124

5.25 £ vs concrete strain for Beam IV 125

5.26 mQvs concrete strain for Beam I 126

vi

FIGURE

LIST OF FIGURES (continued)

PAGE

5.27 mQ

vs concrete strain for Beam II 127

5.2 8 mQ

vs concrete strain for Beam III 128

5.29 m vs concrete strain for Beam IV 129

5.30 Average f from Eq. (4.1) and (4.2) vs. strain 130for Beam II

5.31 Average f from Eq. (4.1) and (4.2) vs. strain 131for Beam IV

vii

Chapter 1

INTRODUCTION:

The term high strength concrete is relative to any

assemblage of concrete technologists. Most concrete

technologists classify concrete as follows:

2500 - 6000 Psi > Normal strength concrete;(13.8 to 41.5 MPa)

6000 - 12000 Psi > Higher strength concrete;

(41.5 MPa - 83 MPa)

> 12000 Psi > High strength concrete.

(> 83 MPa)

In this report higher strength concrete implies

concrete of strength 6000 Psi (41.5 MPa) and above.

A few years ago concrete of 5000 Psi (34.5 MPa) and

above was considered high strength concrete. But, today

with the development of new materials, admixtures and

processers, even concrete of 6000 Psi (41.5 MPa) is

considered normal strength concrete.

PRESENT USAGE:

There are many advantages in using higher strength

concrete over normal strength concrete.

a) Higher strength concrete has greater load carrying

capacity than normal strength concrete. This helps in

reduction of member dimensions and hence dead weight is

greatly reduced.

b) Higher strength concrete is very advantageous in

compression members. In columns, if the section is

decreased the self weight decreases, and the stress due

to self weight is much lower, hence greater external

load can be taken.

c) The' precast and prestress industry is the largest user

of higher strength concrete. In prestressed concrete

members, the moment due to self weight is decreased,

higher external moment can be taken, and the efficiency

of the structure is increased. In prestressed concrete,

concrete strengths of up to 10000 Psi (68.9 HPa) have

been used.

d) It is not particularly advantageous to use higher

strength concrete in slabs, as thin sections have more

deflections.

Higher strength concrete is being used extensively in

New York and Chicago. In the Chicago area concrete of

10,000 Psi (18.9 MPa) was used in at least six different

buildings (21) . Nuclear power plants require concrete in

the 8000 Psi (55 MPa) range.

OBJECTIVES

a) To study the compressive and flexural behavior of higher

strength concrete [11,000 Psi or 75.8 MPa] made with

Kansas aggregate, in order to verify proposed stress-

strain relationships and to study the shape of the

stress block.

b) To compare the cylinder and flexural stress-strain

curves.

c) To find Poisson's ratio.

d) To determine strain at f ' and at rupture.

e) To determine modulus of elasticity.

CHAPTER 2

SELECTION M materials

INTRODUCTION

Material selection is an important factor in the

production of higher strength concrete as it has to meet

requirements for workability and strength development.

Locally available materials should be used to fullest

advantage, subject to suitability, to make the production

of higher strength concrete economical. Once the materials

are selected, the remaining variables should be studied to

develop an optimum mix on the basis of strength, cost and

field performance.

CEMENT

The strength of a mix is proportional to the strength

of the cement. As reported in the state-of-the-art report

of high-strength concrete [24] published in the ACI Journal

"The choice of Portland cement for high strength concrete is

extremely important. Unless high initial strength is the

objective, such as in prestressed concrete, there is no need

to use a Type III cement. Furthermore, within a given

cement type, different brands will have different strength

development characteristics because of the variations in

compound composition and fineness that are permitted by ASTM

C 150".

Walker and Bloom (see Ref. 5) presented values of

compressive strength for concrete from different cement

suppliers and offered the following conclusions:

a) Average strength for different cements varied

substantially.

b) At least one-half of the individual cement differs from

shipment to shipment sufficiently to have an effect on

uniformity of concrete strength.

c) There is a need for better control of uniformity of the

cement within individual sources.

Figure 2.1 (20) shows the effect of various cement

types on concrete compressive strengths based on mixes

having the same workability (5) . Type I (standard) or Type

II (low heat) cement should be recommended for high strength

concrete mixes and the same brand of cement is to be used

throughout the job (5) . Type III cement is too finely

ground and results in rapid setting, accelarated heat of

hydration, and high early strength development with rela-

tively moderate strength gains at 91-day or one year inter-

vals (5) .

In the present investigation, a mix developed by

Nikaeen (11) was used. Type I cement was used. Figure 2.2

shows that Type I cement gives the highest compressive

strength at all ages.

COARSE AGGREGATE

As coarse aggregate comprises a major portion of the

volume, it is very important to use proper coarse aggregate

in developing higher strength concrete. Kaplan (8) has

reported that the use of different coarse aggregates has a

marked effect on the strength of concrete with the same mix

proportions. Depending on the aggregates, a difference of

40% in flexural strength and 29% in compressive strength

were obtained in Kaplan's experiments.

The strength, up to 5000 Psi (34.5 MPa) depends

essentially on the quality of the hardened cement paste that

holds the aggregate particles together. The aggregate at

this strength level and above always has a much greater

strength than the cement paste. The following factors

should be considered in selecting a coarse aggregate for

higher strength concrete.

a) Strength

b) Maximum size and gradation

c) Particle shape and surface texture

d) Cleanliness

e) Mineralogy and formation

f) Aggregate-cement bond

STRENGTH

The aggregates chosen for higher strength concrete

should have a crushing strength at least equal to the

hardened cement paste. Most good quality aggregates have

crushing strength in excess of 12000 Psi (83 MPa) and,

hence, strength is not a major problem in the production of

higher strength concrete.

MAXIMUM SIZE AND GRADATION

Several researchers (3, 4, 10) have shown that in

higher strength concrete the compressive strength increases

when the maximum size of aggregate decreases. However, it

is obvious that there should be some limitation in order to

keep drying shrinkage and creep to a reasonable and

practical value. A maximum size of 0.4 in. (10 mm) is

recommended for most cases (10). Figures 2.3 and 2.4 show

the size effect of coarse aggregate on compressive strength.

In Figure 2.4, it is indicated that the smaller size

aggregate provides the most efficient use of cement in

higher strength concrete. This higher strength efficiency

which is obtained when using smaller aggregate is due to

greater bond between the cement and coarse aggregate because

of the increase in surface area. However, there is a

different optimum size for each aggregate and for each level

of strength desired, depending on the economics of the

situation. Therefore final batching for each job is highly

recommended.

PAKTICLE SHAPE AND SURFACE TEXTURE

In the research done by Carrasquillo, Ramon and Nil son

(5) , it is indicated that the ideal coarse aggregate for

high-strength concrete appears to be clean, cubical, angu-

lar, 100% crushed stone with minimum flat size and elongated

particles. It is also stated that if all other factors are

equal, crushed stone coarse aggregate produces higher

strength concrete than does a rounded aggregate. In

Figures 2.5 and 2.6, the compressive strength of concrete

using different types of coarse aggregate is shown.

CLEANLINESS

Coarse aggregate used in higher strength concrete

should be free of dust coating, because the dust content

causes an increase in the fines, and consequently an

increase in the water requirement of the resulting concrete

mixture. Hence, the water-cement ratio increases and the

strength decreases. Washing the crushed stone coarse

aggregate may not always be necessary, but is always

recommended (22) .

MINERALOGY AND FORMATION

As discussed above, the compressive strength of

concrete increases when using crushed stone as the coarse

8

aggregate. This is not only because of the shape of the

aggregate but also due to its mineralogy. The Waterways

Experiment Station has done some work on the effect of

mineralogy on concrete strength. They achieved 17 000 Psi

(117 MPa) using granite rock (14) .

AGGREGATE-PASTE BOND

When high-strength concrete is desired, the aggregate

paste-bond should be strong, a good quality aggregate with a

suitable surface texture should be used.

The aggregate-paste bond is known to decrease with

increasing water-cement ratio and increasing the size of

aggregate. Alexander (1) found that the cement-aggregate

bond to a 3 inch particle was almost 1/10 of that to a

corresponding 1/2 inch particle.

FINE AGGREGATE:

The gradation and particle shape of fine aggregate are

very important factors in the production of higher strength

concrete. Fine aggregate has a great effect on the water

requirement and consequently the strength of the concrete

mix. In sand of the same grading, a 1% increase in fine

aggregate voids may cause a 1 gallon per cubic yard increase

in water demand (6) .

One of the important functions of fine aggregate in

conventional concrete is its role in providing workability

and good surface finishing. Since higher-strength concrete

contains a large amount of cement paste, the role of fine

aggregate in providing workability and good finishing is not

so crucial. Fine aggregates with a fineness modulus between

2.7 and 3.2 have been most satisfactory (9). Blick (2)

reported that fine aggregates with fineness modulus below

this range might produce low strength and "sticky mixes". A

reduction of the amount passing the No. 50 and No. 100 sieve

on the lower side of the specification limit from ASTMEC-33

is suggested. Such reductions have shown an increase in

compressive strength of 500 to 1000 Psi (3.5 to 7.0 MPa)

(15).

In this investigation, Kaw River Sand with a maximum

sieve size of No. 4 was used.

WATER

Studies (6,20) have shown that water meeting specifica-

tion ASTM C-94(19) has no harmful effect on high-strength

concrete. Therefore, water meeting ASTM C-94 is adequate.

ADMIXTURES

Due to the low water-cement ratio, high-strength

concrete has low slump and workability is not adequate. A

chemical admixture called superplasticizer or "super water

reducer" can be used to improve workability and slump. This

admixture actually reduces the friction angle between the

10

water and surface of contact and causes the mix to be more

workable. It is important to note that this effect is for a

limited time and the mix changes to its original property

after a short time. Therefore superplasticizer increases

the slump within a limited time without altering the

compressive strength, because the slump will go back to its

original position as if no admixture was used in the mix.

Figure 2.7 shows the effect of superplacticizer on the slump

versus the time on a mix with water cement ratio of 0.35

(11). It is believed to have a similar effect on all other

mixes. In this investigation, the brand of superplacticizer

used was Sikament. The quantity of superplacticizer used

was in accordance with the mix proportion developed by

Nikaeen (11) . As slump decreases rapidly with time, it is

important to consider this effect on the job site.

"

£HAEIE£ 1

LITERATURE SURVEY FOR COMPRESSIVE STRESS BLOCK

The ultimate strength of reinforced concrete members

subjected to flexure and axial loading can be predicted

using the equivalent rectangular stress distribution in the

concrete compressive zone. In designing, the ratio of the

depth of the equivalent compressive stress block to the

depth of the actual one, and « , the extreme fibre compres-

sive strain, are to be known. Based on beam tests, using

concretes with compressive strength less than 6000 psi,

(41.3 HPa) the American Concrete Institute (ACI) (17) sug-

gested values for p. , depending on the concrete strength.

In 1975, a minimum value of p , equal to 0.65, was sug-

gested by the ACI. The validity of this stress block, has

always been in question with regard to higher-strength con-

crete. Hence, there is a need to study the shape of the

stress block in higher strength concretes.

Many views have been expressed concerning high strength

concrete, but the views are not consistent and they

contradict each other.

In the work done by Leslie, Rajagopalan and Everard

[9], they concluded:

12

a) The ACI building code rectangular stress block does

not predict the behavior of beams with f ' above

8000 psi (55 MPa)

.

b) Further research is warranted with respect to

maximum strain in concrete for f' exceeding 8000

psi (55 MPa)

.

c) Pending further tests, a triangular stress block

with an extreme fibre stresses of f' and zero

stress at the neutral axis is recommended as a

conservative model for predicting the behavior of

beams with f'c above 8000 psi (55 MPa).

In the work done by Wang, Shah, and Naaman [16] , the

following conclusions were made:

a) The rectangular stress distribution gives a suffi-

ciently accurate prediction of ultimate loads and

moments for reinforced concrete beams and columns

made with high strength concrete.

b) The value of maximum concrete compressive strain at

ultimate was always higher than 0.003 in/in.

In the work done by Nikaeen [11] , he concludes:

a) The shape of the stress block at ultimate changes

from rectangular to parabolic type as the strength

increases, because he found that the position of

the concrete internal reaction force is k_ = 0.37

at ultimate condition for higher strength mix

13

(i.e., £'c

= 9500 psi, 65 Mpa) . This value is very

close to 0.375 which is the center of gravity of a

parabolic stress block rather than 0.5 which is the

center of gravity of a rectangular stress block,

b) The strain behavior of high strength concrete is

different from normal strength concrete, because

strain at the ultimate condition is less than

0.003 in/in. proposed by ACI code (17) or it might

be greater than that proposed by ACI, but it

decreases with time drastically. Therefore, a more

conservative value of 0.002 in/in. was recommended.

14

CHAPTER A

TEST SPECIMEN AND METHOD OF TESTING

In 1955, Hognestad, McHenry and Hanson [7] formulated

the ultimate strength design criterion for concrete. In

their work, they used a special C-shaped structural element.

Tensile stresses were completely eliminated in the test

region of the specimen, by applying loads at two points and

varying them such that the neutral axis remained at the face

of the test specimen throughout the test.

The test specimen had a central unreinforced test

region, and reinforced brackets at the end of the test

specimen which accomodated the two thrusts.

The test reported here was done on a similar specimen

and the dimensions were as used by Nikaeen [11] (Fig 4.1)

.

In this test, the method and analogy adopted by Nikaeen

are very closely followed in order to investigate the

behavior and shape of the stress block in the higher

strength concrete beam section.

METHOD OF TESTING

Suitable tension, compression, and shear reinforcement

were placed in the end bracket to assure that failure would

occur in the central, unreinforced test region. A major

thrust P1 was applied and the neutral axis was maintained at

15

the face of the test specimen throughout the test. Strains

were measured over the test region.

In investigating the behavior and shape of the stress

block, the method and equations developed by Hognestad,

McHenry, and Hanson [7] were used. To formulate the

ultimate strength design criterion, they derived some

equations and used a C - shaped structural element. The

present ACI code for ultimate strength theory is based on

their work. They formulated stress in concrete fibres as a

function of strain in those fibres. They also showed that

stress-strain relationships for concrete in concentric

compression are indeed applicable to flexure (Figure 4.2

shows this fact) . The following equations which permit

stress to be expressed in terms of measured strain and other

known parameters were developed by them [see 7] .

dfQ

fc " 8cd^ + f

o (4.D

dmfc = 8cdT + 2m

o (4.2)c

Pl + p2

fo " —be"

(4-3)

plal

+ ?2 a2

be 2(4.4)

where

16

f = Concrete compressive stress in outer fiber ofthe beam;

» = Concrete strain in outer fiber of the beam;c

P. = major thrust;

P, = minor thrust;

a, and a_ are lever arms;

b is the width and c is the depth of the testingregion.

The details are shown in Figure 4.1.

Figure 4.3 gives the reinforcement layout of the test

specimen. Figures 4.4 and 4.5 give the sections at A-A and

B-B respectively (see Figure 4.1).

STRAIN GAGES

Strain gages were used to measure strains. The strains

were monitered by an automated data acquisition system

controlled by a model 2E Apple Computer.

Ten of Micro Measurement EA-06-7 50DT-120 electrical

resistance strain gages were used on the test beam of the

locations shown in Fig. (4.1).

17

CHAPTER 5_

EXPERIMENTAL WORK AND RESnT.TS

CASTING

The specimens were cast horizontally on a level

surface. Vibration was used to consolidate the concrete.

There was a minimum cover of 3/8 in. The mix proportions

used for all the 4 specimens are shown in Appendix Ila.

Some 3x6 in. cylinders were cast at the same time with each

mix.

CURING

Twenty-four hours after casting, the specimen was taken

out of the mold and then placed in a 100% humidity curing

room. All the specimens were in the curing room for 52

days. The specimens were then taken from the curing room

and were kept in a 50% humidity environment for 7 days in

order to prepare them for the test. Table 5.0 shows details

of the test specimens. Cylinders corresponding to each

specimen were taken out at the same time as the specimen.

INSTRUMENTATION AND APPARATUS

The location of the strain gages used to measure the

strain can be seen in Fig. 4.1. The major load P^, Figure

4.1, is applied by a 300,000 lb, load-controlled testing

18

machine through a system of bearing plates and rollers. The

minor load P„ is applied by a hydraulic jack through a steel

frame shown in Appendix II-c. A pressure gage system,

calibrated prior to the test, was used to control the load.

TEST PROCEDURE

Tables 5.1, 5.2, 5.3, and 5.4 give the average compres-

sive stress for the 3x6in cylinders corresponding to the 4

test specimens. Two strain gages were mounted on some

cylinders and strain was recorded as a function of load up

to failure. Cylinders were tested on the same day that the

corresponding structural element was tested in order to

determine the strength of the mix. The data are in the

tables 5.5 and 5.6 corresponding to beam I, 5.7 and 5.8

corresponding to beam II, 5.9 and 5.10 corresponding to beam

III, and 5.11 corresponding to beam IV. Cylinders corre-

sponding to structural elements I and II and some of the

cylinders of structural element III were tested on a machine

which was not functioning properly. Hence, the strength of

the mix corresponding to specimen IV (f' = 11100 psi or

76.49 MPa) was used in all the 4 specimens in computing the

stress factors.

The test specimen was loaded in such a manner that the

neutral axis always coincided with the outside face of the

specimen. The location of the neutral axis is shown in

Figure 4.1. After each increment of major thrust P., the

19

corresponding increment was such that the average strain

across the neutral surface was approximately zero. This

was monitored by connecting the strain gages on the neutral

face to a wheatstone bridge and maintaining zero potential.

The load and strain at each level were recorded and the

procedure repeated up to failure. The zero strain surface

represents the neutral axis of the specimen and the opposite

side of the cross section represents the extreme compressive

surface. The recorded data for specimens I, II, III, and IV

are shown in 5.12, 5.13, 5.14, and 5.15 respectively. The

average strain at each level is given as a function of load

and is in Tables 5.16, 5.17, 5.18, and 5.19 corresponding to

the 4 specimens.

RESULTS AND DISCUSSION OF THE RESULTS

The average values of cylinder stress-strain data of

Tables 5.6, 5.7, 5.10, and 5.11 corresponding to Specimens

I, II, III, and IV are plotted in Figures 5.1, 5.2, 5.3, and

5.4 respectively. Using the corresponding cylinder stress-

strain curve and the strain values for the flexure test

(indicated in Tables 5.16, 5.17, 5.18, and 5.19) the stress

values at each loading level could be directly determined.

The variation of strain along the depth is shown for beams

I, II, III, and IV in Figures 5.5, 5.6, 5.7, and 5.8 respec-

tively. The shape of the stress block for beams I, II, III,

20

and IV are shown in Figures 5.9, 5.10, 5.11, and 5.12

respectively. The average values of flexural strain at the

inner face (maximum compression) are plotted against load in

Figures 5.13, 5.14, 5.15, and 5.16 for beams I, II, III, and

IV respectively.

To determine the ultimate strength factors, k^k^ and k2

the equilibrium concept is used. The details are shown in

Appendix Il-b. By equilibrium of forces and moment from

Figure 5.21, k^k^r and k 2can be determined.

C p l + p21 J bcf£ bcfc

Pi a-, + P, a,

*' l T ^\ P2)c

< 5 - 2 >

k2

The values of k n k-,, k, and -r—.— can be measured directlyklk3

from zero up to failure. The individual value of the coef-

ficient kg is the ratio of maximum compressive stress in the

beam (i.e., f cmax flexural test) to the corresponding

average cylinder strength f,!,. To calculate k^, which is the

shape factor, k^^ and k2are to be evaluated. The values

of ultimate strength factors are given in Tables 5.20, 5.21,

5.22, and 5.23 for beams I, II, III, and IV. The ultimate

strength factor k2 indicates the position of the resultant

reaction force which is produced by concrete.

21

Figures 5.17, 5.18, 5.19, and 5.20 show the values of

^1^2 and ^2 computed by equations (5.1) and (5.2) as a

function of strain e c at the compression face for Beams I,

II, III, and IV. The term k 2 indicates the position of the

resultant force in the concrete from the outer fiber of the

beam. Figure 5.18 shows that the position of the resultant

force is 0.32 at lower loads and 0.36 at ultimate condition.

In beam II, k2changes from 0.32 at lower loads to 0.36 at

ultimate (Figure 5.19). From Figure 5.19, 5.20, and 5.21,

one can see that k2

changes from 0.31 to 0.35. This indi-

cates that the stress block is more triangular than rectan-

gular or parabolic, but it tends to become closer to para-

bolic towards the ultimate. Beam I unlike the other three

beams had k2changing from 0.37 to 0.43 which indicates the

stress block appears to be close to a rectangular shape.

The curve pattern for beam I looks exactly like the curves

for the other three specimens but it seems to be shifted.

This may be due to some error in conducting the experiment.

The f,!, value corresponding to the beam IV cylinder was

used in calculating k3 for all the four specimens as the

cylinders corresponding to the other beams were tested in a

defective machine. The outer fiber stress is calculated

using equations 4.1 and 4.2 for all the specimens. The

results are given in the Tables 5.24, 5.25, 5.26, 5.27,

5.28, 5.29, 5.30, and 5.31 corresponding to beams I, II,

22

Ill, and IV. A computer program was used to find the dif-

ferential parts. Tables 5.28, 5.29, 5.30, and 5.31 give

flexural stresses using both the methods i.e., Equations

4.1, 4.2, and the cylinder stress- strain curve, for compari-

son purpose for beams I, II, III, and IV.

Figures 5.22, 5.23, 5.24, and 5.25 show the values of

f Q (Equation 4.3) as a function of strain b at the compres-

sion face for the four beams respectively. Figures 5.26,

5.27, 5.28, and 5.29 show the values of mQ (Equation 4.4) as

a function of strain e c at the compression for the four

beams respectively. Figures 5.30 and 5.32 give average f c

from Equation (4.1) and Equation (4.2) vs. strain for beam

II and beam IV respectively.

Comparing the area of the ACI stress block (17), with

the area of the parabolic, stress block, we have

0.85 f^O-^Jb = k-^f^bc or

klk3

?1. 85

The values of fly, so calculated are 0.88, 0.8, 0.8, and 0.72

for beams I, II, III and IV respectively. This is higher

than the ACI recommended value of 0.6 5.

The modulus of elasticity corresponding to cylinders of

Beam III and Beam IV were 6.71 x 10 6 psi (0.462 GPa) and

6.67 x 10 psi (0.459 GPa) respectively. This was found to

be nearly equal to the modulus of elasticity calculated from

23

American Concrete Institute formula (17), which was

6.36 x 10 6 psi (0.438 GPa). The Poisson's ratio

corresponding to a cylinder from Beam II was found to be

0.23.

STRAIN BEHAVIOR

The maximum recorded strains corresponding to Specimens

I, II, III, and IV are 0.00330 in/in., 0.00303 in/in,

0.002 80 in/in. 00021 in/in. respectively. Maximum recorded

cylinder strains at stress near ultimate corresponding to

Specimens I, II, III, and IV are 0.0021 in/in., 0.00243

in/in., 0.002116 in/in. and 0.00203 in/in. respectively.

The cylinders had maximum recorded strain values less than

0.003 in/in. Two of the beams had maximum recorded strain

values greater than 0.003 and two of them had maximum

recorded strain values less than 0.003 in/in. Therefore, a

more conservative value of 0.0025 in/in. is recommended.

Reference (12) has also suggested a similar value. Nikaeen

[11] has suggested a very conservative value of 0.002 in/in.

24

CHAPTER 5.

SDMMARY MD CONCLUSIONS

SUMMARY

Four "dogbone" shaped [7] beams of higher strength

concrete were tested to investigate their structural bending

properties. The beams were loaded such that zero strain was

maintained at the outer face. The beams were made out of

concrete of 11000 psi (76 Mpa) . A number of 3"x 6" cylin-

ders corresponding to each beam were tested in concentric

compression. The main objective of this experiment was to

study the shape of the stress block of failure for higher-

strength concrete. The following conclusions were made from

the test results.

CONCLDSIONS

1. Higher-strength concrete has a brittle mode of failure

and failure is sudden without any formation of cracks

before failure. The failure line passes through the

stone and not through the interface of mortar as in

normal strength concrete. [This was also observed by

other investigators (12, 11)].

2. The values of k, near the ultimate condition for beams

II, III, and IV were 0.36, 0.36 and 0.35. These values

25

are very close to 0.375 the value for a parabolic

stress distribution. Thus higher strength mixes appear

to have a parabolic stress distribution of failure in

bending.

3. Strain-behavior in higher strength concrete is

different from normal strength concrete and is observed

to be less than the ACI proposed value of 0.003 in/in.

A more conservative value of 0.0025 in/in. is recom-

mended.

4. The modulus of elasticity may be estimated by the ACI

formula (17) (Eq. 10).

5. The value of Pi in higher strength concrete is greater

than the ACI recommended value of 0.65.

RECOMMENDATIONS FOR FUTURE WORK

For higher-strength concrete to be effectively used in

the future, intensive research should be conducted. Higher-

strength concretes of ranges 11,000 psi to 15000 psi (75.8

MPa to 103.4 MPa) should be tested to determine the

structural behavior in bending. Also, experiments should be

done on a larger scale i.e., more beams should be tested to

get consistent results and some definite code requirements

should be developed.

26

APPENDIX J

references'

1. Alexander, K. M. , "Factors controlling the Strength andthe Shrinkage of Concrete," Construction Review . Vol. 33,No. 11, Nov. 1960, pp. 19-29.

2. Blick, Ronald L. , "Some Factors Influencing High StrengthConcrete," Kederji Concrete . April 1973.

3. Bloem, D.L. and Gaynor, R.D. "Effect of AggregatesProperties on Strength of Concrete," American ConcreteInstitute Journal . Vol. 60, No. 10, October 1963, pp. 1429-1455.

4. Burgess, A. J., Ryell, J., and Bunting, J., "High-StrengthConcrete for the Willows Bridge," American ConcreteInstitute Journal . Vol. 67, No. 8, August 1970, p. 611.

5. Carrasquillo, Ramon L. , Nilson, A.H. , and Slate, Floyd 0.,"The Production of High-Strength Concrete," Research ReportNo. 7 8-1 . Department of Structural Engineering, CornellUniversity, Ithaca, May 1978, p. 2.

6. Freeman, Sidney, "High-Strength Concrete," Publication No.15176, Portland Cement Association, 1971 (Reprint from

.aoii£in Concrete . 1970-71) , p. 19.

7. Hognestad, Eivind, Hanson, N.W. , and McHenry, Douglas,"Concrete Stress Distribution in Ultimate Strength Design,"Journal .Of ihs American Concrete Institute . Dec. 1955, Vol.27, No. 4, pp. 455-47 9.

8. Kaplan, M.F. , "Flexural and Compressive Strength of Concreteas Affected by the Properties of Coarse Aggregate," AmericanConcrete Institute Journal . Vol. 55, May 1959, pp. 1193-1208.

9. Leslie, Keith E. , Rajagopalan, K. S. , and Everard, Noel J.,"Flexural Behavior of High-Strength Concrete Beams," A£IJournal . Vol. 73, No. 9, September 1976, pp. 517-521.

10. Mather, Katherine, "High-Strength Concrete, High DensityConcrete," (ACI Summary Paper), American Concrete InstituteJournal . Proceedings, Vol. 62, No. 8, August 1965, pp. 951-962.

27

11. Nikaeen, Ali, "The Production and Structural Behavior ofHigh-Strength Concrete." A Master's Thesis . Department ofCivil Engineering, Kansas State University, 1982.

12. Nilson, Arthur H. and State, Floyd 0., "StructuralProperties of Very High-Strength Concrete," Second ProgressRePP ft £113^. 76-Q8752, Department of Structural Engineering,Ithaca, New York, January 1975.

13. Perenchio, W. P. , "An Evaluation of Some of the FactorsInvolved in Producing Very High-Strength Concrete, " BulletinJSfij. RD014. Portland Cement Association, 1973.

14. Saucier, Kenneth L. , "High- Strength Concrete, Past, Present,Future," Concrete International. American ConcreteInstitute, Vol. 2, No. 6, June 1980, pp. 46-50.

15. Smith, E. E. , Tynees, H.O. , and Sauciet, K.L. , "HighCompressive Strength Concrete, Development of ConcreteMixtures, " Technical Documentary Report No^. XDS 63-3114 .

D.S. Army Engineer Waterways Experiment Station, VicksburgMississippi, February 1964, p. 44.

16. Wang, Pao-Tsan, Shah, Surendra P., Naaman, Antoine E.

,

"High-Strength Concrete in Ultimate Strength Design,

"

Journal Qi ihS Structural Division . ASCE, No. 197 8, pp.1761-1773.

17. "Building Code Requirement for Reinforced Concrete (ACI 318-83)," American Concrete Institute, 1983.

18. "Cement; Lime; Ceilings and Walls (including manual ofcement testing)", Annual Book of ASTM Standard . Part 13,1976.

19. "Concrete and Mineral Aggregates (including Manual ofConcrete Testing) ," Annual Book of. ASTM Standard , part 14,1974.

20. "High-Strength Concrete," First Edition, Manual q£ ConcreteMater ials-Aggregates . National Crushed Stone Association,January 1975, p. 16.

21. "High-Strength Concrete in Chicago High-Rise Buildings,"lasJs Force Report No^. 5_, Chicago Committee on High-RiseBuildings, Feb. 5, 1977.

22. "High Strength Concrete-Crushed Stone Makes the Difference,"Third Draft, Presented at the January 1975 National Crushed

28

Stone Association Convention in Florida, November 1974, p.31.

23. "New York Gets Its First High-Strength Concrete Tower,"Engineering News- Records. Vol. 201, No. 18, November 2,197 8, p. 22.

24. "State-of-the-Art Report on High-Strength Concrete,"Reported by ACI Committee 363, Ad jlojirjial, Vol. 81, No. 4,July-August 1 9 84 .

25. Tentative Interim Report on High-Strength Concrete, AmericanConcrete Institute Journal . Proceedings, Vol. 64, No. 9,September 1967, pp. 556-557.

29

APPENDIX II

a. MIX PROPORTIONS (Based on trial mix done by Nikaeen

(ID)

Proportions used for 1 cu ft.

Materials Quantity

Cement, Type I 31.93 lb

Quartz ite 49.65 lb

Sand 60.88 lb

Water 8.28 lb

Super pi asticizers 340 ml.

1 lb. = .45 kg

30

b. Derivation of ultimate strength factors k.k,, k 2

b,

V%V

(\

—

• • •

c

/ lc=,

T=Asfy

L)bc

Figure a: Force Couple System

Without any assumptions we obtain the equilibrium equation offorce and moment:

Fx = 0, T = C > T » k

1k3

fj. be

M1*3 - fj. be(1.1)

M = T(d ^2°1 C(d - k,c)

2 " c " Tc c Cc (1.2)

Applying the same concept to the test specimen shown in figure

5.21, we have C= P, + p..

Substituting in Equation (1.1) we obtain

*ilc, -

Pl + P

21*3 " bef ' (1-3)

31

where f ^ i s the cylinder compressive strength.

Mu " pl al

+ p2a2

Substituting in Equation (1.2)

kd P

l al + p2a2

2 * c (P1+ P2)c

d = c for the test specimen

Therefore:P1&1 + P2 a2

K2 = 1 "

(Pl + p2 ) c

32

c. STEEL FRAME

The minor load P2 is applied by a hydraulic Jack through

a steel frame shown below (Adapted from Ref. 11).

^xCw^JT

CUT SECTION

_BEAM

-ROLLER

.MOVABLE PLATE

.RAM

1 in = 25.4 mm

33

d_ cmo df o. Computer Program For Calculating -rr-, -— , f_, m~ and f,

Q£ C u C *

10 INPUT -NO. OF DP.TB POINTS N"?-)N20 DIM X (N) , YIN) ,fl(31 , DYIN1, Z (N)

M N2 m N - 24a dim Bi:,3),aB(3,;i,sf(3)so FOR I 1 TO NSa PRINT"DP.TO POINT NO. " 1

1

70 INPUT"X(I)«?-|X<I)SO INPUT-Y(II-?")Y(II30 INPUT"Z(I)-?"|ZU>100 NEXT I

lie FOB T-l TO 212« FOR I - 3 TO N2IS* FOR J • 1 TO 3ite b<j, : )- i

ise u - I - 3 • Jise B(J,2> - XIIJ) .- X(I)178 B<J,3> " 9<J,2)-2ise next j190 FOR J 1 TO 32ee FOR K - 1 TO 32i

e

BS(J, k> • e220 FOR L • 1 TO S230 SB(J.K) - BB(J,K) - 3(L,J1- B(L.K)2*0 NEXT L.K.J250 FOR J - 1 TO 3260 BF<J> - e270 FOR K » 1 TO S280 IK - I - 3 * K290 IF T-2 THEN Y(I)-Z<I>200 IF T-2 THEN Y(IK>-Z(IK1310 BF(J) • BFCJ) » B<K,J) Y(IK)320 NEXT K, J330 - 3B(l.l) • (BBI2.2) • 3B(3,3) - 9B<£. 21 • BB (3, 2) > - 56(1,21 • 96(2, 11 *

BB(3, 31

340 - D » BBC1.21 • BB(2,31 • BB(3, 1) - BB<1,3: • (BB(3,2) • BB (2. 11 - BB(2,2)• BB(3. 1)

>

330 E - BB(l.l) • <BF(2)»BB(3, 3) - BB (2. 3> »BF (3) 1 - BF ( 1 UMBB (2, 1 1 »BB (3, 2) -

3B(2. 3) • 3B(3, 11 1

360 E - E • 3BC1.31 • (BB(2, 11 • 3F(31 - 3FI21 • 3B(3, 111

370 C(21 - E/D380 £ - BB(l.l) • (BB(2. 21 • BF<3) - BF<21 • SB(3,21) - BB(1,21 * (BB(£. 11

• BF(3) - BF(2) BB<3, 111390 I E - 3F(11 • CBS (2. 11 • 3BI2.21 - 3B(2, 21 • 38(3, 11!400 CI31 - E/D410 IF I - 3 THEN 4SO420 IF I - N2 THEN 310430 DY(I) -0(2)440 SOTO 360430 DY(I - 21 - C(2) • 2 • C<3) • (X<1) - X (31 1

460 LPRINT"DY/DX (11-" ;DY(11470 0Y(21 - C(21 • 2 • CC31 • (X(21 - X(3)l4aO LPRINT "OY/DX (21-" ?DY(2)490 DY(I) . C(21300 GOTO 360

34

3ia OYU) » C<2)32a DY(I l) - C<2) * a • C<3> * <X(I » 1) - XII))33a LPRINT "DY/DXCjI * lt")-"|DY<I + 1)

3*a OY<I • 2) - C(2) • 2 • C(3) * <X(I * 2) - XI I))330 LPRINT "DY/DXC'lI * 8) ")«" lDY< I * 2)36a LPRINT "OY/DX(', |I|")-")DY(I)373 NEXT I

38a IF TO 1 THEN sia39a LPRINT "STRAIN AT COMP FACE Fa DF/0saa GOTO 638Sia LPRINT "STRAIN AT COMP FACE ITO OM/062a FOR M-l TO N53a IF T-2 THEN Y<M>«2»Z<M>64a F»X <M)*OY<M)-fY(M)63a IF T-2 THEN Y<M)-Z(KI)Sea LPRINT TABC3) X <K1> TAB(23) YCM) TAB(4a) DY(M) TAB(Sa) r

673 NEXT Kl

saa next tesa END

35

e. Calibration of the minor load P2

Pressure (Psi) Load (lb)

75 500

150 1000

235 1500

315 2000

395 2500

475 3000

550 3500

625 4000

700 4500

750 4850

800 5200

1 psi =6.89 kPa1 lb = 4.45 N

36

APPENDIX III

TABLES

AND

FIGURES

37

jjCO

<uEl

3

2

S. o•HJJ10

suo*Jc

oI

.a

a9JJ —

.

n

10 10

aV —

>t-H(0 "O5<HUJ 3o m

o o

oi 5

10 «a<u co -H

M• 3U

Zc

ai

ch aijj jja aa a

sai

S „ai oCJ~j

jju ia

v IXjj(0

3

oto

ojj

r? *r Tf COCO 00 CO V.

ovo

oJJ

oJJ

r* VO

CO CO

\fi in oOH iH (N

tp a CO *3- CO GO«3> ^ .

co mvo c-»

m-s.

VO CO

CN HVO CO

en ihCM IN

VO CO

CO

0>

a o

o

S 0. H H5a> -h H HJ H

4•H O E E1 gO 0] 10 10 a

&£ CD Hi 0) 0)

ca a (3 caCO CO

38

Table 5.1

Cylinder Data Corresponding to Beam I*

No. of 3x6 in. Load Stress

Cylinders Area in lb. psi

1 7.07 55,000 7779

2 7.07 56,500 7991

3 7.07 55,000 7779

4 7.07 53,500 7567

5 7.07 50,000 7072

6 7.07 53,500 7567

SZ£»f ' average = —t^c b

- 4*P = 7626 psi

1 in = 25.4 nun

1 lb = 4.45 N

1 psi =6.89 KPa

* while testing these cylinders, the machine was notfunctioning properly and hence the strength indicatedis not the true strength of the mix.

39

Table 5.2

Cylinder Data Corresponding to Beam II*

No. Of 3x6 in. Load Stress

Cylinders Area in lb. psi

1 7.07 63,000 8911

2 7.07 69,000 97 60

3 7.07 65,000 9194

4 7.07 64,000 9052

5 7.07 66,000 9335

6 7.07 64,000 9052

f ' Average =6

= Z*f = 9220 psi

1 in = 25.4 mm

1 lb = 4.45 N

1 psi = 6.89 kPa

* while testing these cylinders, the machine was notfunctioning properly and hence the strength indicatedis not the true strength of the mix.

40

Table 5.3

Cylinder Data Corresponding to Beam III*

No. of 3x6 in Load Stress

Cylinders. 2

Area in lb. psi

1 7.07 76,000 10,750

2 7.07 68,500 9,6 89

3 7.07 68,500 9,6 89

4 7.07 68,000 9,618

5 7.07 67,000 9,477

6 7.07 75,000 10,622

7 7.07 67,500 9,547

fc

Avei age =71 f

c 695927 " 7

= 9920 psi

1 in '= 25. 4 mm

1 lb = 4.45 N

1 psi = 6.89 kPa

* while testing these cylinders, the machine was notfunctioning properly and hence the strength indicatedis not the true strength of the mix.

41

Table 5.4

Cylinder Data Corresponding to Beam IV

No. Of 3x6 in Load Stress

Cylinders Area in lb. psi

1 7.07 78,000 11,033

2 7.07 72,000 10,184

3 7.07 80,000 11,315

4 7.07 82,000 11,598

5 7.07 80,000 11,315

6 7.07 78,600 10,750

7 7.07 81,800 11,570

f ' Aw,=>r ^n<a _£ fc m 777CC = 111 1 n «r..'

1 in = 25.4 mm

1 lb = 4.45 N

1 psi =6.89 kPa

42

Table 5.5

Stress-Strain Relationship for Cylinder #1

Corresponding to Beam I

Load(Lb)

Stress(psi)

StrainReading in

Gage #1

(m)

StrainReading inGage #2

(m)

Average(m)

5,000 707 - 32 - 216 - 124

10,000 1420 - 57 - 420 - 23 9

15,000 2120 - 117 - 615 - 366

20,000 2 830 - 220 - 7 93 - 506

25,000 3540 - 360 - 965 - 662

30,000 42 40 - 52 4 -1140 - 831

35,000 4950 - 707 -1330 -1019

40,000 5660 - 937 -1560 -1248

45,000 6370 -1190 -1820 -1500

50,000 7070 - 240 - 276 - 258

1 lb = 4 .45N

1 psi = i5.89 kPa

43

00 « •* a —

.

as «*-t > 3J -« Wu O m H *a a.u a +J as —

'

> oa W «<

H•

Of) » tSa G U %r* iH u -->.

«J *o c >. «u <a *4 IA BO 1-w <D

m OSHH

00 W i-l

a S U =tfc ~tfa a > o 3,^ « -H w oo —

-

d «e -" e ooeo «0 -rt *-.

01 3 « -H U0\ iH ^

00 00 <Na P B % ^^.

•H O (U

03 a -J 0) aU OS oO

a a at

n OS •n »

00 00 iHs s a 4fc ,-*

«OS T3 hJ « =1

h RS 00*j V a aBfl OS ^ ©

•a ~

33

44

Table 5.7

Stress-Strain Relationship For Cylinder #2

Corresponding to Beam II

Strain Strain Average Strain Strain

Load Stress Reading Reading of Reading Reading

(Lb) (psi) in Long in Long Longitudinal in in Poisson's

Gage #1 Gage #2 Strain Transverse Transverse Ratio(ue) (ue) Readings Gage #1 Gage #2

(ue) (|ie) (ue)

4 2 2

5,000 707 37 - 280 - 158 30 19 .155

10,000 1420 - 56 - 290 - 173 18 28 .133

15,000 2120 - 102 - 368 - 234 48 43 .194

20,000 2830 140 - 570 - 355 82 55 .193

25,000 3540 - 184 - 785 - 485 125 70 .201

30,000 4240 - 255 - 938 - 596 161 81 .203

36,200 5120 - 344 -1080 - 713 200 97 .208

40,000 5660 - 474 -1258 - 866 250 117 .212

43,000 6080 - 570 -1370 - 972 288 134 .217

46,000 6510 - 645 -1460 -1050 317 145 .220

49,000 6930 - 717 -1540 -1130 346 158 .223

52,000 7360 - 805 -1640 -1220 381 177 .229

55,000 7780 - 901 -1750 -1330 415 196 .230

58.000 8210 - 996 -1860 -1430 449 218 .233

61,000 8630 -1110 -2000 -1560 496 244 .237

63,000 8910 -1240 -2160 -1700 559 271 .244

64,000 9050 -1422 -2440 -1930 627 284 .236

1 lb = 4.45N

1 psi = 6.89 kPa

45

Table 5.8

Stress-Strain Relationship For Cylinder #1

Corresponding to Beam II

Load(Lb)

Stress(psi)

StrainReading in

Gage #1(lie)

StrainReading in

Gage #2

(at)

Average(us)

- 508 - 13

5,000 707 - 110 - 253 - 121

10,000 1420 - 219 - 253 - 236

15,000 2120 - 315 - 402 - 358

20,000 2830 - 403 - 546 - 474

25,000 3540 - 516 - 689 - 602

30,000 4240 - 627 - 816 - 721

33,000 4670 - 707 - 904 - 806

36,000 5090 - 784 - 988 - 886

39,000 5520 - 865 -1080 -.970

42,000 5940 - 945 -1160 -1054

45,000 6370 -1020 -1240 -1130

48,000 6790 -1110 -1347 -1230

51,000 7210 -1200 -1450 -1325

54,000 7640 -1290 -1550 -1424

57,000 8060 -1390 -1670 -1530

60,000 8630 -1490 -1797 -1645

62,000 8770 -1576 -1930 -1750

64,000 9050 -1670 -2060 -1870

66,000 9340 -1770 -2080 -1930

1 lb = 4

1 psi =.45 NS.89 kPa

46

Table 5.9

Stress-Strain Relationship For Cylinder #1Corresponding To Beam III

Strain StrainLoad Stress Reading in Reading in Average(Lb) (psi) Gage #1

(lie)

Gage #2(lie)

(|18)

6 3 4

2,000 280 - 54 - 45 - 50

5,000 707 - 115 - 116 - 115

10,000 1420 - 239 - 224 - 231

15,000 2120 - 359 - 356 - 357

20,000 2830 - 463 - 484 - 474

25.000 3540 - 570 - 623 - 597

30.000 4240 - 673 - 753 - 713

33.000 4670 - 738 - 830 - 784

36,000 5090 - 810 - 916 - 863

39,000 5520 - 879 - 999 - 939

42,000 5940 - 950 -1080 -1020

45,000 6370 -1023 -1165 -1090

48,000 6790 -1110 -1260 -1180

51,000 7210 -1200 -1360 -1280

54,000 7640 -1290 -1460 -1380

57,000 8060 -1390 -1571 -1480

60,000 8490 -1510 -1700 -1600

63,000 8910 -1650 -1830 -1740

66,000 9340 -1840 -2010 -1930

67.500 9550 -2090 -2150 -2120

1 lb - 4.45 N1 psi = 6.89 kPa

47

o w h ta a!H U B *• at wH O 1 « §

* *a a <o a.

•MO «CO OS (9

u so c« a -~

-

Ph -H i.H aA »<a %<A <a a ~a M « w <o wo 4-1 •O '" ao 3.H &n S on 3 -•*-> © fl 9St PC O

hJ

oa

a ,_,

•H a a<3 •h ah M -*4 tHw w> -a %V3 CO a o —1 h •H ** fl> «

[A d at a w

2d

i-( fH »H fH CS

I I

1* •* *o o oiH O t- v> en

1 1 1

oI-l

1

O 00 oo\ o* hO i-i CO

48

« a «h m —«

<a w u T3 a.

aoU

a> ao « a w<

>3

a •* <s•h so<d %<tf tt *>*

u m -m a

(3 Q

w is .-1 so a.

3d

Zin oo

49

Table 5.11Stress-Strain Relation for Cylinder #1

Corresponding to Beam IV

Load(lb)

Stress

(psi)

1 psi = 6.89 kPa

StrainReading in

Gage #1

(lis)

StrainReading in

Gage #2

(ue)

Average(lie)

5,000 707 - 36 - 161 - 99

10,000 1410 - 161 - 238 - 200

15,000 2120 - 284 - 313 - 298

20,000 2830 - 401 - 432 - 417

27,000 3820 - 517 - 603 - 560

30,000 4240 - 655 - 609 - 632

35,000 4950 - 740 - 698 - 719

39,000 5520 - 811 - 881 - 846

43.000 6080 - 868 - 960 - 914

46,000 6510 - 941 -1051 - 969

49,000 6932 -1010 -1120 -1050

52,000 7360 -1140 -1120 -1130

55,000 7780 -1290 -1190 -1240

58,000 8210 -1350 -1260 -1310

61,000 8630 -1430 -1340 -1390

64,000 9050 -1500 -1460 -1480

67,000 9480 -1600 -1570 -1580

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o « -^"•^ 3 1-1,0« h & rH

s~

H3

so W-) 0\ *o <*\ rHr- o o r- Q\ mso so v» CN o\ 00

H rl rH H © ©

so o * m so enCS SO o o\ r- Olso *Ti ifi rH 00 vo

<N M a cn rH r-l

O O o © © ©t^ -<r 00 (N W -tCO so 00 rH en •^en en m <r W *

£o O o © © © .mCO C\ O Os 00 o\o\ tr> 00 rH «n r* ONTf IT) in so vo so 00

so

11

o o o © © o Mso o VJ fsl d m aso 00 CT\ fH m *rrH

1 1

H1 1

CN1 l

rH

o o o o © ©00 w o M en «nK-l 00 o C* -<r m^r v m *n m >n 2

«n

o o © © © o <*o o o © © ©o o © «n © -*r II

o o o OA c* « An m •» <r VI so

83

Table 5.28: Load and Stress Data for Flexural Test of

Beam I Using Eq. 4.1 and Eq. 4.2 and

Cylinder Stress-Strain Curve

Maj or

Thrnst P,

(lb)

AverageMinor Strain at f

cAverage of

Thrnst Pj Compression Eq. (4.1) and (4.2)(lb) (ue) (psi)

fcUsing

Cylinder StressStrain Curve (psi)

10,000 - 64 - 528 - 458

10,000 183 - 118 - 683 - 840

20,500 550 - 247 - 1580 - 1740

30,000 850 - 363 - 2430 - 2530

40,400 1180 - 488 - 3280 - 3370

50,400 1410 - 599 - 4030 - 4090

60,400 1720 - 726 - 4840 - 4900

70,000 2020 - 845 - 5620 - 5630

80,000 2300 - 963 - 6350 - 6330

90 , 000 2590 -1080 - 7010 - 7010

100,000 2840 -1230 - 7680 - 7810

110,000 3180 -1370 - 8480 - 8560

120,000 3430 -1500 - 9220 - 9180

125,000 3530 -1570 - 9660 - 9490

130,000 3630 -1650 -10600 - 9870

135,000 3750 -1710 -11000 -10000

140,000 3820 -1750 -11300 -10200

145,000 3950 -1820 -11000 -10500

150,000 4070 -1910 -10500 -10800

155,000 4130 -2000 -10400 -11100

160,000 4230 -2100 - 9660 -11400

84

Table 5.28 Continued

MajorThrust P,

(lb)

MinorThrust P,

(lb)

AverageStrain at

Compression(us)

feAverage of

Eq. (4.1) and (4.2)

(psi)

fcUsing

Cylinder StressStrain Curve (psi)

162,000 4220 -2160 - 9750 -11500 »

162,700 4220 -2200 - 9660 -11700

165,000 4230 -2260 -10200 -11800

170,000 4280 -2370 -10900 -12000

175,100 4350 -2490 -11300 -12300

179,600 4350 -2610 -11400 -12400

185,000 4350 -2730 -11500 -12500

190,000 4330 -2880 -11300 -12500

205,000 3170 -3340 -10200

lib = 4.45 N

1 psi = 6.89 kPa

* values of stress from this point on obtained from extrapolated curve.

85

Table 5.29: Load and Stress Data for Flexural Test of Beam II

Using Eq. (4.1) and Eq. (4.2) and Cylinder Stress-Strain Cnrve

MajorThrust P,

(11>)

AverageMinor Strainat f Average of f Using

Thrust Pj Compression Eq. (4.1) and (4.2) Cylinder Stress(lb) ((is) (psi) Strain Curve (psi)

9,500 - 129 - 894

19 , 800 600 - 240 - 1770

30,000 670 - 301 - 2320

40,500 1380 - 495 - 3240

50,000 1900 - 631 - 4080

60,100 2360 - 764 - 5060

70,000 2670 - 885 - 5850

80,000 2970 -1000 - 6600

90,100 3330 -1140 - 7350

100,000 3670 -1270 - 8010

119,800 4320 -1560 - 9310

129,500 4570 -1720 - 9880

139,500 4830 -1860 -10300

149,400 5050 -2040 -10700

154,500 5150 -2150 -10900

159,500 5170 -2250 -10900

164,500 5220 -2350 -11200

169,400 5230 -2480 -11200

175,000 5320 -2610 -11100

180,000 5320 -2740 -10800

185,000 5320 -3070 - 9580

- 915

- 1690

- 2100

- 3410

- 4300

- 5130

- 5870

- 6570

- 7350

- 8040

- 9410

-10100

-10700

-11300

-11600 *

-11800

-12000

-12200

-12400

-12500

-12500

lib = 4.45 N 1 psi - 6.

* values of stress from this

89 kPa Failure at 195,000 lb

point on obtained from extrapolated curve.

86

Table 5.30: Load and Stress Data for Flexnral Test of Beam III

Using Eq. (4.1) and Eq. (4.2) and Cylinder Stress-Strain Cnrve

MajorThrnst P<

(lb)

MinorThrnst P,

(lb)

Aye rageStrainat f Average of fusing

Compression Eq. (4.1) and (4.2) Cylinder Stress(us) (psi) Strain Carve (psi)

10,000 - 57 - 490 - 408

10,000 400 - 149 - 785 - 1060

20,000 800 - 273 - 1650 - 1920

30,000 1150 - 389 - 2550 - 2700

39.000 1500 - 513 - 3320 - 3530

50,000 1875 - 650 - 4210 - 4420

60,000 2220 - 776 - 5050 - 1680

70,000 2500 - 895 - 5800 - 1940

80,400 2875 -1030 - 6560 - 6740

90,000 3200 -1160 - 7210 - 7450

100,000 3580 -1310 - 7970 - 8240

110,000 3920 -1460 - 8620 - 8980

120,000 4166 -1600 - 9290 - 9590

130,000 4170 -1750 - 9800 -10200

140,000 4733 -1930 -10400 -10900

150,500 5000 -2130 -10800 -11500 *

160,000 5270 -2320 -11000 -12000

165,000 5280 -2430 -11500 -12200

170,000 5330 -2570 -11600 -12300

175,000 5350 -2690 -11600 -12500

185,000 5270 -2810 -11800 -12500

lib = 4

* valnei

45 N 1 psi =

of stress from

6.89 kPa

this point on

Failure at 195,

obtained from extr

000 lb

apolated cnrve.

87

Table 5.31: Load and Stress Data for Flexure Test of Beam IV

Using Eq. (4.1) and Eq. (4.2) and Cylinder Stress Strain Curve

MajorThrust P,

(lb)

MinorThrust P,

(lb)

AverageStrainat f Average of f Using

Compression Eq. (4.1) and (4.2) Cylinder Stress(ue) (psi) Strain Curve (psi)

7,000 - 23

10,000 - 39

11,500 200 - 117

20,000 650 - 203

30,000 1010 - 329

40,000 1520 - 482

50,500 1860 - 613

60 , 100 2300 - 755

70,300 2890 - 934

80 , 000 3150 -1050

90,000 3370 -1160

100,00 3870 -1330

110,100 4120 -1470

120,000 4580 -1660

130,000 4820 -1800

140 , 000 5000 -1950

149,500 5220 -2120

159,000 5430 -2320

164,400 5330 -2420

- 324

- 477

- 782

- 1480

- 2360

- 3193

- 3960

- 4790

- 5880

- 6580

- 7400

- 7960

- 8560

- 9450

-10100

-10700

-10900

-11000

-10900

- 173

- 286

- 834

- 1430

- 2290

- 3330

- 4180

- 5080

- 6160

- 6830

- 7450

- 8310

- 9030

- 9860

-10400

-11000

-11500

-12000

-12100

lib - 4.45 N 1 ps

* values of stress from this

L = 6.89 kPa Failure at 170,000 lb

point on obtained from extrapolated curve.

88

12,000

Type I

11,000

en

a.

X© 10,000zUJcrKwtu>CO

S 9.000Ea.3oo

8,000 -

stone

admixture

3 ± 1/2" slump

J_28

AGE DAYS

56 90

Figure 2.1: Effect of Various Brands of Type I Cementor. Concrete Compressive Strength (20)

89

12,000

10,000

8,000l-

Age, days

Figure 2.2: Effect of Various Cements onConcrete Compressive Strength (2)

90

1 1

. nr iiJ— 1 ' --»^^^z ^^^ S^

u/fef £ 1 -

o .roA 9/ -o e v•V jP/o •! 5 .5 o t»«<* K *

^ 5^=°—•*?!-Ik yS^ >E^Uu {/ 4 5 < -

1

1/8/1/ 600

iaM

'mi<D

-V.I*

— tn\m

*-i«

J'«i« E

c

aoc

oa

o2

*— -E -

cs -< u

c

isd 'q;5uej|s 3A!ss3jduioo

91

10

a

e

<r>

o

inQ.

JOs

ca>

c75

4-

el

7,000 PSi

6,000 P

5,000 PSi

2,000 PS"

3,000 PSi

4,000 PS'

Slump - 3.8 to 5.8 in.

Cure 28 days , moisl

Max. size aggregate, in., log scale

Figure 2.4: Maximum Size Aggregate for StrengthEfficiency Envelope (15)

92

<0Q.

oZUJcrl-cn

UJ>en<nUJ(XQ.

Oa

9,000 -

8,000

7.C00

6,000-

5,000

823-Jb CEMENT/ c-y

I00-* FLY ASH/c-yWATER REDUCER

Figure 2.5: Compressive Strength of Concrete Using TwoSizes and Types of Coarse Aggregate for7,500 psi Concrete (1 )

93

I-ohicrHCO

Hi>COCO_

11,000-

- 10,000 -

9,000 -

8,000-

7,000

Figure 2.6: Compressive Strength of Kigh-StrengthConcrete Csing Three Sizes and Typesof Coarse Aggregate

Q 4

1

1s

i

1•n o

6Mo

s

o

I

LU

S E

o04

O « a>' r- « o t t)

("'.) dwms

95

96

in

in

en_

UJ

zoo

.001 .002 aoz .004 o .001 .002 .003 .004 .005STRAIN -FLEXURAL TESTS STRAIN -COMPRESSION TESTS(5BY8BY 16-IN PRISMS) (6 BY I2-IN CYLINDERS)

Figure 4.2: Concrete Scress-Strain Relations (.7)

97

l /a OIA- WtR£

ALL OIMEMSONSIN INCHES

Is 23.4 mm

"Igura 4.3: aaiaiorcamanc Layout (11)

98

h-6/8'

Cos 45*>a

WO. 3 BARS AT 3.3

SECTION A -A (SEE FIG. 4.2,4^l"» 22.4mm ' '

a - the lag siza of stirrup

Figure 4.4, An Arbitrary Section A-A «WtRebar Arrangement at Each Lag (11)

99

NO. 4 BAflS

3.3*

/0.373 11

3.25* ,1/8 01A-1

t I0.875"

1

0.373 3.23* ,_0.373 "

3*

SECTION B-8(SSS PIS •*. 2)

I "=23.4mm

"igure 4-5: An Arbitrary Section of Coliarn ParrWith Rabar Arrangement (XI)

100

7000-

MAX STRESS 7567 PS I

(52 MPa)

Note: Testing machine not functioning properly.

200 400 S00 300 1000 1200 1400 ISOO 1800 2000

STRAIN IN (MICRO IN/IN)

1 Psi - S.S9 kPa

Figure 5.1: Cylinder Compressive Stress-StrainCarve for Beaa I, Data from Table 5.6

101

Note: Testing machine not functioning properly.

200 100 600 800 1000 i 200 1*00 1600 1600 2000

STRAIN IN frtlCRH IN/TNI

1 Psi =6.89 IcPa

Figure 5.2: Cylinder Compressive Stress-StrainCurve for Beam II, Data from Table 5.7

102

I I 000-

I 0000-

9000

aooo-

7000sT

R

E sooo.ss

I 5000-N

S 4000-

3000

2000-

1000'

1AX STRESS.| og

|

a pS]

(73 MPa)

200 «00 600 800 i 000 1200 MOO 1600 ,800 2000

, _ .

STRAIN IN (MICRO IN/IN!1 Psi » 6.89 kBa

Figure 5.3: Cylinder Compressive Stress-StrainCurve for Beam III, Data from Table 5.10

103

I I 000'

I 0000'

3000-

r rooo-4

5 5000-

S000

3000

200 400 600 900 1000 1 200 HOO iSOO 1800 aooo

STRAIN IN (MICRO IN/IN)

Figure 5.4: Cylinder Compressive Stress-StrainCurve for Beam IV, Data from Table 5.11

104

soc! ooo i 500 coao

STRAIN (MICRO IN/IN]

2500

1 in. = 25.4 mm

1 lb = 4.45 N

Load in lb

Figure 5.5: Location of Strain Gages vs. Strain for Beam I

105

SOO 1000 1S00 2000 2500 30C

STRyUNCMCRO IN. /IN.)1 in. » 25.4 nun Load in lb

1 lb = 4.45 N

Figure 5.6: Location of Strain Gages vs. Strain for Beam II

106

in. 25 .4 mm

iooo isoo jooo

S'SAIN (MICRO \H. 'JM.'

2SO0

Load in lb

1 lb - 4.45 N

Figure 5.7: Location of Strain Gages vs. Strain for Beam in

107

30000so; oo30000

I 200001-19300

1 in. » 25.4 nun

1 lb = 4.45 N

900 i 200 1 600

STRAIN (MICRO IN. /IN.

I

2000 240

Load in lb

Figure 5.8: Location of Strain Gages vs. Strain for Beam IV

10 8

3.0

2.5

SOO0 3000

s t »ess(°s:>

1 in = 25.4 mm, 1 psi 6.89 kpa

1 lb = 4.45 N

Load in lb

Figure 5.9: Stress Block of Beam I

109

s.o-4

i.5

4.0-

3.5

2000 4000 S000 8000

STRESS (PSD

1 in = 25.4 mm, 1 psi =6.89 kpa

1 lb = 4.45

Load in lb

Figure 5.10: Stress Block of Beam II

110

2000 *000 6000 8000i 2000

STRESS 'PS I!

1 in = 25.4 mm, 1 psi =6.89 kpa Load in lb

1 lb = 4.45 N

Figure 5.11: Stress Block of Beam III

111

2000 4000 0000I 2000S000 8000

STRESS (PSD

1 in = 25.4 mm, 1 si = 6.89 kpa Load in lb

1 lb = 4.45 N

Figure 5.12: Stress Block of Beam IV

112

320000

200000-

1 30000

Failure at 205,000 lb

J00 1000 IJOO 2000 2«0O 3000 3J00

AW STRAIN IN SASCS «UmCM IN. /IN.)

1 lb - 4.45 N

Figure 5.13: Axial Load vs. strain for gaces 4 and 5for Seam I

113

MO 1200 1S00 2000 2S00

*VS STHAIM IN GAOSS *«<mCSO tM./W.I

1 lb - 4.45 H

3000

Figure 5.14: Axial Load vs. strain for gages 4 and 5for Beam II

114

Failure at 195,000 lb

soo 1000i soo 2000 asoo

*.VE STSAIN [N SACES 4«(MCS0 IN. /I?).

I

joeo

1 lb - 4.45 N

Figure 5.15: Axial load vs. strain for gaces 4 and 5for Beam Iir'

115

(1*)

1 lb

100 800 1300 ISOO 2000

VE STRAIN IN SACSS 4*5MOW IN. /IN.)

4.45 H

2«50

Figure 5.15: Axial load vs. strain for gages 4 and 5for Beam iv

116

Vl/3

1 OOO 1 500 2000

STSAINIMICHO IN. /IN.

I

2500 3000

Figure 5.17: Stress Factors k, , k_k. Vs. Concrete StrainFor Beam I * J. J

117

SCO ,00° >50O 2000 2S00 3000

STSAImmcRO IK. 'IN.)

Figure 5.18, Stress Factors ^ ^ vs. Concrete StrFor Beam IIam

118

500 ,000 1500 2000 2500 3000

STRAIN tPIICRO IN./1N.)

Figure 5.19: Stress Factors k,, k^ltj Vs. Concrete Strain

For Beam III

119

0.50-

0.00-

400 300 I 200 I 600

STRAIN (MICRO IN./1M.)

Tl^ure 5.20: Stress Factors k2< k1k^

v3/8

Vl/3

2400

Figure 5.20: Stress Factors k_, lt.lt Vs. Concrete StrainFor Beam IV

2' 13

120

STRAIN

b»3i i

i

i

I"* 25.4mm

Figure J. 21: Condition ac Ulclmace Load in Test Specimen

121

a iooo

3000

500 1000 iSOO 2000 2500 3000 3S00Vlt STRAIN IN CAGES 44S!MICS0 IN. 'IN.'

1 psi = 6.89 kPa

Figure 5.22: f vs concrete strain for Beam I

122

9000-f

rooo

5000

5000

3000-

2000-

500 1000 ,500 2000 2500 3000

WE STHA1N IN SAGES US (MICRO iN.'IM.'

1 psi =6.89 kPa

Figure 5.23: f vs concrete strain for Beam II

123

9000-1

7000

sooo-

5000

'00 1000 ,500 2000 2500 300

AVE STRAIN IN SAGES 4*5 (MICRO iN.'lN.'

1 Psi = 6.89 kPa

Figure 5.24: f vs concrete strain for Beam III

124

7000-

8500.

WOO

?ooo

4500

3500-

3000

2100

2000

;500-

500-

100 90" 1200 ;S00 2000 2100

AVE STRAIN IN OAGES *4S (MICRO if).'!M.>

1 psi =6.89 kPa

Pigure 5.25: fo vs concrete strain for Beam IV

125

3 WO ,000 ,500 2000 2500 J000 35

AVE STRAIN [H 5A0ES «4301tCS0 iN.'t*.'1 psi = 6.89 kPa

Figure 5.26: m vs concrete strain for Beam I

126

5000-

4S00-;

3500-

3000-

lpsi =6.89 kPa

3 500 1000 ;500 iOOO 2500 300;

*VE STRAIN IN GAGES **S(MICSO iN.'IN.'

Figure 5.27: siq vs concrete strain for Beam II

127

2500

*VE STRAIN iN GAGES *« (MICRO IN. 'IN.'1 psi =6.89 kPa

Figure 5.28: n^ vs concrete strain for Beam III

128

1Q0O-

won

2000-

,500'

500

3 100 900 I 200 i 600 2000

AVE STRAIN IN GAGES '45 (MICRO IN. MM."

1 psi = 6.89 kPa

Figure 5.29: mQ vs concrete strain for Beam IV

129

1 3000-

1 2000-

a^.

1 1 000-

10000-

9000-

3000-

Sr

R 7000^£S

5000-

S

3000-

/

4000-

3000-

2000-

1000-/

a-

500 1000 1500 200C 23CO

AVE STRAIN IN GAGES 445 (MICRO III. 'IN.'

1 psi » 6.89 ka

Figure 5.30: Average f c from Bq (4.1) and (4.2) vsstrain for Beam II

130

800 1200 i600 2000 2400

WE STRAIN IN GAGES **SCMCS0 IN. 'IN.!

1 psi = 6.89 ka

Figure 5.31: Average f c from Eq (4.1) and (4.2) vsstrain for Beam IV

131

APPENDIX IV

NOTATION

e = strain.

e c= strain at compression face.

a = level arm.

b = width of beam.

c = depth of beam.

d= distance for outermost fiber to center of gravityof steel.

f c = stress in concrete at different levels of loading.

f'c= concrete cylinder strength.

f , m cross-section stress parameters.

fy

= estimated allowable strength in reinforcement.

h = height of beam.

kj, k2 , IC3 = ultimate strength factors (k,, k, areshape factors and k, is the position ofconcrete internal force from outermostcompresion fiber.

C = concrete internal force.

As= area of reinforcement.

M moment.

Mu = ultimate moment.

Pj = axial load applied by testing machine (i.e., major thrust)

P2 = eccentric load applied by hydraulic ram system(i.e., minor thrust).

T = force carried by reinforcement.

y = depth of stress block.

132

ACKNOWLEDGEMENTS

The author is greatly indebted to his major professor, Dr.

Staurt E. Swartz, for invaluable advice and guidance during his

study at Kansas State University and carrying out this

investigation.

Sincere thanks are due to Dr. Robert R. Snell, Head of the

Department of Civil Engineering, for providing financial support

for this project. Sincere thanks are extended to Dr. Cecil Best

for his guidance. Appreciation is also due to the other

committee member, Dr. Teddy 0. Hodges.

Special thanks to Mr. Ali Nikaeen who helped me in this

project.

Special appreciation is expressed to Russell L. Gillespie,

Civil Engineering technician, for his guidance and help in using

the facilities.

Special thanks also to Ms. Christy Schroller, Word

Processor of the Electrical and Computer Engineering Department,

for typing this thesis.

133

The Structural Behaviorof Higher Strength Concrete

by

Narayan Babu D. Hireraagalur

B.E. , Bangalore University, 1982

AN ABSTRACT OF A MASTER'S THESIS

submitted in partial fulfillment of the

requirements for the degree

MASTER OF SCIENCE

Department of Civil Engineering

KANSAS STATE UNIVERSITYManhattan, Kansas

1985

ABSTRACT

An experimental investigation into the structural

bending properties of plain beams of higher strength

concrete was performed.

The objective of this study was to obtain further

information on the shape of the compressive stress block at

failure for higher strength concrete beams.

Four "dogbone" shaped beams were cast with an unre-

inforced test region. Loads were applied so that bending

was produced in the unreinforced test region and the neutral

axis maintained at the outer surface of the test region.

Experimental data were obtained up to the failure of

the test specimen and the k 2 values of beams II, III, and IV

were 0.36, 0.36, and 0.35. These were very close to 0.375,

the k2 value for a parabolic stress distribution.

-