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SHEAR STRENGTH AND BEARING CAPACITY

OF REINFORCED CONCRETE DEEP BEAMS

by

Kam Kau WONG B. Sc. (Eng. )

A thesis presented in fulfilment of the requirements

of the Degree of Doctor of Philosophy

at

Department of Civil Engineering

The University of Leeds

August. 1986

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1

ACKNOWLEDGEMENT

The author mould like to take this opportunity to express

his appreciation and gratitude to all those who in some may

have contributed to the research work herein presented. In

recognition of their individual contribution to this study the

author would additionally like to express a special thanks to:

Professor A. R. Cusens, Head of the Department of Civil

Engineering, forgranting

theauthor

theopportunity

to

undertake this study and for his supervision, guidance,

comments and patience given throughout it;

The Oversea Research Council and Tetley/Lupton for

supporting this work through the award of research studentship

and scholarship;

Hr. V. Lawton and his technical staff for preparing and

arrangement of the experiments;

Hr. R. Vuxbury, Hiss H. Y. N. Lam and Hiss C. Dahl for

preparing and processing the photographs;

Hy brothers Hr. S. Wong, K. N. J. Hong and K. N. H. Flong for

developing their own word processing package, COHET - Customers

Orientated Mathematical Expression Text-formater, particularlyfor this dissertation.

Finally, the author would like to thank his parents fortheir encouragement and supports.

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11

ABSTRACT

Reinforced concrete deep beans with small span/depth

ratios usually fail by crushing of concrete in the bearing zone

above the supports. In order to increase the load carrying

capacity of deep beans, bearing strength around the supports

should be enhanced.

The first part of this study involved the investigation ofbearing capacity of plain and reinforced concrete blocks.

Effects of edge distance, footing to loading area ratios,heights, base friction and size effect are studied with plain

concrete blocks. Bearing capacities of reinforced concreteblocks with different forms, diameter and spacing of

reinforcement are also investigated. It is found that

interlocking stirrups at small spacing are the most effectiveform of reinforcement. A failure mechanism for a concreteblock in bearing is proposed and found to give the best

estimate as compared with other models by different

researchers.

The second part is concerned with the behaviour of

reinforced concrete deep beans with span/depth ratios rangingfrom 0.7 to I. I. These beans were tested under uniformlydistributed load at the top. It is found that a shear crack is

formed along the line joining the inner edge of the support to

the third point at the top level of the bean. The concreteblock on the outer side of the crack rotates about the centre

of pressure in the compression zone. Shear strength is

determined by shear in the compression zone, aggregateinterlock of the shear crack and dowel action and the

components of forces of reinforcement across the crack. Based

on these observations, a model of the failure mechanism in

shear is proposed which gives excellent results in comparisonwith other models proposed.

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111

TABLE OF CONTENTS

INTRODUCTION................................................ 11.1 GENERAL................................................. 1

1.2 OBJECTIVE............................................... 2

1.2.1 BEARING CAPACITY OF CONCRETE BLOCKS...............

21.2.2 DEEP BEAMS

........................................ 31.3 OUTLINE OF THESIS

....................................... 3

2 LITERATURE REVIEW ON THE BEARING CAPACITY OF CONCRETEBLOCKS

...................................................... 52.1 INTRODUCTION

............................................ 52.2 BEARING CAPACITY OF PLAIN CONCRETE

...................... 62.2.1 STRESS DISTRIBUTION AROUND THE BEARING ZONE

.......4

2.2.2 INTERNAL FRICTION THEORY OF SLIDING FAILURE...... 142.2.3 EMPIRICAL FORMULA................................ 21

2.2.4 PLASTIC ANALYSIS................................. 2b

2.3 BEARING CAPACITY OF REINFORCED CONCRETE................

282.4 SUMMARY

................................................ 4c)

3 EXPERIMENTAL INVESTIGATION ON THE BEARING CAPACITY OFCONCRETE BLOCKS

............................................ 443.1 INTRODUCTION........................................... 443.2 PLAIN CONCRETE BLOCKS

.................................. 443.2.1 SERIES E

......................................... 453.2.2 SERIES R-H

.......................................

45

3.2.3 SERIES S ......................................... 463.2.4 SERIES B......................................... 46

3.3 REINFORCED CONCRETE BLOCKS-SERIES R................... 46

3.4 MATERIALS & THEIR PROPERTIES........................... 50

3.4.1 MATERIALS........................................ 5D

3.4.2 MIX DETAIL....................................... 50

3.5 CASTING AND CURING..................................... 52

3.6 CONTROL SPECIMENS...................................... 52

3.7 INSTRUMENTS AND TEST PROCEDURE......................... 53

3.8 BEHAVIOR OF TEST....................................... 57

3.8.1 GENERAL.......................................... 57

3.8.2 SERIESE......................................... 583.8.3 SERIES R-H

....................................... 603.8.4 SERIES S

......................................... 6D3.8.5 SERIES B

......................................... 663.8.6 REINFORCED CONCRETE BLOCK - SERIES R............ 69

4 EXPERIMENTAL RESULTS - BEARING CAPACITY OF CONCRETEBLOCKS

..................................................... 744.1 INTRODUCTION........................................... 744.2 BEARING CAPACITY OF PLAIN CONCRETE BLOCKS

..............74

4.2.1 EFFECT OF EDGE DISTANCE...... ..... ..........

744.2.2 EFFECT OF HEIGHT AND LOADING AREA RATIO

..........

79

4.2.3 SIZE EFFECT ...................................... 844.2.4 EFFECT OF BASE FRICTION

.......................... 854.3 REINFORCED CONCRETE BLOCKS

............................. 884.4 CONCLUSION

............................................. 96

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iv

4.5 PROPOSED SOLUTION..................................... 102

4.6 COMPARISON WITH TEST RESULT...........................

1054.6.1 PLAIN CONCRETE BLOCKS

.... ......................105

4.6.2REINFORCED

CONCRETE BLOCKS ...................... 128

5 LITERATURE REVIEW ON THE SHEAR STRENGTH OF REINFORCEDCONCRETE DEEP BEAMS

.......................................131

5.1 INTRODUCTION..........................................

1315.2 ELASTIC SOLUTION...........................

.............5.3 EXPERIMENTAL ANALYSIS OF REINFORCED CONCRETE DEEP

BEAMS................................................. 1375.4 RECOMMENDATIONS FOR THE DESIGN OF REINFORCED CONCRETE

DEEP BEAMS............................................ 149

5.4.1 PORTLAND CEMENT ASSOCIATION [72,19461........... 1495.4.2 DE PAIVA AND SIESS [69,1965]

.................... 151

5.4.3 RAMAKRISHNAN AND ANANTHANARAYANA 1737........... 1525.4.4 COMITE' EUROPEEN DU BETON - FIP (17]........... 1545.4.5 ACI COMMITTEE 318

............................... 1555.4.6 KONG 140,45]

.................................... 1575.4.7 CIRIA GUIDE 2 E68,1977]

......................... 1585.4.8 AL-NAJJIM [63]

.................................. 1605.5 SUMMARY

............................................... 165

6 EXPERIMENTAL INVESTIGATION ON THE SHEAR STRENGTH OFREINFORCED CONCRETE DEEP BEAMS

............................ 1696.1 INTRODUCTION

.......................................... 169

6.2 DESCRIPTION OF TEST SPECIMENS ......................... 1696.3 MATERIAL AND MIX DETAIL............................... 170

6.4 CASTING AND CURING..... .............................. 172

6.5 INSTRUMENTS AND TEST PROCEDURE........................ 175

6.6 BEHAVIOUR OF TEST..................................... 181

7 EXPERIMENTAL RESULTS - SHEAR STRENGTH OF REINFORCEDCONCRETE DEEP BEAMS

....................................... 1937.1 INTRODUCTION

.......................................... 1937.2 STRAIN DISTRIBUTION ON CONCRETE SURFACE

...............193

7.3 STRAIN DISTRIBUTION IN REINFORCEMENTS.................

1997.4 DEFLECTION............................................ 202

7.5 SHEAR TRANSFER BY AGGREGATE INTERLOCK AND DOWELACTION................................................ 203

7.6 PROPOSED FAILURE MECHANISM OF REINFORCED CONCRETEDEEP BEAMS

............................................ 2157.7 COMPARISON WITH TEST RESULTS

.......................... 2237.8 RECOMENDATIONS FOR THE DESIGN OF REINFORCED CONCRETE

DEEP BEAMS............................................ 240

7.9 SUMMARY............................................... 242

O CONCLUSION AND SUGGESTION FOR FURTHER RESEARCH............

2458.1 BEARING CAPACITY

......................................245

8.2 DEEP BEAMS ............................................ 2478.3 SUGGESTION FOR FURTHER RESEARCH.......................

249

APPENDIX A.........." .................................... "

250APPENDIX B

................................................300

REFERENCES................................................ 332

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V

PRINCIPAL NOTATION

A Cross-sectional area of the beam.

a Width of the concrete blocks.

ai Width of the loading plates.

Ab Sectional area below the reference plane.

A Sectional area of main forcement.S

A Areaof vertical web reinforcement.wv

Awh Area of horizontal web reinforement.

b Breadth of the concrete blocks.

bi Breadth of the loading plates.

by Apparent width of the modified end-blocks.

C Cohesion of concrete.

Co Cohesion of concrete at effective pressure, p=O.

D Diameter of the footing of the concrete blocks.

d Effective depth.

Df Dowel force.

ATs Loss of tensile force towards the support due to the

present of vertical reinforcements.

ea Eccentricity of loading along the side width a.

eb Eccentricity of loading along the size width b.

fa Aggregate interlocking stress.

fb bearing strength of the concrete blocks.

f, Cylinder strength of concrete.

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vi

fcu(100) Cube strength of concrete obtained from 100 mm cubes.

fcu(150) Cube strength of concrete obtained from 150 mm cubes.

ft Tensile strength of concrete estimated by splitting

cylinder tests.

ft(pri.)

Tensile strength of concrete estimated by rupture

tests.

Ec Young Modulus of concrete.

H Overall heightof

thespecimens.

hb Depth of the section below the reference plane

Hw Asw/b-Sh

fL Restraining stress.

fxx Direct stresses along the direction of the x-axis.

fyy Direct stresses along the direction of the y-axis.

fxy Shear stresses.

L Span.

Lc Clear span; distance between the inner edges of the

supports.

i Second moment of inertia of the section of the beam.

Ib Second moment of inertia of the section below the

reference

Ie Influence factor of bearing capacity of concrete

block width eccentricity loading.

M Bending moment at critical section of the beam.

Sh Spacing of horizontal web reinforcement.

Sv Spacing of vertical web reinforcement.

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vii

T Tensile force of the main reinforcement.s

V Shear force at the critical section of the beam.

v Shear strength.

vc Shear strength taken by concrete.

vs Shear strength taken by steel.

V Ultimate shear force.u

VA /(b-S )W 5V V

p Effective normal pressure on the shear plane.

Pc Cracking load.

Ph Uniform horizontal pressure along the wedge form

below the bearing plate.

P Ultimate load.u

R Footing to loading area ratio.

Wa Distance of the loading position from the edge of the

blocks.

X Shear span.

Xc Clear shear span; clear distances between the outer

edge of the bearing plate and the inner edge of the

supports.

Xe effective ahear span.

y Depth of the bar, measured from theto the point where it interests theinside edge of the bearing blocks ofthe outside edge of that at the load

y0 Depth of the compressive zone from

beam.

top of the beam

line joining the

the supports to

ing point.

the top of the

z Lever arm at which the reinforcement act.

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viii

E Yield strain of the reinforcement.Y

E Shear strain.xy

Txy Shear stress.

w Angle of internal friction.

a Semi-apex angle of the wedge formed beneath the

loading plate,

p Volumetric % of lateral steel.

AS -fY/a -b -fc

Coefficient of friction.

47ex

Applied direct stresses.

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i

1 INTRODUCTION

1.1 GENERAL

CHAPTER 1

A deep beam may be defined as a structural member whose

depth is of the same order of-magnitude as its span. Various

investigators have shown that the elastic behaviour is

different from that of the more common flexural mambers. This

difference in behaviour is mainly attributed to the significant

effects of vertical normal stresses and shear deformations in

these members. The strength of deep beams is usually

controlled by shear, rather than flexure, provided normal

amounts of longitudinal reinforcement are used. On the other

hand, the shear strength of deep beams is significantly greater

than that predicted using expressions developed for slender

beams. As reinforced concrete structural members are nowadays

being increasingly designed on the basis of their ultimate

strength, there is a need to know the ultimate behaviour and

strength of deep beams as well.

Although a clear division between ordinary flexural member

and deep beam behaviour does not exist, most literature [457

dealing with this subject recognizes deep beam action at

span/depth ratios less than 2.0 and 2.5 for simply supported

and continuous members respectively. However, for beams with

span/depth ratios less than 1, its load carrying-capacity is

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2 CHAPTER 1

usually determined by the bearing strength in the region above

the supports 112,13,533 rather than shear.

In order to investigate the shear strength and behaviour

of deep beams with span/depth ratios less than 1, the behaviour

of bearing zones above the supports should be understood first.

This thesis is divided into two parts for this purpose, the

first part deals with the investigation of bearing capacity of

concrete blocks and the second part is concerned with the shear

strength of deep beams with span/depth ratios less than 1.

1.2 OBJECTIVE

1.2.1 BEARING CAPACITY OF CONCRETE BLOCKS

The behaviour and ultimate strength of bearing capacity of

plain and reinforced concrete blocks is studied. Special

attention is paid to the following:

(1) Effect of the loading position: position of the loadingpoint from the edge of the block (edge distance).

(2) Effect of footing to loading area ratio, R.

(3) Effect of the height of the concrete block.

(4) Effect of the size of the specimen (scale effect).

(5) Effect of the friction at the base or supporting edge of

the concrete block.

(6) Effect of form, diameter and spacing of reinforcement used.

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3 CHAPTER 1

Based on the experimental behaviour a failure mechanism is

proposed for the load-bearing concrete blocks.

1.2.2 DEEP BEAMS

Deep beams with span/depth ratios ranging from 0.7 to 1.1

are studied. Attention is focussed on the crushing of concrete

above the supports. The investigation is concentrated on the

following areas:

(1) Surface crack formation and development of crack width.

(2) Distribution of strain on the concrete surface.

(3) Distribution of strain in the reinforcement.

(4) Verticaland

horizontal deformation.

1.3 OUTLINE OF THESIS

The first part of the thesis, concerned with the bearing

capacity of concrete blocks, is dealt in Chapters 2 to 4. The

second part about the shear strength of reinforced concrete

deep beams is covered in Chapters 5 to 7.

For a better understanding and to provide a background

knowledge of the subject, a literature review is necessary.

Chapters 2 and 5 are respectively the literature review of the

bearing capacity of concrete blocks and of the shear strength

of reinforced concrete deep beams.

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4 CHAPTER 1

Chapters 3 and 6 are concerned with the manufacture,

instrumentation and testing of specimens for bearing capacity

of concrete blocks and deep beams respectively.

Results obtained from experiments are detailed in

Chapters 4 and 7 together with discussion and a proposed model

of failure mechanism both in bearing capacity and shear

respectively.

Chapter 8 is a summary of the findings of this

investigation with a number of suggestions for further

research.

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6

way to improve its capacity.

The bearing strength of plain and reinfoi

received attention from researchers since

elastic analyses are limited in value by the

of concrete and the complexity of stress in

Different researchers have different ways of

problems. The methods used will be discussed

2.2 BEARING CAPACITY OF PLAIN CONCRETE

CHAPTER 2

'ced concrete has

1888. However,

brittle behaviour

the bearing zone.

approaching these

below.

2.2.1 STRESS DISTRIBUTION AROUND THE BEARING ZONE

The state of stress in the bearing zone is of an

exceedingly complex three dimensional nature. This stress

distribution is due to the very high compressive stress, and is

influenced by many factors, such as the relation between the

area over which the load is applied and the size and shape of

the cross-section of the unit. For the designer, a knowledge

of the distribution of stresses in the bearing zone is

essential for detailing, to ensure that adequate steel is

provided and properly placed to sustain these stresses, as well

as any other bearing or shear stresses that may be present.

The firstapproach

to thecalculation of stresses

in

blocks subjected to concentrated loads was based on some tests

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7 CHAPTER 2

preformed by Marsch in 1924. The following assumptions were

made:

<1> The stress due to a concentrated load are uniformly

distributed at a distance equal to the width of the

prism.

<2> The curvature of the trajectories causes tensile

stresses, the latter being distributed according to a

parabolic law.

The distribution of the compressive stress trajectories

deduced by Mörsch is shown in fig. 2.1. According to the

figure, compressive stresses are uniform over the loaded area

and the remote end of the end-block. It can be seen that

Z= P(a-al)/4H (2.1)

and if the tensile stresses are distributed according to a

parabolic law, then the maximum tensile stress for a

rectangular prism of breath b is

ft = 3Z/2ab (2.2)

However, the assumption of a parabolic law is based on the

measurements of transverse strain by Kruger [20) but Kruger

measured the strains at three positions only, from which he

constructed a parabola representing his view of the stress

distribution. Since any curve can be drawn through three

points, this assumption may not be true. To obtain the

cracking load according to the above formula, the actual

tensile strength of the material should be found. MGrsch also

advises a correction of the depth of block, h, as shown in

fig. 2.1(c), and he suggests that it is more important to use

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p 04

a-I a)

Fýg. 2.1.

c hri. i7a

1

0

va

17

C)i0ö

a

.925a

, r0.875 a

01-0 0.386a 0.5a 0.675a 1.00

vertscol stressdi.strt, but: on

(b) (c3

Stress tra jectorL. es Ln Morsch' s theory.b Tenstie stress dtstrLbutt, on.c Ret ati on between h and a.

4X- width b

45; 1

______a______

e

1 transverse

1 stress

11

I1

I öl

shear -stress

öl H

tenstion1I

on,

a

t al I bl

Ft, g. 2.2 (a) Structural model proposed by Magnet.(b) Transverse and shear stress dt.strLbutt, ons.

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9 CHAPTER 2

high-strength concrete for the blocks than to employ large

amounts of reinforcement.

Another method of computing the principal tensile stresses

in the end of a prestressed concrete beam is given by

Magnel (55,56] as shown in fig. 2.2. He assumed that at a

particular reference plane AB, the transverse stress, fxx due

to the bending moment M, and shearing force S is distributed in

a parabolic curve of the third degree as in fig. 2.2(b). By

means of the boundary condition, the transverse stress can be

calculated and will be a maximum at 0.5a from the contact area.

Similarly, the shear stress can be calculated using the

appropriate boundary conditions. On the assumption that the

pressure under the anchorages of the cables disperses at an

angle of 45 degrees into the end of the beam, the distribution

of longitudinal stresses can also be calculated. In this way,

the principal stresses can be found. The beam will fail in the

condition that the principal stress reaches the maximum tensile

strength of the concrete. Fig. 2.2(b) is an example of the

stressdistribution

ofthe

anchorage block estimated by Magnel.

Bortsch (81,82] made one of the earlier theoretical

approaches to the problem of bearing capacity as well as stress

distribution in structural units under concentrated loads. He

assumed the load distribution on the contact area of the

loading plate as a cosine function as shown in fig. 2.3. From a

stress function analysis, the transverse, longitudinal and

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10 CHAPTER 2

shear stresses can be calculated. The maximum transverse

tensile stress f

yy

occurs at a distance of 0.2 to 0.3a from the

contact area which is different from those predicted by Magnel,

and it is in a range of 0.38 to 0.45P/a for R between 10 and

20. At a distance of 1.7a from the end of the block, the

tensile stress disapear, being 0.055P/a at x/a=1.0 .Bartsch

deals with large values of R>20 and does not give any

indication as to whether his theory can be used for values of R

approaching unity.

Another theoretical approach to the problem of calculating

of the stresses in the anchorage zone is by Guyon (26,27].

Fig. 2.4 represents the sectional elevation of the end beam with

bearing surface AB and plane CD. They are in equilibrium under

the action of forces on CD distributed linearly, and the forces

on AB, concentrated on small area with P1 and P2 as resultants.

In addition, the following conditions must be satisfied for

equilibrium to be maintained.

<1> According to the St. Venant principle and from

experimental verification by photoelasticity thatbeyond a certain distance from the end of the beam

approximately equal to the depth of the beam, the

stresses are almost entirely longitudinal, the

transverse stresses can be neglected.

<2> The resultant of the stresses fyy along EF must be

zero.

<3> The sum of the moments of the stresses fyy about a

point in EF must equal the sum of the moments of theforces acting on EB and FC.

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11

U

a

Ft.g. 2.4

C

Sectt. onal, el, evatt on of end beam by Guyon.

tenat. e

0.5p

0 0.4p

ö -r-- -r- -r- - --r

°a 0.3p'r -I-ý- --r--- -----I-------I-

äi ýý6 Iyl

i0.2p 4- t

° O. p--ý-

ýy- --1-

o4(s

3o/2 2a

Ft.g. 2.3 Load- dt.strtbutLon Ln Bortsch's theory

q/2 aCa)

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12 CHAPTER 2

<4> The resultant of the shear stresses, fxy must equal the

resultant of the horizontal forces applied to BEFC.

Considering the above boundary conditions by using Fourier

series as a stress function, Guyon (5,6) gives six tables for

the calculation of stresses fyy, fxx and fxy caused by the

forces in the anchorage zone. The variation of fyy along the

axis for various value of ai/a is shown in fig. 2.5(a). The

tensile stress contours for diffferent degrees of concentration

of the applied force are shown in fig. 2.5(b) to (d). It is

interesting to note that, in addition to the tension produced

deep in the block along the line of action of the force, there

are tensions near the surface in the two corners; this will be

referred to as the spalling zone and the tensile region along

the axis as the bursting zone. However, recent photo-elastic

tests [83) as well as the tests on concrete units show that

Guyon under-estimated the stresses.

Bleich [81,82] made use of an Airy stress function F and

considered the boundary conditions. For a two-dimensional

problem, he was able to calculate the vertical, horizontal and

shear stresses successfully. In the case of the applied load

shown in fig. 2.6, the tensile stresses calculated are shown in

fig. 2.7. Sievers 19] presented an approximate formula for the

three dimensional condition shown in fig. 2.8 which satisfied

the boundary condition. He modified the two-dimensional stress

distribution developed from Bleich's accurate solution with the

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T0.7

0.6

0.5

0.4

c 0.3

0.2

0. i

0

X

P

2b1t

13

V

IIa) (b1

Ft.g. 2.6 LoadL.ng condttt, on tn 6Let. ch's theory.

IiI

1I Iý1ýI

611

- -1_II__

.O" 1--L -I-

1--I- --L -I-1I I

-I-

--II-

-1-I-

-I--1-1---i--t-L--I-

IIIIII1I

--I--t -1- lg-1 -- -ý1--I-i-ý

-1-

--1 -L-I-iI11.

d. 9

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Ot.etance fron anchorage end

Ftg. 2.7 Tensile stress di.strt. butt. on by BLet.ch's Theory.

r

C.

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14

following assumptions:

CHAPTER 2

<1> The actual inner stresses at a distance of (a-x) from

the beam end are taken equal to those in a modified

end-block having the apparent width of b=bx obtained by

two-dimensional analysis.

<2> The apparent width of the modified end-block is given

by

-O. Birn

bx = b1 - (b-bl) (i+2.5nq) eq (2.3)

<3> The applied load is considered to be uniformlydistributed on the area ai. bx

It has been confirmed by three dimensional photoelastic

tests that this formula agrees fairly well with the

experimental distribution.

2.2.2 INTERNAL FRICTION THEORY OF SLIDING FAILURE

A number of tests have been carried out by

Meyerhof E59,1953] to investigate the bearing strength of

concrete and rock. The results indicate that the material

generally fails, depending on the magnitude of the confining

pressure, by splitting or shear along one or several rupture

surfaces. The failure condition can approximately be

represented by the relation for the shearing strength C, of the

material.

C= Ca +p -tan'r (2.4)

where Co = shear resistance per unit area for p=O.

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15 CHAPTER 2

p= effective normal pressure on the shear plane.

-e = angle ofinternal friction.

Consider a strip load of width ai acting concentrically on

a concrete block as shown in fig. 2.9. On failure, a wedge of

material is found immediately beneath the footing with a

semi-apex angle equal to a, fig. 2.9(b).. By considering the

equilibrium of half of the wedge and assuming that the

horizontal pressure, Ph causing the splitting of the block is

uniformly distributed along the wedge. The horizontal

splitting pressure can be obtained as

Ph = fb2"tan2a - 2-C0-tans, (2.5)

assuming a triangular distribution of tensile stresses to

resist the bending moment produced by the horizontal splitting

pressure, fig. 2.9(a). Substituting in Eq. 2.5, the unconfined

prism strength is

f' = 2C -cotes (2.6)co

which can be simplified to obtain the ultimate bearing

stress, fb

fb 2H/a1-cotta " ft. cota

-=i+ (2.7)f" (OH/ai-cota) -f'

For large ratios of H/al

fb/fc =1+ ft-H/(4Co-al) (2.8)

By differentiating Eq. 2.7 with respect to a, the minimum value

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16

of fb/fý can be obtained as

CHARTER 2

fb/f' =2+O. I5H-ft/ (ai -fl) (2.9)

which is the lower bound for bearing strength to cylinder

strength ratio. It can be seen that the bearing capacity of

surface footings is directly proportional to the ratio of block

thickness to footing width H/ai. Moreover, experimental

results show that the bearing capacity of the mass blocks is

somewhat greater than the theoretical estimates for a small

ratio of block thickness to footing diameter (H/D); this

difference may be explained by the lateral confinement of the

material due to frictional restraint on the base of the blocks,

which had been rejected in the analysis. For large ratios of

H/D, the ratio of the bearing capacity to the prism strength

(fb/fý), tends to a limiting value of 7 which is given by the

present analysis for shearing failure with w=45 degree. If the

width is increased the bearing capacity of the mass blocks is

less than the theoretical estimate on account of premature

failure bysplitting.

Wheresplitting of the material is

prevented, the bearing capacity can be estimated from the

theory. It increases rapidly with the size of the block and

approaches the limiting value of 24 times the cylinder strength

for a footing on a semi-infinite solid. However, tests carried

out by Muguruma 0627 and Niyogi 1657 indicate the opposite

result: bearing capacity decreases as the height of the block

increases, particularly for those with small values of the

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17 CHAPTER 2

ratio R (footing area/loading area). Probably this is due to

the non-linear distribution of tensile stresses, Ph as the

height of the block increase.

Tests were conducted by Tung Au [6,7,19603 to determine

the bearing capacity of concrete blocks with R ranging from 2

to 16 and with depth equals to either full (series A) or half

(series B) width of the block. It was found that, in series A

at failure, a vertical crack which started at the top of the

block progressed downward indicating splitting due to sliding

failure. The maximum load was reached after the formation of

an inverted pyramid. For those in series B, the blocks were

split radially and in most cases, no clear-cut pyramids were

observed. Cracks usually appeared first at the bottom of the

sides and progressed upward. This indicate that splitting was

caused by radial pressure resulting from large deformation of

concrete under the base plate and the depth of the block is not

enough for the formation of an inverted pyramid. He assumes

that the block will split diagonally as it is loaded with a

square plate, fig. 2.1O. Based on Meyerhof's proposal, Eq. 2.5

can be obtained. Again, with the assumption of the uniform

horizontal pressure Ph along the wedge, the horizontal

splitting force F can be calculated. This force produces

combined direct tension and bending in the concrete block with

a stressdistribution

as shownin fig.

2.10(e). The maximum

tensile stress at the top of the block can be computed as

8/8/2019 uk_bl_ethos_373477


Ph

-4

LqJ re°,2

18

°14_Ph0/2octo

!

P-h

octalftý'+ýOOcc!

urIfs

ý-a

c ai c bi c a)

FLg. 2.9 Structural model proposed by Myerhof.

1-

9

wii

l o)

02/2a,

1

FU

IIN

PH

II ýI0

y01

0t b)-

(c)

direct stress bendLn8 stress

-bendi. ng stress

Ftg. 2. i0 Structural model proposed by Tung Au.

tb°h

Co

2stnoCF0

i_ý_

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19

ft = Ph/K*coto

where

'' a1Z 1y- a1/6 cotoc

k4AY

CHAPTER 2

(2.10)

(2.11)

and A= area of the diagonal section of the block except the

wedge,

y= position of the neutral axis from the top,

I= second moment of inertia of area A about the neutralaxis.

By substituting in Egs. 2.9 and 2.5, the bearing strength

to cylinder strength ratio can be obtained as

fb1fý =t 2C0/fý -* K-ftIf ') -cots (2.12)

It was found from experiment that the half apex angle a varies

from 19 to 25 degs. approximately. As both cots and K are

sensitive to small changes in the value of a, the results from

Eq. 2.12 are too scattered to justify. Moreover, at high

pressure, the stress distribution along the depth of the block

becomes non-linear and Eq. 2.1O no longer applies. Therefore,

Eq. 2.12 only gives an approximate solution. Nevertheless, it

provides a rational basis for relating the empirical constants.

A dual failure criterion for concrete is adapted by

Hawkins 130-32]. For regions subject essentially to tension,

the governing factor is assumed to be maximum tensile stress.

For regions subject essentially to compression, failure is

assumed to be due to sliding along planes inclined to the

direction of principal stress. The limiting stress on the

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20 CHAPTER 2

failure plane is again taken as

C= Co + p-tan' (2.4)

Consider a specimen of rectangular section loaded as shown in

fig. 2.11. A failure wedge ABC is formed and punched down into

the crack. For the equilibrium of the wedge

fb = fý + 2Fo-cotm/al (2.13)

The force F0 depends upon the resistance offered by the block

to the penetration of the crack. Lenschow and Sozen £521

assume that

2.41ft, Wa + 11.8M0 /Wa

. -f 4R%"Q 'c. a-t,

l1.8L/W + 7.84a

where L is the measure of the crack length shown in

fig. 2.11(b). Mo is the moment about the crack line DE. The

magnitude of Mo depends on the position of spalling crack FG.

Mo is approximately given by

Mo = fb -ai -Wa2 /2H - fb -a12/a (2.15)

By substituting Eq. 2.14,2.15 into 2.13 gives

if2.41ft-Wa+5.9a1 -Wa-fý/T-1.49a12-fc/Wa

fb = fc +-

al L (11. GL/Wa+7.84)tanm/2-5.9Wa/T+1.49a1/Wa

(2.16)

At collapse the rate of change in the force F0 with increase in

length L equals the tensile strength ft. Differentiation of

Eq. 2.13 gives

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21 CHAPTER 2

d (fb)2d

(Fo) 2ft

_- tans- = tans (2.17)d (L) a1 d (L) a1

Differentiation of Eq. 2.16 w. r. t. L and substitution into

Eq. 2.17 give a quadratic expression for L/Wa for which the

positive root is

L Wa 0.25a1

WT -tanoc W -tansaa

F a1-fc (a1\2 14.5

+ 0.291 2.41+5.91 -1.48(-) 1-0.664

1 T-ft \Wa! 1 (2.18)

The value of T can estimated by

` 3W W < a/6

a aFor sym. load....

T=i

l a/2 W > a/6a

SW W < a/3a a

ecc. load....

T=i1a W > a/3 (2.19)

a

2.2.3 EMPIRICAL FORMULA

Shelson E76] has carried out tests on twenty-one Bin.

cubes loaded through a mild steel base 1/4 in. thick and 1.0,

1.41,2.0,2.93 and 3 in. square respectively. He found that

the maximum bearing pressure increased as the ratio of footing

area to loading area increased as shown in fig. 2.12. For a

relatively low value of R, the bearing capacity increases

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22 CHAPTER 2

considerably as the ratio R incrased. As R becomes larger the

ratio of bearing capacity to the compressive strength tends to

have a limit of 5, which corresponds to the case of loading of

a semi-infinte footing. Fig. 2.12 has been plotted together

with a comparable curve obtained from the specifications of the

ACI Code with a factor of safety of 4. It indicates that the

Code provides a more than ample margin of safety at the higher

ratio of R, but for low values of R, which are more common in

practice, the margin of safety is not good enough. A more

reasonable design formula has been proposed by Shelson 1767 as

(fig. 2.12)

T

fb/fc = 0.25 F7 .3 (2.20)

It follows the actual failure curve more closely than the ACI

Code requirement. At the lower end, this curve provides a

permissible stress in accordance with the Code and for higher

values of R, the curve remains quite conservative but certainly

represents an improvement.

Tests have beencarried out

by Kriz[49], through 39 plain

concrete specimens loaded with different edge distances and

plate sizes. He found that the bearing strength was

proportional to the square root of f which in turn is related

to the concrete tensile strength. Bearing strength was

influenced by the width of the bearing plates and by the

distance of the bearing plates from the edge of the specimen

(edge distance, Wa). Splitting failure occurred when the

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23 CHAPTER 2

distance was more than 40 mm., otherwise there was shear

failure along an inclined plane extending outward from the

inner edge of the bearing plate. For plain concrete specimens,

a proposed formula was used to calculate the bearing strength

as

fb=5.73 f'05

(Wa/al)1/3 (2.21)c

To investigate the effect of height of concrete block upon

bearing capacity, concentrated loading tests were carried out

in two series by Muguruma [627. Series I had rectangular

section 250 x 150 mm, with three different heights of 500,250

and 150 mm. Specimens having 200 x 200 mm section were adopted

with five different heights, from 100 to 400 mm, in series II.

Series I specimens were loaded with a rectangular plate so that

load was distributed uniformly throughout the thickness of the

block b, while in series II a square plate was used for

applying concentrated load. An empirical formula was derived

from the results of these tests,

7.61H/a-3.54

16.44H/a-6.65fb 1/(6.67H/a-2.91) + 0.71 ) -ft-R (2.22)

This empirical equation is applicable to the prediction of the

bearing capacity of concrete of relatively high compressive

strength about 40N/mm2. It was noted that the effect of height

becomes important as the value of R become smaller.

It is suggested by Niyogi [65,66] that the bearing

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24 CHAPTER 2

capacity of a specimen is influenced by

<1> The geometry of the block and loading condition,(a) the dimensions of the loaded surface of specimen

relative to those of bearing plate,(b) relative height of specimen defined as ratio H/a,

(c) eccentricities of loading, expressed as e/a and

e/b,<2> The bearing area,<3> Mix proportions and strength of concrete,<4> Size of the specimen.

Tests of over 100 blocks with dimensions varied from

0.5x8x4 to 24x8x24ins, under strip load and eccentric load,

with rectangular and square bearing plates, were conducted.

The effect of eccentricity on bearing capacity of the concrete

block was also investigated by loading the concrete block with

unaxial and biaxial eccentric load. As a result of the tests,

it is seen that the cube-root formula considerably

underestimates the bearing strength for square loading and

somewhat overestimates the strength for strip loading.

Fig. 2.13 gives a plot of results of the experiment. It can be

seen that for R less than 8 the bearing strength decreased with

increasing height of the specimens. This was probably due to

<1> The reduced influence of base friction as the height ofthe specimen increased and,

<2> the size effect.

But with R greater than 8, shallow blocks had lower

bearing strength. This reduction was perhaps caused by

increased concentration of vertical reaction at the bottom of

specimens, leading to an equivalent localized loading condition

from both top and bottom. Finally, the ratio of the bearing

8/8/2019 uk_bl_ethos_373477


6

r5p

p4

3

Läz

ii

0

25

11111111

--y-t--I

r Test curve

I J.F 'T -F

11Pro Deed

ý-wt th softy factor B4

ICI Code T

- -i - r- w th. aafty factor 4i

20 40 60 80 100

Foott.ng area/Loaded area

Ft,g.2.12 Bearing strength to area ratio. Shelson.

3.0

R-162.5

R-12

2.0R-8

0

_1.5 R-6

R-4

!. 0R-2

0.5

00.5 1.0 1.5 2.0 2.5 3.0

H/a

( a) ( b)

Ft.g. 2. i3 Influence of bearLng strength by thehet.ght of the specimen. Nt.yogt..

8/8/2019 uk_bl_ethos_373477


26 CHAPTER 2

capacity to the compressive cylinder strength of a concentric

load concrete block can be estimated by

fb [abi[ /a b110.5

-=0.421- +-+11-0.291 (-- -) + 5.06fc [a1 b1 jL

\a1b1/ 1 (2.23)

The influence of eccentricity on the bearing capacity of the

concrete block can be represented by the influence factor Ie

I /e eb, 2 10.5 ea eb

ie = 2.36 [ 0.83 - ka b/ )j-Ü. 44 (b)-1.15a

(2.24)

2.2.4 PLASTIC ANALYSIS

Coulomb's failure hypothesis was presented in 1773, in

which it was assumed that the internal cohesion is constant and

the internal friction is proportional to the normal pressure on

the sliding surface £21]. This assumption was formulated

mathematically by Mohr (1882) as

C= Co - p-tanw (2.25)

and can be represented diagrammatically as shown in fig. 2.14(a)

For uniaxial compression

fc = 2CQ"cot' where a= 45-w/2 (2.26)

Coulomb's -failure hypothesis can be supplemented by another

hypothesis (the separation failure hypothesis) in which, the

failuresurfaces move away

fromeach other perpendicular to the

failure section, provided the biggest tensile stress is equal

8/8/2019 uk_bl_ethos_373477


27 CHAPTER 2

to the separating resistance ft, i. e. of=ft. This hypothesis

can also be represented diagrammatically in fig. 2.14(b).

Consider a plane homogeneous deformation field occurring

in a narrow zone of height 6 between two rigid parts, marked I

and II in fig. 2.15. Part II moves V in relation to part I

making an angle p to the direction of crack. The internal work

per unit length along the line of discontinuity is

f'-V(1-sing)/2..................... oe<e

W=i sins-sineI f'-V(1-sinp)/2 + ft-V-

.... o>e

L 1-sine (2.27)

A block is loaded with strip load as shown in fig. 2.16(a). A

wedge of materialwith an apex angle

2m is formed beneath the

loading surface, it fails by sliding along the surface.

Splitting failure is found along the vertical crack.

The-internal work corresponding to the

wedge is W. = f*-V-ai (1-sines)/2 (2.28)ILW

vert. crack is Wie = 2fß-V(H-ai-cot(K/2)-sin(p+v) (2.29)

External work done by the load is

We = fb. ai. V. cos(a+v) (2.30)

By equating the internal and external work we find

fb =

f.[1-sin'r,

/2 + ft-sin(a+, r)[2H-since/a1-coso

(2.31)

sins - cos((X+w)

For minimum fb

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28 CHAPTER 2

2H-cosw/al 10.5

cots = tans + secaI1 -1-f'(1-sin(x)/2f

t-sin'r"j

fb

(min)= ft(2H-tan(2p+v)/a1 - 1) (2.32)

If the loading plate is too near to the edge of the specimen it

fails by shearing off the corner as shown in fig. 2.16(b).

Similarly, by considering internal and external work, we obtain

fb = f'-(2Wa

+a1)/2a1 (2.33)

(min) c

2.3 BEARING CAPACITY OF REINFORCED CONCRETE

Shizuo Ban C9] has performed tests of eighteen specimens

to investigate the effect of transverse reinforcement upon the

cracking and ultimate loads. He used mortar blocks 20.8 ins.

in length and 7.1 by 4.75 ins. in cross-section, loaded with

an anchorage plate of 0.5 ins. thick from 2x2 to 6x4 inches in

plan. The permissible stress for concrete in tension was

assumed as 1/3 of its tensile strength determined by tensile

splitting tests of 6x12 ins. cylinders. Spiral reinforcement

was arranged in the tensile overstress region based on Bleich's

two-dimensional solution. It was found that spiral

reinforcement was the most effective way to increase the

bearing capacity of concrete particularly as the size of

anchorage plate becomes larger. The initial cracking load (not

the ultimate load) was approximately proportional to the

8/8/2019 uk_bl_ethos_373477


29

11

ýv II

p;

TII

ýý

oc

a -90

ta)

1.4

I .2

ea

ia

1.0

z':

t

x

Ft.g.

2.15 Deformatt.on

ft, eLd tn pLastLc anal, sts.

( b)

Ft.g.

2.16 Loodtngcondttt. on in ptastt. c andysi. s.

1IIi ý'

1

1Iý

1I1

1

0 0.2 0.4 0.6 0.8 1.0

P- ULtt.eate Load ofRet.nforced concrete.

Pa - ULti.nate Load of

PLa.n concrete.

8/8/2019 uk_bl_ethos_373477


30 CHAPTER 2

cylinder strength fC of concrete.

For the purpose of obtaining the effects of spiral

reinforcement on the bearing capacity of concrete block,

concentrated loading tests on the 200mm cube specimens having

different percentages of spiral reinforcement were tested by

Muguruma [62]. The ultimate bearing capacity became larger

with increase of spiral diameter of reinforcement but there was

no obvious increase in the initial cracking load. However,

when the losing area A' became smaller in comparison with the

concrete sectional area A" inside the spiral reinforcement

little increase of bearing capacity was to be expected, because

sliding failure would take place or there would be shear

failure of the concrete just under the base plate. The

ultimate bearing capacity increased in proportion to the

percentage of reinforcement as shown in fig. 2.17. The use of

spiral reinforcement with a smaller diameter of steel was more

effective in increasing the ultimate load capacity as well as

the initial cracking load. Moreover, circular spiral

reinforcement was more effective than square spiral

reinforcement to resist bearing and cracking.

Niyogi E67] also performed tests in reinforced concrete

blocks. All tests were with 8 in. concrete cubes which were

reinforced with either spiral steel or reinforcing mesh. Two

spiral sizes were used of large and small diameter extending to

full or part depth of the cubes. The numbers of turns for the

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31 CHAPTER 2

spirals were varied. Nominal vertical reinforcement was

provided to hold the lateral steel. The percentage of lateral

steel for the specimens was calculated on the basis of total

volume of the lateral steel against the volume of the cubic

specimen. Different types of reinforcement are shown in the

diagram below, fig. 2.1B. It is noted that the cracking

strength in general improved with the provision of

reinforcement. Large spiral (B, BH) appeared to be more

effective against cracking than other forms. Spiral of small

diameter (S, SH) did not increase the resistance of the

specimens against initial cracking. Cracking loads of

specimens with larger bearing plates were influenced to a

lesser extent by the provision of reinforcement than with

smaller plates. In general, the higher the volumetric

percentage of lateral steel the greater was the increase in

bearing capacity by reinforcement for a particular ratio R.

The increase in the bearing strength was probably the result of

the effective spreading of the concentrated load over the

concrete. With spiral reinforcement the increase was due to

the increase in compressive strength of the confined core of

concrete induced by the lateral steel under load. Thus, the

bearing strength of spirally reinforced concrete compared to

that of plain concrete of similar quality may be expressed as

n(reinft)/n(plain) =1+ K-p (2.34)

where p= volumetric % of lateral steel,K may be taken as 55 for all variation of R.

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32

Q0 0 QoQ-aI

29t

8-Type S-Type BH-Type SH-Type

0oDDBS-Type BSH-Type

GS-Type DG-Type

F ,g. 2.18 Forms of ret, of orcement tested by N .yogi. .

^--1

1------ -

P

Mo

(V I

Y,

Ya

V,

4&-bs'. szone

t-Referenceone

P

Pz

V

1

( b)

X

(c)

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33 CHAPTER 2

Lenschow C52] proposed a failure mechanism for the

concrete block subjected to concentrated load. The

distribution of transverse stress in the anchorage zone of a

beam subjected to a concentrated load acting parallel to the

longitudinal axis is pictured in fig. 2.19(a). The deflections

of fictitious springs inserted across the longitudinal cuts in

the beam were related to the transverse stresses. The

transverse tensile stress across the. axis of the load was

referred to as the 'bursting stress' while the transverse

tensile stress across any other longitudinal plane was called

the 'spalling stress'. The physical analogue for the anchorage

zone of a beam is shown in fig. 2.19(b) to (d). The prismatic

beam shown in fig. 2.19(b) is subjected to a concentrated load P

and could be represented by the beams in figs. 2.19(c) and (d).

Fictitious springs inserted represented the concrete and

resisted the deflection of the outer parts of the beam.

According to the physical analogue, the maximum spalling stress

for a rectangular section is

f= -2 "d-Mo/b

"hb2 (2.35)

and the maximum bursting stress is

fbc = Mo/b-hb2 (2.36)

The force of a single concentration of transverse reinforcement

at the surface of the spalling zone with tensile strength of

the concrete neglected was expressed as

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34 CHAPTER 2

r11

Fo -M 1ý (2.37)

L 9Ec - Ib y/Ab "G - 3W/Mo j

where Ec = Young Modulus of concrete,

Ib = Second moment of inertia of the section below the

reference plane,Ab = Sectional area below the reference plane,

hb = depth of the section below the reference plane.

The effect of the concrete tensile strength can be

recognized by modifying F0 as

F1 = Fo[1-

Ec. Ib/K- (b "ft/Mo)2]

(2.38)

The effect of transverse reinforcement on the bursting crack

varies drastically with the position of the reinforcement. It

is advisable to use light stirrups at close spacing. The force

in the reinforcement in terms of force per unit length f0, can

be expressed as

fo = Mo (1-ft/fb) /hb2 (2.39)

Jensen Cab] has considered the problem of an upper bound

plastic solution using a failure mechanism. This type of

failure is frequently observed in lightly reinforced blocks and

known as splitting failure. He made a number of tests on

200 x 200 x 400 mm blocks with reinforcement perpendicular to

the direction of the load as in fig. 2.20. A sliding failure

occurred alongthe

sides ofthe

wedge and a separationfailure

along the vertical line. By considering the external work done

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35

-I

V2 V2

0

(a) (b)

Ft,g. 2.20 SLi at ng and separatLon faLLuremec an sm y nsen.

q_j F

r- 7A, f

Fýg. 2.2i reent orcedf cconcreteel, oca.

for

-1 ýI-A, fu

AA

cß/2

8/8/2019 uk_bl_ethos_373477


36 CHAPTER 2

by the load and

reinforcement, the

could be expressed

and the angle of in

fb

f'C

where

the internal work by the concrete and

bearing capacity of the reinforced block

in terms of the degree of reinforcement I

ternal friction w as

4.1-sin(ß+') -sing + (1-sinv')

(2.40)

2cos(p+') -sin©

11 + 4"j1-cosy/(1-sinn) - sin', ']0.5

tang = (2.41)4"ý/(1-siný") + cos'r

For high I the above equation can be estimated by a straight

line

fb/f' = 2.6.1 + 1.2 (2.42)

Nielsen 164] considered the rotational equilibrium of a

quarter block acted on by vertical load and uniformly

distributed reaction at mid-height of the original block, and

maintained in equilibrium by horizontal compression near the

load and tensile forces in the transverse steel as in fig. 2.21.

It can be calculated that the ultimate load can be expressed as

Pu = t-bi-hi/(a-ai)2 where t= 2As-fy/bi-hi (2.43)

He concluded that with the light reinforcement provided, the

carrying capacity depended on the compressive strength and not

the tensile strength of the concrete.

Al-Nijjam [63) proposed a model based on Nielsen's model

with some modifications. Fig. 2.22(b) shows the state of

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37 CHAPTER 2

internal forces assumed as an equilibrium system with the

vertical load at the top and a triangular stress di stribution

on AA at mid-height of the original block, instead of the

uniform distribution stress proposed by Nielsen. When the

block was heavily reinforced, an upper limit of the bearing

capacity could be expressed as

fb/fý = (a/a1)1/3

(2.44)

When the reinforcement was lighter, the equilibrium conditions

of fig. 2.22(b) could be maintained with a2<a and for these

cases the bearing capacity could be related to the reduced

dimensions

fb/f' = (a/a2)1/3

(2.45)

Referring to fig. 2.22.

cote = e/z = (a2/b-a1/4)/z (2.46)

and also cote = 2A -f /P (2.47)5yu

Equating these, a` = 12z-As-f /P + 3a1/2 (2.48)y u

By substituting in Eq. 2.45

/(a12-bi "f b) + 3/2]1/3b/fý =[12z-As-f

(2.49)y

Therefore the bearing capacity can be expressed in form of a

4th degree polynomial.

(fb/f ')4-

3fb/2f' =1 (12z/al )c

(2.50)

with the limitation fb/f' < Ca/a111/3 (2.51)

It is noted that the influence of reinforcement at a distance

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38 CHAPTER 2

greater than a/2 from a load is very small and may be neglected

from the calculation.

Tests by Kriz 1493 were made of 185 reinforced columns

subjected to loads distributed across their width through steel

bearing plates. The specimens were divided into seven groups

with different forms of reinforcement.

Group I pecimens were reinforced with 4 no. 5 bars ofintermediate grade. The vertical reinforcement

was tied with no. 2 ties spaced 8 ins. centre to

centre. The lateral reinforcement at the top ofthe column consisted of a welded grill with two

cross bar and 2,3,4 or 5 lateral bars as shownin fig. 2.23(a).

Group II -Vertical colum reinforcement consisted of 4

no. 11 bars with fy = 90,000psi.

Group III -Ties were omitted.

Group IV -Both vertical column reinforcement and ties wereomitted.

Group V Two to three layer of lateral reinforcement were

provided with spacing of 2 ins., fig. 2.23(b).

Group VI -Ties and bars are bent as in fig. 2.23(c).

Group VII -Specimens were reinforced laterally by 5 no. 4deformed bars welded to two bearing plate as infig. 2.23(d)

.

Specimens with bearing plates at the edge of the column

failed along an inclined plane similar to those observed in

plain concrete specimens. Group II to VI failed by crushing of

concrete underthe bearing

plates.The

modifications madein

the reinforcement in group II and VI had only a small effect on

the behaviour of the specimen. Omitting of ties or vertical

8/8/2019 uk_bl_ethos_373477


SomeStag Bawrl

8

!. 5'

Saga Sýza B

w"

L

Group I. Wetded Group V.Welded Lateral. Group VI. BentLateral. Rei.nt Rei.nt. *.n Layers Laterd Ret.nt.

(a) (b) (c)

[ «1iI

'

C1 1=7

3/2'

I

39

Ti

Ti... 8'cc

i/4' x0' x3'L

2 PLomng

Anchorage

5Bars / 5Bars/

i/4'x2 HR

iý

I

42'--!

I

5/8 Butt

Wetd

li"

E----12'--I

8/8/2019 uk_bl_ethos_373477


40 CHAPTER 2

bars resulted in increased propagation of cracks, while

additional layers of lateral reinforcement contained the

cracking in the top of the columns. The lateral reinforcement

apparently had no effect on the bearing strength when the edge

distance was less than 40 mm. Empirical formulae were derived

from the tests to be

fb = 5.73f'0.5_(Wa/a1)1/3.11+0.198C1(As1/b)H/V](1/16) (2.52)c

where

F0..... Wa < 40mm.

c1=1l2.5

.....Wa > 40mm.

H= Horizontal force,V= Vertical force,

Ast = Cross-section area of lateral steel.

This agreed with the experimental results with a slight

under-estimation.

2.4 SUMMARY

(1) Magnel 155,567 found that the stress distribution of an

anchorage block in the direction of the anchorage force

was as shown in fig. 2.2b.

(2) Bartsch [81] assumed a cosine function of load

distribution on the loading plate and found that the

maximumtransverse tensile

stress occurred at a distance

of 0.2 to 0.3a from the loading surface. Tensile stress

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41 CHAPTER 2

gradually diminished further away from the loading plate.

(3) Guyon [26,273 made use of Fourier series to obtain a

stress distribution of the anchorage zone as shown in

fig. 2.5a. He also constructed six tables for the

calculation of stress under different loading conditions.

(4) Bleich (82] used Airy stress functions to find the

distribution of anchorage stress in fig. 2.7. By

introducing an apparent width (Eq. 2.3), Sievers modified

Bleich's two-dimensional solution for three-dimensional

used.

(5) Based on the shear strength of concrete, Eq. 2.4 and the

assumption of uniform distributed horizontal splitting

pressure along the wedge, tleyerhof [59] and Tung Au [6,7]

worked out their failure model of reinforced concrete

bearing blocks (Egs. 2.7 and 2.10 respectively). Meyerhof

stated that experimental bearing strength was greater than

theoretical estimates, especially in blocks with small H/a

ratios, due to the presence of base friction which had

been neglected in the analysis. From his calculations,

bearing capacity is directly proportional to height/width

ratio, however tests carried out by Muguruma and Niyogi

indicated the opposite result; this is probably due to the

non-uniform distribution of tensile stress Ph.

(6) Tung Au's [6,77 formula only gives approximate bearing

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42 CHAPTER 2

strength of concrete blocks, because of the variation of

a.

(7) A Dual failure mode of tensile separation and shear

sliding was adopted by Hawkins E30-323. Bearing strengths

of concrete block can be estimated by Eq. 2.16.

W) Empirical solutions were used by Shelson E76], Kriz (49],

Muguruma 162] and Niyogi (65,66]. Muguruma's formula is

the only one which takes account of the height of the

specimens.

(9) Plastic analysis can be used to find the bearing strength

of concrete by equating the internal energy and external

work, Egs. 2.32 and 2.33.

(10) Shizuo Ban 19] and Niyogi (65-67] stated that spiral

reinforcement is the most effective way to increase the

bearing capacity of concrete blocks.

(11) Myguruma (62] suggested that spiral reinforcement with

smaller diameter of steel and with comparable area inside

the reinforcement to the loading area is an effective way

to improve bearing capacity.

(12) Lenschow 152] proposed a failure mechanism for concrete

blocks subjected to concentrated load and arrived at a

solution for the maximum spalling and bursting stresses to

be calculated by Eqs. 2.35 and 2.36 respectively. Forces

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43 CHAPTER 2

on the transverse reinforcement are given by Eqs. 2.37 and

2.38.

(13) Nielson E64] and Al-Nijjam E63] proposed a model based on

the equilibrium of internal stresses in the bearing

blocks.

(14) Kriz's 149] empirical solution (Eq. 2.52) is based on the

tests of a large number of specimens with different forms

and amount of reinforcement.

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44 CHAPTER 3

3 EXPERIMENTAL INVESTIGATION OF THE BEARING CAPACITY OF

CONCRETE BLOCKS

3.1 INTRODUCTION

In this investigation, an attempt has been made to study

experimentally the factors affecting the bearing capacity of

concrete blocks. Experiments comprised two phases; plain and

reinforced concrete blocks.

3.2 PLAIN CONCRETE BLOCKS

The bearing capacity of plain concrete blocks is mainly

dependent on:

(1) The distance of the load to the nearest edge of the block(edge distance, Wa),

(2) The ratio of the footing area to the loaded area, R,(3) The height to width ratio of the blocks, H/a,(4) Size of the blocks,(5) Effect of base friction, and

(6) Strength of the concrete.

The effect of the strength of the concrete on the bearing

capacity of the blocks was not specially investigated in these

experiments. Twenty six concrete blocks were subjected to

concentrated load applied over their full breadth by a steel

bearing plate. The block tests were divided into four groups.

The first (series E) was designed to investigate the effect of

edge distance. In the second group (series R-H), specimens

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45 CHAPTER 3

were used to explore the relationships between the ratio R and

H/a and the bearing capacity of the blocks. The effects of

size and base friction were studied by the third (series S) and

the fourth group (series B) respectively.

3.2.1 SERIES E

This series consisted of three blocks with constant

dimensions, 100 mm. thickness, 1000 mm. depth, and 1260 mm.

overall length. They were placed vertically and loaded with a

steel bearing plate of 50.8 mm. width, 100 mm. long, and

50 mm. thick. Each block was loaded twice, once on each edge

of the block. Edge distance, Wa varied from 30 to 280 mm.

3.2.2 SERIES R-H

Sixteen blocks with constant width 400 mm. and thickness

100 mm. were cast. Their heights were varied with 200,400,

800, and 1000 mm., which corresponded to H/a ratios of 0.5,

1.0,2.0, and 2.5. They were loaded concentrically with 4

different sizes of steel bearing plate across their full

breadth. The widths of the bearing plates were 6.35,25.3,

50.8 and 101.6 mm. which give values of R as 62.99,15.81,

7.87 and 3.94 respectively.

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46

3.2.3 SERIES S

CHAPTER 3

Concrete blocks of 3 different sizes were included in this

series of tests. They varied from 100 mm. to 200 mm. square

with corresponding thicknesses from 12.5 to 50 mm. They were

loaded concentrically with a bearing plate size from 12.7 to

25.4 mm. so as to give a constant ratio R equal to 7.87. For

each size of

the specimen, three blocks were tested and the

average of their ultimate load was taken.

3.2.4 SERIES B

The effect of the base upon the ultimate bearing strength

is believed to depend on (1) the footing to loading area ratio,

(2) the height of the blocks. These effects were demonstrated

by the testing of 4 blocks. They had constant width, 400 mm.

and thickness 100 mm. but with two different heights of

1000 mm. and 200 mm. They were loaded with two sizes of

bearingplate

101.6 mm. and 6.35 mm. width. The friction at

the base was reduced by using a sheet of 2.4 mm. thick PTFE

placed at the bottom of the specimen when it was tested.

3.3 REINFORCED CONCRETE BLOCKS-SERIES R

A series of 8 blocks with dimensions 1260 x 1000 x 100 mm.

were cast. They were reinforced at two corners with different

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47 CHAPTER 3

forms of reinforcement. Each corner of the block was loaded

separately, with a steel plate 101.6 x 100 x 50 mm., one after

the other. Blocks were denoted as RI to R8 as shown in

fig. 3.1. In order to distinguish between each end, R1/1 and

R1/2 were used to represent block R1 with END 1 and END 2

repestively. A similar arrangement was used for the other 7

blocks.

RI/1, R1/2 and R2/1 were reinforced in such a way as to

investigate the effect of the diameter of the stirrups on the

bearing capacity of the concrete. R2/2 showed how the block

behaved if the reinforcement was placed closer to the surface.

R3/1 and R3/2 had almost the same cross-sectional area of steel

as in R2/1. They were reinforced with closely spaced thinner

steel and were used to study how the spacing of the

reinforcement affected the bearing capacity of the blocks.

Forms of the stirrups were studied in R4/1 and R4/2. The

effect of the spread of reinforcement was investigated by R5/1

and R5/2.

The eccentricity of loading did

strength of concrete. For plain concri

already been done with blocks El to

concrete, it was investigated by blocks

unreinforced in order to gain an idea of

the reinforcement.

affect the bearing

ate, experiments had

E3. For reinforced

R6 to R8. RO/1 was

the effectiveness of

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50 CHAPTER 3

3.4 MATERIALS & THEIR PROPERTIES

3.4.1 MATERIALS

Cement Ordinary Portland cement conforming to theBritish Specification was used throughout.

Coarse Aggregate North Notts quartzite gravel with a

maximum size of 10 mm., 'irregular' shape

and 'smooth' surface texture as classified

by British Standard, BS 812.

Fine Aggregate Air-dried sand from the same quarry as the

coarse aggregate was used. It wasclassified zone 3 according to BS 882.

The grading curve for the fine and coarse aggregates are

shown in Fig. 3.2.

Reinforcement Although deformed bars were commonly used

in practice, plain round mild steel barswere chosen as their strain can be

measured easily and more accurately. Ifdeformed bar is used instead of plainbars, a safer structure will b- resulted.A typical stress-strain curve and strengthproperties are shown in fig. 3.3.

3.4.2 MIX DETAIL

The first specimen, El was cast using mix proportions by

weight i: 2.68 : 3.85 with water/cement ratio of 0.65. This

gave 20 mm. slump, a V-B time of 4 secs. and a compacting

factor value of 0.885. The rest of the specimens were cast

using mix proportions by weight 1: 1.96 : 2.83 with a

water/cement ratio of 0.54. Average values of 125 mm. slump,

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51

100

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52 CHAPTER 3

V-B time less than 1 second and compacting factor of 0.96 were

obtained.

3.5 CASTING AND CURING

In order to avoid the variation of strength of concrete

along the height of the specimen (Besser 1983, (24)), all the

test specimens were cast horizontally. An oiled steel mould

1260 x 1000 x 100 mm. was used throughout. Smaller specimens

were obtained by partitioning the mould with wooden blocks

which were held firmly by clamps. All the specimens were

compacted on a vibrating table.

Control specimens were cast with each mix and also

compacted on a vibrating table. They were stripped from the

moulds and placed in the curing room at 20 degs. C., relative

humidity of 95-100 percent, 24 hours after casting. The test

specimens were covered with damp hessian for 3 days, watered

constantly and then transferred to the curing room.

3.6 CONTROL SPECIMENS

Control specimens consisted of five 100 mm cubes, five

150 mm. cubes, six 300 x 150 mm. cylinders and three

100 x 100 x 500 mm. prisms. Compressive strength of concrete

was providedby 100

mm. cubes,150

mm. cubes and three

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53 CHAPTER 3

cylinders while the tensile strength of concrete was given by

splitting tests of thethree

cylinders and rupturetests

of two

prisms. One of the prisms was also used to obtain the Young's

Modulus and Poisson's ratio of the concrete. The concrete

properties of each specimens are listed in Table. 3.1

In order to have a better indication of the strength of

concrete in the test specimens from tests on the control

specimens, the same procedure was applied for casting and

curing on both. The control tests were made at the time when

the relevant blocks were tested.

Control specimens were tested in accordance with BS 1881.

3.7 INSTRUMENTS AND TEST PROCEDURE

Strains on the surface of the concrete were measured by

Demec gauges. For specimens in series E, in order to obtain

the local strain around the compression zone of the test

blocks, Demec gauges with 50 mm. gauge length were used and

they were more concentrated under the bearing plate. At each

position, 6 Demec points were fixed to create a rectangular

rosette of 45 degrees.

For test specimens in series R-H, S and B. only the

transverse strain along the line of loading and the compressive

stains at mid-height of the specimen were measured. In

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56 CHAPTER 3

general, a 50 mm. gauge length was used for the transverse

strain while a 100 mm. gauge length was used for vertical

strain.

For reinforced blocks, strain of the steel was measured by

electrical resistance gauges and recorded by a data-logger,

while strain on the concrete surface was measured by Demec

gauges. Load was increased in 5OkN increments and at each

stage ofloading, cracks were observed by means of a hand

magnifying glass and marked with ink.

When the specimen was ready for test, it was taken out

from the curing room and a thin coat of white emulsion was

painted on the surface after it had dried. Demec points were

fixed into position. A layer of plaster of paris was

introduced in the bottom of the specimen and between the

bearing plate and concrete. These allowed a good contact area

between steel and concrete. The specimen was then checked for

position vertically and loads were applied in steps of 50 or

100 KN. Strain measurements were made at each increment of

load. The mechanism of loading is shown in Fig. 3.4.

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57

I-Bean

Hydrauti

Roll,

Steel Bear

Bedded wt. hPLaster of Ports

Bedded ui, ttiPlaster of Paria

I- Bernur

CHAPTER 3

Ft..g. 3.4 Lod&ng Mechant.sm for Bearing Capact. y Expt.

3.8 BEHAVIOR OF TEST

3.8.1 GENERAL

The behaviour of a majority of the unreinfarced specimens

was characterized by the suddenness and explosive nature of

their failure which was often accompanied by an audible report.

A wedge was formed beneath the bearing plate with an apex-angle

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58

ranging from 30 to 40 degrees.

CHAPTER 3

Reinforcedspecimens

behaved ina more controlled manner;

they usually cracked along the line of load and then failed by

subsequent widening of the cracks.

3.8.2 SERIES E

The general patterns of cracks and modes of failure are

shown in Fig. 3.5. With the exception of block E3/2, the

specimens split vertically into two halves along the axis of

the bearing plate. For loads with small edge distance Wa, the

block was lifted up beyond the point which is about 3 times the

edge distance (Fig. 3.5. (a)).

In block E3/2, failure was by shearini

with vertical cracks penetrating almost to

specimen. As load was increased to lOOkN, a

at the top of the specimen, 100 mm. from

This propagated at an angle 90 degs. to the

loadwas

increased (Fig. 3.5. (c)).

3 off the corner,

the bottom of the

crack was observed

the loading edge.

horizontal as the

These failure wedges formed below the loaded area were

pyramid-shaped with an apex angle ranging from 30 to 40 degs.

All the specimens failed audibly immediately after the

formation of vertical cracks below the loaded surface. This

originated at about SOmm from the top.

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59

: L'

1

1

il,

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60

3.8.3 SERIES R-H

CHAPTER 3

The patterns of cracks and modes of failure are shown in

figs. 3.6. to 3.9. Vertical splitting of the blocks into 2

halves along the axis of the bearing plate was the usual mode

of failure for the majority of specimens. However, for a small

relative size of bearing plates, i. e. large and deep blocks,

R4-H1, R3-H1 and R4-H2, splitting occurred from the top of the

specimen andterminated

ontheir

.two

sides, resulting in the

splitting of the block into 3 parts as shown in figs. 3.6(c),

3.6(d) and 3.7(d). On the other hand, for shallow blocks,

h/a<O. 5 and larger bearing plates, R<63, splitting usually

occurred from the bottom. This is due to settlement of the

supporting beam at high loads creating bending of this slender

block which initiated cracks at the bottom.

Except for loading over comparatively shallow specimens,

cracking and failure of the specimens took place

simultaneously. Cracks originated further away from the loaded

surface with larger bearing plates. Failure of these specimens

is sudden and associated with a loud noise.

3.8.4 SERIES S

Crushing of the concrete beneath the bearing plate and

subsequent splitting of the block into two halves was the mode

of failure of Series S specimens. Although failure took place

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61

(a

(c)

(b )

(d)

F .g. 3.6 Appearance after f aLLure of blocksLn sere,es R-H wi. h hei,gh - 1000mm.(a RI-Hi, (b R2-Hi,

c R3-HI, (d R4-HI.

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62

RI H2

(a)

ccI

(b )

(d)

Fi.,g. 3.7 Appearance after fai. Lure of blocksin serees R-H with hei,gh - 800mm.

a RI-H2. (b R2-H2,

c R3-H2, (d R4-H2.

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63

14w4-ý6 äYä&\Il 1

itrC «"r"1fi

v7"

R2-IF3

(a)

r

dg ýý"ý6ä"ä ä Pf

7"

(b )

r4

7)

cc) (d)

F ,g. 3.8 Appearance after f aLLure of bLocks

Ln serf es R-H wLh het,gh - 400mm.

a RI-H3, (b R2-H3,

c R3-H3, (d R4-H3.

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64

(a)

cc)

(b )

(d)

F , g. 3.9 Appearance after f aLLure of bLocks

Ln sere es R-H wLh hei,gh = 200mm.a RI-H4, (b R2-H4,

c R3-H4, (d R4-H4.

8/8/2019 uk_bl_ethos_373477


65

w Lx

I all *º)+1 J' i-.

ý7 s

Z.

a' lb

U

. (v)Cn ý

W -U

-)0

Lm mN

C V

N v

U0

-CuW

YO

mL MJ

.o r3

v

L(D

O C!

U VC 0

v ->MO

a

aa

-

0

(T)

C6

U-. -ýv

_.

8/8/2019 uk_bl_ethos_373477


66 CHAPTER 3

almost immediately after the formation of vertical cracks, a

more controlled failure was observed. They failed without a

loud noise.

The patterns of cracks and failure mode are shown in Fig

3.10.

3.8.5 SERIES B

Fig. 3.11 indicates the type of failure of Series B

specimens tested with a sheet of PTFE under them.

For deep blocks B3 and H4 the mode of failure was similar

to those in R1-Hi and R4-Hi respectively (Fig. 3.11(c) and (d)).

Theseshow

that thepresence of

thesheet of

PTFE had littleor

no effect on the behaviour of these specimens.

However, with shallow blocks, H1 and B2, a different mode

of failure was observed. It was obvious with specimen B1, that

the block was split into two halves without crushing of the

concrete under the bearing plate (Fig. 3.11(a)). For specimen

82, splitting occurred from the bottom instead of from the top

as in specimen R4-H4.

Fig. 3.12 shows the mode of splitting for specimens Bi and

B2. The PTFE at the bottom of the specimen was acting as a tie

while the two halves of the concrete block behaved as two

compressive struts. Failure of these blocks were taking

place

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67

(a )

C

(b )

ýý...., ý

114

r

T

(d)

F ., . 3.11 Appearance after f ai.. ure of bLocks Ln sere,es B.(a) Bi. (b) B2, (c) B3, (d) B4.

8/8/2019 uk_bl_ethos_373477


68 CHAPTER 3

as the tie ceased functioning, either by breaking the PTFE into

two pieces (block Bi) or by the sliding of the struts on the

surface of it (block B2). The mechanism of the failure is

shown in fig. 3.12(c). All the specimens in this series failed

with a loud noise.

Compresswe

400

400 X ),. 200

SpLi. tLng J(a) BLock Bi

Crack

conPreasi. ve conPreaatve

strut strut

\-TLeaction by the PTFE

(cl

SpUttmg

Crack

Ft.g. 3.12 (a) & (b) Crack pattern before fai, Lure.(c) FatAure mechantsm of block Bi & B2.

(b) Block 82

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69 CHAPTER 3

3.8.6 REINFORCED CONCRETE BLOCK - SERIES R

All of the reinforced blocks (Series R) failed in local

bearing at the concentrated load. The presence of the

reinforcement gave the block ductility as it failed.

Fig. 3.13 shows the crack patterns for all the specimens in

this series. Despite the differences in the form and amount of

reinforcement used, their crack patterns were similar. The

first crack which appeared was in the centre of the bearing

plate and originated at about 100 mm. from the loading edge.

This crack propagated downward as the load was increased.

Occasionally, spalling cracks appeared around 150 mm. from the

inner edge of the bearing plate and extended downward at an

angle of approximately 70 degrees to the horizontal.

At higher loads, radial cracks appeared, which originated

from the edge of the bearing plate and radiated downward as the

load increased. Finally, failure was predominantly by local

compression with flakes of concrete spalling off below the

loading plates, as can be seen in the photographs in fig. 3.14.

8/8/2019 uk_bl_ethos_373477


\MP

Iüýý

56 bo

2Block Ri

EN

70

-4648

so4

's

so4so5w

49p5'55

e

T

72 to

TT

6{ 66

77

28Lock R3

EN

aj62 62

61+

60

62 jj

ý1 fblý

11«602

61

i2

63 ý

2 ENStock R5

72 n7s

?S

n

r71 r-ibo to

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2B6ock R2

EN

2 ENBlock R4

so49 45

ý

53ao

ý44 ae

48

4 soso

f45

Jsoß

81

62

2 EN

Block A6

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71

rEöE

cj.t

lY jö3

0C. c

C.

oc

v00

J

J

7

a

'OOO

.C

3

t

,. a-a

3

.ýtU

vL0

O

cO

a.O

Lv

.0m

c*i0)

U-

-NN UI POO

Cl 0000000003 n to U

8/8/2019 uk_bl_ethos_373477


72

I1i

Uh/_P./I

1ý

(a )

cc)

ri

21

rr-

(b )

(d )

FLg. 3.14 Crack pattern after faLLure for block Ln series R.(a) RI, (b) R2, (c) R3, (d) R4.

8/8/2019 uk_bl_ethos_373477


73

R. /Z

(a )

R5/

cc)

1?i/>

I ýi' ý

iI.J

ý(.

11

?i/I

(b)

Fi,g. 3.15 Crack pattern after fat .Lure for block t,n series R.(a) R5, (b) R6, (c) R7.

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74 CHAPTER 4

4 EXPERIMENTAL RESULTS -- BEARING CAPACITY OF CONCRETE BLOCKS

4.1 INTRODUCTION

In the previous chapter, the general behaviour of the

specimens during the tests has been described. In this

chapter, more experimental results will be shown and discussed

in detail, such as the stress and strain distributions in

concrete and steel, and the factors affecting specimen

behaviour will be analysed. Finally, a proposed model of

failure is drawn up and compared with existing theories.

The values of cracking load Pc, and ultimate load Pu are

tabulated together with the dimensions and material properties

of the specimens in table 3.1.

4.2 BEARING CAPACITY OF PLAIN CONCRETE BLOCKS

4.2.1 EFFECT OF EDGE DISTANCE

The deflected shapes of the specimens analysed by the

finite element method (FEM) with edge distances of 30 and

280 mm. are given in fig. 4.1(a) and (b) respectively. It can

be seen that the specimen with concentrated load near the edge

failed by shearing off the corner as a result of less

8/8/2019 uk_bl_ethos_373477


75

J"tfr :' -ýýýI J

fftJItfJtJ ''y-. 1_..

1/+J55I+J

ft`

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

+ t t r t s

1 i i

t t r tI + t

l t = t7 j _{_--

: is

, +

i

Ft.g. 4. la Deflected shape ofblocks

with edgedistance 30mm.

i_.

I_.

,

I-

,

t.

s

~S 11

1 ý ý1 / 1

f ýI 1

1 ýýýý. -_

y1

1

1

1

1 11

1I

1:

. 1 t1

_I 1 1

'.

i.

i 11

/I

" ýrý 1 j

7 t ...i" 1 /

f__

11

11 1 t l

t 1 1 1 1

-1 / t

ý . -t ýýý"ý 11 l

1 1 1

_. 3_ ... t. - ;ý ! j

7 1 1 ý1 1 t

1 ' 1 1 '.. .

. -

1 1 11 [

71

j 1 1

1

8/8/2019 uk_bl_ethos_373477


76 CHAPTER 4

confinement on one side by the concrete near the edge.

However,specimens with

a larger edge distance failed by

splitting the concrete block into two halves.

The variation of transverse stress with depth along the

loaded line in blocks El to E3 is given in fig. A. 1-3. The

experimental stresses are plotted together with the stresses

analysed by FEM. It can be seen that the general trends of the

stress distributions are similar in all three blocks, each

contains a high compression zone near to the loaded surface,

followed by a tension zone which causes the splitting of the

blocks. The depth of the tension zone varies with edge

distance, loads further away from the edge creating a larger

and deeper tensile zone. The maximum transverse tensile stress

occurs at around 30 mm below the loaded surface in blocks with

a small edge distance of 30 mm. but at 130 mm in a block with

a large edge distance of 280 mm.

From fig. A. 1-3 indicate that stresses obtained from the

tests tend to fluctuate. This is probably due to the use of

the Demec gauge with 50 mm gauge length, which is not sensitive

enough to detect small strains. However, the experimental

stress distributions still follow the stresses given by the FEM

with three discrepancies. High compressive stresses were not

recorded in blocks E2 and E3 due to the difficulties in putting

Demec points close to the loaded surface. At high loads,

tensile stresses obtained experimentally are higher than those

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77 CHAPTER 4

estimated by FEM. This can be explained by the presence of

micro-cracks which are not considered in the FEM analysis.

Stresses at the bottom of the blocks were recorded as tension

during the experiment while compression is suggested by FEM

analysis. The tensile stresses in the experiment are generated

by the deflection of the supporting beam which is assumed to be

fixed in the FEM analysis.

Fig. 4.2 shows the variation of the bearing strength of the

concrete block with edge distance; the values are plotted as

two dimensionless ratios: fb/fý against Wa/O. 5ai. It can be

seen that the graph is composed of two straight lines, they

meet each other at Wa/O. 5a1=3.5 which corresponds to the edge

distance of 90 mm. It is suggested that with edge distances

smaller than 90 mm, decrease in edge distance will result in a

dramatic loss of confinement by the surrounding concrete which

leads to a large decrease in the bearing strength of the

concrete block. On the other hand, with edge distances greater

than 90 mm, the increase in edge distance will steadily

increase the confinement by the surrounding concrete and a

higher bearing strength is obtained. The bearing capacity of

the concrete block under concentrated strip load can be

estimated by

fb 0.12Wa/al + 1.16 Wa/0.5a. > 3.5

=i (4.1)fl I 0.47W /al + 0.55 Wa/0.5a1 < 3.5

8/8/2019 uk_bl_ethos_373477


78

2.0

1.9

1.8

ý-° 1.7

a 1.6

Lr 1.5

1.4a

1.3

!. 2

i. 1

1.0

0.9

0.8

0.7-t0

FIG. 4.2 Effect of edge dLstance Ln blocks of series E.

H (mm. A-3.94 R-7.87 R- i5.7 R- 63.0

H- 1000 0.96 0.96 0.95 0.96

H- 800 0.95 0.95 0.95 0.95

H- 400 0.97 0.98 0.98 0.98

H- 200 1.57 1.62 1.64 1.64

Table 4.1 Ratt, o of compressLve stress at Loaded Leneto average stress at the bottom of theblocks.

(30) (80) (130) (180) (230) (280)

123456789 10 it i2

Edge di.etance to halt' beart.ng w.dth ratio, W,0.5a,

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79 CHAPTER 4

4.2.2 EFFECT OF HEIGHT AND LOADING AREA RATIO

Fig. A. 4-19 show both the transverse strain distributionsI

along the loaded line and the vertical strains across the

mid-height of the specimen. They are platted together with the

values obtained by FEM analysis.

In general, the experimental data agrees well with the

theoretical values and with less fluctuation because a Demec

gauge of larger gauge length, 100 mm is used thus increasing

the sensitivity. Again, in observing the figures, it can be

seen that all the specimens have a compression zone at the top

immediately below the loading plate. Their sizes vary with the

size of the bearing plate. Below this compression zone is a

region of tension, usually described as the bursting zone.

Again, the size depends on the size of the bearing plate;

maximum tensile strain occurs at 100,50,25, and 10 mm below

the loading surface when loaded respectively with 101.6,50.8,

25.3 and 6.35 mm width bearing plates. This bursting zone

extended to a depth of 350 mm below the loaded surface for

specimens with large bearing plates and 300 mm for those with

smaller bearing plates. Nevertheless, below this bursting

zone, is a virtually unstressed region especially for high

specimens. For shallow specimens, tensile strains are recorded

at the bottom of the specimens. These are believed to come

from two sources. Firstly, they are actually the tail of the

bursting zone; this is particularly important for specimens

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8p CHAPTER 4

with large bearing plates, as the bursting zone for these

specimens extends to a greater depth. Secondly, they are

caused by the settlement of the supporting beam, this creating

a bending moment at the bottom and consequent tensile stresses

and strains. Settlement is increased with the magnitude of the

load; large bearing plates usually takes more load thus

producing a larger tensile zone at the bottom of the specimens,

4 ig. A. 16-18.

Compressive strain at mid-height agrees well with that

obtained by FEM analysis with a few exceptions according to

fig. A. 4-19. In some circumstances, fig. A. 12-15, the

experimental compressive strains tend to be smaller than those

estimated by FEM. This is probably due to the estimation of

Young's Modulus of the specimen.Blocks R1-H3, R2-H3, R3-H3

and R4-H3 exhibit this discrepancy as they were cast from the

same batch of concrete. Apart from this, R1-H4 recorded a

particularly high compressive strain around the middle of the

specimen, this is in fact due to the presence of a crack across

a pair of demec points. This should be ignored in reading this

figure.

As shown in fig. A. 4-7, compressive strain was almost

uniform at mid-height for high specimens with 1000 mm height.

As the height of the specimen decreases, compressive strain is

increasingly more concentrated below the load position. This

is obvious in specimens with 200 or 400 mm height

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81 CHAPTER 4

(fig. A. 12-19). Moreover, within the two specimens of equal

height, the one loaded with a smaller bearing plate had a

higher concentration of compressive strain than the others.

Table 4.1 tabulates the ratios of the vertical stress at loaded

line to the average compressive stress for all the sixteen

specimens.

The ratio of bearing strength to cylinder crushing

strength is plotted against the footing to loaded

area ratioin

fig. 4.3. It is noted that results for all the specimens with

height other than 200 mm come very close to each other.

Shorter specimens have higher bearing strength but this is not

very significant. Thus for specimens with 200 mm in height,

there is a 30% increase in strength in comparison with others

of similar loading condition but greater in height. This is

probably due to the disturbance of the tension zone by the base

of the specimen. The restraint at the bottom of the specimen

by the base contributes a compressive force which delays the

splitting of the specimen along the loaded line and thus a

higher bearing strength results. The bearing strength ratios

estimated by Shelson E23 and Kriz 118] are shown on the same

graph in fig. 4.3. It can be recognized that Kriz's estimates

are conservative for all values of loaded area ratio while

Shelson's estimates are conservative for a high loading area

ratio, R but become unsafe as the values of R falls below 13.

Fig. 4.4 shows how the ratio of bearing to cylinder

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82

9

® --- Hl a-0.5V -- Hl a-1.0®- H/ a-2.0

X --H/a-2.58

/

/f

f1 30%Lncr.

7 / 1

fö

/

/öa f x' 6 /

d

.3

/

30% incr.

coL 5 /

/I

SheLson : tp/PQ-3

J

q

/

/

/ 30% cr.

C/

m 3 //

s

2

Knz : fp/te - 5.73 Joy / fe

I

0 10 20 30 40 50 60 70

Footong to Loo&ng area rat*. q R.

FLg. 4.3 Bearing strength rati. o vs Loadt ng area ratio.

8/8/2019 uk_bl_ethos_373477


83

iExpt. Mugurumo9' ® --- R= 63.0

® --- R= 15.8

---R= 7.9

4

7i

U

.JV

\

CDcm 5 _. -L

W

c

U4

01\\

c

m

i \\

2 --_ -- -_

1

0 0.5 1.0 1.5 2.0 2.5

HeL. ht to width ratio. H; a of the blocks.

3.0

Ft.g. 4.4 Bearing strength ratLo vs height to width rat o.

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84 CHAPTER 4

strength varies with the height of the specimens. It can be

seen that the bearing to cylinder strength ratio is unchanged

with height for specimens higher than 600 mm. Below this

height the bearing strength begins to increase slowly and then

more rapidly at heights less than 400 mm. This is coincident

with the way that compressive force is concentrated at the

bottom of the specimen (table 4.1). The values of bearing to

cylinder strength ratio estimated by Muguruma (8I are also

shown in fig. 4.4. Muguruma also gives a similar trend of

increase in strength with short specimens as found in the

experiments. However, he appears to over-estimate the ratio

for the specimens with a bearing area ratio less than 16 and

under-estimating those greater than 63.

4.2.3 SIZE EFFECT

The distributions of transverse and vertical strain of

blocks Sf-S3 are very similar to those in series R-H

(fig. A. 20-22). They agree. well with those obtained by FEM

analysis.

The size effect was first introduced by Niyogi

C(1974), (19)]. He stated that the bearing strength falls as

the size of the specimen is increased. If the loading area

ratio, R remained constant, the bearing capacity would decrease

with the increase in size according to table 4.2. AI-Nijjam

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85 CHAPTER 4

((1981), (23)] stated that the bearing strength of concrete is

dependent on a factor of

( a'/a )1/4

(4.2)

where a' = width of the original specimen

according to his test results.

Fig. 4.5 shows how bearing strength varies with the size of

the specimens. It was found that as the size of the specimens

decreases, their strength increases in an exponential nature.

On the same figure (fig. 4.5) the way in which bearing

strength increases with the decrease in size as suggested by

Al-Nijjam, is plotted along with what obtained in this

investigation. Al-Nijjam's estimation has a more gentle

increase in bearing strength as the size decreases. It is in

fact more suitable for larger specimens. For this range of

scale phenomena there is adequate aggreement with the following

expression

fb/f' =k[1.45

e-a/(30 + 0.9] (4.3)

where a= the width of the specimen measured in mm.k= proportional constant.

4.2.4 EFFECT OF BASE FRICTION

In the presence of a sheet of PTFE at the bottom of the

specimen, the mode of failure for blocks 81 and B2 changed. As

has been described in chapter 3, splitting occurs from the

8/8/2019 uk_bl_ethos_373477


2.7

2.6

' 2.50

a 2.4

d

2.3

r2.2

2.1

2.0

c-)

01.9-

co

1.8

i. 7

1.6

1.5

1.4

0

X of size LncreaseX of decrease Lnbearing strength

3.0 25

2.0 !5

1.5 12

1.3 4

TobLe, 4.2 St.ze effect suggested by Ni.yogi (i9).

! 00 200 300 400

Block atze a ma.

F .9.4.5 fb/ a vs size of the blocks a.

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G7 CHAPTER 4

bottom of the specimen and failure of these specimens took

place without actually crushing the concrete below the bearing

plate. This is confirmed by fig. A. 23. The tensile strain

recorded in block Bi was very much larger than expected. In

fact, it was not purely tensile strain, because a crack

occurring at the early stage of the experiment, coming from the

bottom of the specimen is included. Moreover, the settlement

of the support becomes more dominant with the presence of a

layer of softer material (PTFE) at the base, thus adversely

affecting the cracking of the specimen from the bottom. Apart

from this discrepancy, the tensile strain distribution agrees

well with the theoretical value by FEM analysis (fig. A. 23-26).

It is interesting to note that the theoretical transverse

strain is not zero at the bottom of th specimen. Tensile

strains at the base indicate that tension force is needed in

order to restrain the base.

Table 4.3 shows how PTFE affects the bearing strength of

the concrete blocks. It can be seen that the bearing strength

decreases with a reduction of the base friction. This

reduction is (22%) for a short specimen loaded with a large

plate, (block Bi). This is understandable in that short

specimens depend on base friction to gain their bearing

strength, especially when loaded with large bearing plates,

because the tension zone is more likely to extend to the base.

A larger bearing plate means that a higher value of load- is

needed for failure; high loads can produce larger settlement of

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88 CHAPTER 4

the supporting beam and thus further reduction in the bearing

strength. Blocks B2 and H3 show similar reductions in strength

of 14 and 16% respectively, because block H2, although short,

is loaded with a small bearing plate and block B3 is high but

loaded with a larger bearing plate. The smallest reduction in

strength is in block b4, only 8%. This is bcause it has the

height and is also loaded with a small bearing plate so that

the tension zone can hardly reach the bottom, and there is not

much bending of the block with a small bearing plate.

4.3 REINFORCED CONCRETE BLOCKS

The crack patterns for all the reinforced concrete blocks

had beenshown

in the previous chapter, fig. 3.13a. Fig-3.13b

gives an idea of how the maximum crack width varies with load.

In this section, the behaviour of each test, the crack

patterns, crack widths and strain distributions in concrete and

steel will be discussed and compared with each others in

detail.

Blocks RI/1, RI/2 and R2/1 are reinforced with steel of

similar arrangement but with different diameters of transverse

steel of 6,10 and 8 mm respectively. These three specimens

failed with similar load of 600,590 and 620 kM respectively,

(table 3.1b). Although their crack loads are quite different

from each other, they are believed to come from two sources:

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89 CHAPTER 4

<1> differences in concrete properties and <2> the recognition

of the presence of cracks during the test. They have similar

crack patterns, the most vital crack is the one along the

loaded line. Blocks R1/1 and R1/2 have a more brittle

behaviour than R2/1 which is again considered to be dependent

on the concrete properties rather than the reinforcement. The

distributions of transverse concrete strain along the loaded

line for these specimens are shown in fig. A. 27-28. They have

similar distributions of strain, a large compressive strain at

the top immediately below the loading plate and then followed

by a tension zone extending to around 300 mm below the loading

surface. Below this tension zone is an unstressed region, but

occasionally, tension is recorded at the bottom indicated a

bending of the supporting beam. Strains in the reinforcement

are shown in fig. A. 35-37. Apart from the yielding of one

particular stirrup in Block R2/1, the remainder have not

yielded even after failure of the specimen. This suggested the

ineffectiveness of this form of reinforcement. In fact, these

specimens failed by buckling of the vertical steel between two

transverse reinforcing bars owing to the lack of restraint.

This can be improved by reducing the spacing between the

transverse steel.

Block R2/2 employs the idea of pushing the transverse

steel upward so that it has stronger reinforcement in the

higher tension region. However, the result is not encouraging:

it has a lower ultimate load at 590 kN than 620 kN in R2/1.

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90 CHAPTER 4

R2/2 has a larger crack width, up to 1.12 mm at 580 kN. It has

a central crack extending from the top to the bottom of the

specimen. At low load the distributions of transverse strain

are similar to Block R2/1. At high load, -(larger than 400 kN),

transverse strain is not confined to the top region but extends

to the bottom of the specimen, fig. A. 28. Most of the

reinforcement along the loaded line does yield, fig. A. 38, this

suggested that the effectiveness of the transverse

reinforcement depends on its spacing. The transverse

reinforcement in R2/2 has been placed too high, and although it

has an effective confinement at the top, an unreinfarced region

is left below, which is still within the tension zone.

Therefore, a large crack width and lower ultimate load has

resulted.

Blocks R3/1 and R3/2 have similar amounts of transverse

reinforcement in terms of cross-sectional area of steel as in

R1/2. They use smaller diameter steel but closer spacing.

R3/1 used stirrups of 6 mm diameter and 26 mm spacing, while

R3/2 used stirrups of 8 mm diameter and 52 mm spacing. Block

R3/1 hasa crack

loadand ultimate

loadof

650and

770 kNwhile

R3/2 has crack and ultimate load of 520 and 643 kN

respectively, which is much higher than the corresponding

values of 400 and 590 kN in block R1/2. The use of smaller

diameter steel and closer spacing increases the ductility as

the specimen fails, fig. 3.13b. Cracks in these two specimens

are spread more radiallyrather than concentrated

alongthe

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91 CHAPTER 4

loading line and have smaller crack width. This indicates that

the reinforcement is effective in distributing the stresses to

the whole reinforced area rather than concentrated in the line

of loading. As shown in fig. A. 29, distribution of strain on

the concrete surface is similar to block R1/2 but is lesser in

magnitude especially in the tension zone in block R3/1.

Fig. A. 39-40 shows that most of the stirrups in block R3/1 and

R3/2 along the loading line, especially those at the top,

yielded. Those at the bottom and next to the loading line have

been stressed quite significantly. Above all these phenomema

suggested that reinforcing with thinner and more closely spaced

steel is an effective way of increasing the bearing capacity of

the concrete block. Moreover, failure of block R3/1 is in fact

not by splitting of the block into two halves, but by sliding

of the bearing plate towards the rear of the block, because of

setting up errors. Strictly speaking, block R3/1 should

withstand a higher load than 770 kN.

Block R4/1 and R4/2 have different forms of

reinforcements, fig. 3.1. Block R4/2 has long stirrups which

enclosed all the vertical stirrups while R4/1 has smaller ones

further away from the loaded line enclosing two vertical

stirrups and has larger stirrups below the loading position

enclosing four vertical stirrups. They were designed to

compare their performance with block R2/1 which has only small

stirrups each of which enclosed two vertical stirrups. As

shown in fig. 3.13, their crack patterns are slightly different

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92 CHAPTER 4

from each other. Block R4/2 has the simplest crack pattern,

only a single crack along the loading line. As the stirrups

decrease in size, in blocks R4/1 and R2/1, more cracks are

formed and they radiate from the loading plate. Strain in the

steel, fig. A. 37,41-42 indicate that when the small loop

stirrups are used higher stress is generated in the steel, this

suggested that this form of stirrup is more effective in

resisting bearing stresses. The ultimate load of block R4/1,

680 kN is greater than that of block R4/2,600 kN, confirming

this idea. However, block R2/1 has a rather low ultimate load

620 kN, and this may be due to the difference in the properties

of the concrete. If the differences of ultimate load and crack

load Pu-Pc are considered, it can be found that block R2/1 has

the largest difference, Pu -Pc=174 kN, and this becomes smaller

in block R4/1,130 kN and smallest in block R4/2,60 kN.

Therefore, it can be concluded that small stirrups are more

effective in resisting bearing stress than large stirrups.

This can be explained as more lateral restraint can be provided

by smaller stirrups.

Blocks R5/1 and R5/2 were designed to compare the bearing

strength with the distribution of the reinforcement, fig. 3.13.

Block R5/2 has the simplest form of arrangement; two vertical

and two horizontal stirrups. Block R5/1 has one more bay of

reinforcement, one on each side of the loading line and one

more row of reinforcement at the bottom. They are used to

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93 CHAPTER 4

compare their performance with block R2/1 which has one further

bay of reinforcement on each side of the loading line and one

more row of reinforcement than block R5/1. From fig. A. 31 and

28 it can be seen that the distribution of strain on the

concrete surface is very similar in blocks R5/1, R5/2 and R2/1.

The strains in the reinforcement, figs. A. 37,44-45, are again

similar, except that yielding of one stirrup is found in block

R2/1. All the others had not reached yield point before the

concrete block failed. However, as their crack patterns are

considered, there is a difference in mode of failure found in

block R5/2, which had its load transferred to the bottom of the

reinforcing matrix and failed as if load is applied at the

bottom horizontal stirrup in a block of plain concrete.

Failure was due to the formation of a wedge of concrete below

the reinforcing matrix, followed by the separation of the block

along the loading line. It failed with the characteristic of

all the plain concrete blocks, a brittle mode of failure.

It is therefore recommended that the matrix of reinforcement

should have a width at least as wide as the loading plate. As

far as the ultimate load is concerned, there is only a slight

different in magnitude between blocks R5/1, R5/2 and R2/1. It

is therefore difficult to decide whether a wider spread of

reinforcement do increase the bearing capacity of the concrete

block at this stage.

Blocks R5/1, R6/1, R6/2, R7/1, R7/2 and RB/2 were

reinforced with the same amount and formof reinforcement

but

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94 CHAPTER 4

with different edge distances varying from 80 to 330 mm. They

were designed to investigate the effect of edge distance on the

reinforced concrete blocks. Figs. A. 32-34 show the distribution

of transverse strain on the concrete surface along the loaded

line. In general, the experimental and theoretical values

agree with each other at low load except near the bottom of the

block. Higher tensile strain recorded at the bottom of the

block during the tests is due to the bending of the supporting

beam which is assumed rigid during FEM analysis. At high load,

the experimantal strains are greater than those given by FEM

analysis due to the presence of cracks. This discrepancy

happens at lower loads as the edge distances decrease. Similar

to plain concrete blocks, the depth of the tension zone below

the loading plate increases as the edge distance increases.

More cracks are developed for blocks with small edge distances,

fig. 3.13. This is because of the earlier formation of cracks

with blocks with small edge distances. Fig. 4.6 is plotted to

show the relation between the difference between ultimate and

crack load "Pu-Pc) and the edge distance Wa of the blocks. It

can be seen that smaller edge distance exhibit a larger (PLL Pc)

value for blocks with edge distance smaller than 180 mm which

correspond to Wa/0.5a1=3.5. Blocks with edge distances larger

than this failed as soon as the block cracked. Therefore, it

can be said that reinforcement is more effective with a block

of small edge distance as it can prevent the block from

shearing at the corners to cause failure. Strains in the

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140

120

100

19

80Y

a60

a40

20

0

s

2.0

1.9

1.8

s-° 1.7

a 1.6

,c1.5

L 1.4

L

1.3

1.2

4)co

i. i

m

1.0

0.9

0.8

0.70

rb/r, - 0.22W,o, + 1.01

10,.

rp/r, -0.12W0/a,+1.ss

vrv

fp/to - 0.47 W0/ai + 0.55

®Xul. . Load

. Ptai. n

I® uLt. Load

0 crack Load%R6ýnft.

i23456789 10 11 12

Edge di.stonce to half bean. ng with ratLq Wa/O. a1

Ft,g. 4.6 Pu- P. vs edge distance W0.

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96 CHAPTER 4

reinforcement are shown in figs. A. 43,45-49. Fig. 4.7 is

plotted of crackingto

cylinder strength ratio against edge

distance to half bearing width ratio. It can be seen that for

Wa/O. Sa1<3.5, the cracking strength agrees well with Eq. 4.1,

and the concrete block cracks as

plain concrete block fails. Foi

strength is higher than that

conclusion, the cracking strength

estimated by

soon as its corresponding

r Wa/O. 5ai>3.5, the cracking

estimated by Eq. 4.1. In

of a concrete block can be

t 0.47Wa/a1 + 0.55 Wa/0.5a1<3.5

fb/fý 0.12Wa/a1 + 1.16 - plain 1 (4.4)

r Wa/4.5a1>). 5

L 0.22Wa/a1 + 1.41 - reft. }

4.4 CONCLUSION

The behaviour of concrete blocks under bearing pressure

can be summarized by the following:

(A) PLAIN CONCRETE BLOCKS

<1> Specimens with concentrated load near the edge

Wa/4.5a1<3.5, failed by shearing off the corner while

specimens with a larger edge distance Wa/O. 5a1}3.5 failed

by spliting the concrete block into two halves.

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97 CHAPTER 4

<2> The depth of the tension zone below the loading plate

increased as the edge distance increased.

<3> The bearing capacity of plain concrete blocks under

concentrated strip load can be estimated by

r 0.12 Wa/a1 + 1.16 Wa/4.5a1 > 3.5

fb/fý =j (4.1)

L 0.47 Wa/ai + 0.55 Wa/0.5a1 < 3.5

<4> When a concrete block is loaded with a bearing plate, a

compression zone is generated at the top immediately below

the loading plate. Below this compression zone is a

region of tension (bursting zone). Both the size of the

compression zone and the bursting zone increase with

increase in size of the bearing plate.

<5> Specimens with heights shorter than 300 mm have their

tension zone confined by the restraint at the bottom of

the specimen. This contributes a compressive force which

delays the splitting of the specimen. The bearing

strength of specimens 200 mm in height is 30% higher than

corresponding specimens of similar condition but greater

in height.

<6> Muguruma E59] describes a similar trend of increase in

bearing strength with short specimens but he appears to

over-estimte the strength for specimens with a bearing

area ratio less than 16 and to under-estimete those

greater than 63.

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913 CHAPTER 4

<7> Kriz's estimates 1491 are conservative for all the values

of loaded area ratio while Shelson's estimates (76] are

conservative for a high loading area ratio, R but become

unsafe as the value of R falls below 13.

<8> Reducing the size of the specimen can increase its bearing

strength and this is known as the size effect (or scale

effect). Al-Nijjam (63] has stated that the bearing

strength of the concrete is increased by a factor of

(a' /a)1/4

(4.2)

<9> A1-Nijjam's estimation [63] has a more gentle increase in

bearing strength as the size increases and is more

suitable for larger specimens. From the present

experiments, the bearing capacity is found to be

proportional to the following expression

fb/f' = 1.45 ea/80+ 0.9 (4.3)

<10> Bearing strength decreases with a reduction of the base

friction. The largest reduction (227.. ) is found with short

specimens loaded with a large plate (block Bi) while the

smallest reduction (07.. ) corresponded to high blocks loaded

with a small plate (block B4).

(B) REINFORCED CONCRETE BLOCKS

<1> Similarstrain

distributionsare obtained

inreinforced

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99 CHAPTER 4

concrete blocks as in plain concrete blocks.

<2> The bearing capacity of the reinforced concrete block is

not dependent on the diameter of the reinforcement.

<3> Reinforcement should be maintained in the whole of the

tension zone. Lack of reinforcement in any part of the

tension zone may result in large crack widths and lower

strength.

<4> Closely spaced smaller diameter reinforcement is much more

effective in resisting bearing force than widely spaced

thick reinforcement. With this form of reinforcement more

restraint can be provided to the vertical reinforcement by

horizontal stirrups. This can prevent the buckling of the

vertical steel at early stages and provide a better spread

of tensile stress to the surrounding block. Radial cracks

with small crack widths are found in place of a single

vertical crack with large crack width.

<5> The use of small interlocking stirrups is more effective

than using a single long stirrup, because more lateral

restraint can be provided by small interlocking stirrups.

<6> From the present tests, it is difficult to tell whether

the distribution of-reinforcement has any effect on the

bearing capacity of the blocks. It is recommended that

the matrix of reinforcement should have its width at least

as wide as the loading plate to prevent brittle failure of

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100 CHAPTER 4

the block below the reinforcement.

<7> Reinforcement is more effective with blocks of small edge

distance, Wa/O. 5a1<3.5 as it can prevent blocks from

shearing off the corners.

<8> The cracking strength of the reinforced concrete block

with Wa/O. 5a1<3.5 can be estimated by the bearing strength

of plain concrete block using Eq. 4.1.

<9> In general, the cracking strength of reinforced concrete

blocks and the bearing strength of plain concrete blocks

can be estimated by

10.47Wa/a1 + 0.55 2Wa/a1<3.5

fb/fý =j0.12Wa/a1 + 1.16 (plain) ) (4.4)

r 2W/a1>3.5

L 0.22Wa/a1 + 1.01 (reinft. ) J

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101

Stockno. fp He

X of reductt, onin strength

8i 1.30

R1-H4 1.67

22 X

B2 7.45

R4-H4 8.41

14 X

B3 0.98

Ri-HI 1. i6

16 X

B4 5.72

R4-Hi 6.22

ex

Tobte 4.3 Reductt. on to strength wt,h PTFE

at the bottom of the blocks.

Pu 2

01/4

zý F!

2ý-F2

3

zi-0.4aß

z3 - 0.75a

02/2

02/4

Pu /2

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102

4.5 PROPOSED SOLUTION

CHAPTER 4

Consider a plain concrete block (axbxH) as shown in

fig. 4.8a subjected to a vertical compressive force along its

centre line at its top surface through a steel plate, width al

and across the full thickness b of the concrete block. At the

bottom, the concrete block is supported over its entire

surface.

Fig. 4.8b shows the concrete block split into two halves

and its internal forces are considered. First of all, it is

assumed that the bearing plate is thick and rigid enough to

have a rectangular stress distribution immediately below the

steel bearing plate. Therefore, after splitting there is a

resultant force Pu/2 acting at a distance a1/4 from the line of

splitting. Along the line of splitting, the distribution of

stresses is rather complex and assumptions are made according

to observations made during the experiments. It is observed

that there is a compression zone at the top of the block

immediately below the bearing plate followed by a region of

tension and then an unstressed region at the bottom. For

reasons of simplicity, they are assumed to be linearly

distributed along the splitting line. As described in

section 4.2.2, it is noted that the depth of the compression

zone is dependent on the width of the bearing plate. According

to the graphs showing the experimental transverse stress

distributions, figs. A. 4-19, it is reasonable to assume that the

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103 CHAPTER 4

compression zone ends at a depth of 0.4ai from the loaded

surface. Below this compression zone is a region of tension,

and similarly the position of the maximum tensile stress is

dependent on the width of the bearing plate. With the help of

figs. A. 4-19, it is assumed that this maximum tensile stress

occurs at a depth a1 below the loading surface. The position

where this tension zone ends is not distinct and it is rather

difficult to determine as there is usually a long tail of small

stress before it finally become zero. However, it is assumed

that the tension zone effectively ends at a distance 0.75a

below the loading surface.

At the moment, only higher blocks H/a>O. 75 are considered

and for high blocks, compressive stress is distributed

uniformly along the base, figs. A. 4-11. A rectangular

distribution of compressive stress is therefore asumed at the

bottom of the block. As demonstrated by blocks B3 and B4,

where the friction at the base of the blocks was released,

behaviour was similar to their counterparts RI-H1 and R4-H1,

This suggests that the friction at the base for high blocks is

negligible and therefore, can be ignored in this model.

Furthermore, the concrete block is assumed to fail as soon as

the maximum tensile stress reaches the tensile strength of

concrete as estimated by splitting cylinder test.

Fig. 4. Gb shows the proposed model and the assumed stress

distribution. The resultant forces and their centres of action

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104 CHAPTER 4

have been calci

block and they

F1

F`

FTJ

F4

ilated based on a unit thickni

are as follows:

= F2 + F3 - F4 zi

= ft- (ziz1

)/2 z2

= ft- (z3-z`) /2 z3

=0 z4

L-ss of the concrete

= O. 4a1

= aal

= O. 75a (4.5)

=H

By taking moments at the position of the resultant force FI and

considering

the free body diagram in fig. 4-8b

Pu-(ate al)/8 =[2z2.

F2 + (z3-z1+2z2)F3]/3

a,, = 4[4z2-F2 +2 (zß z1+2z2)F3]/3Pu + ai (4.6)

Although many researchers (6,7,32,49,76] adopted the cube

root formula

fb/f. = R1/3 (4.7)

this has been demonstrated to be conservative especially for

large values of R (fig. 4.3). It appears that

fb/fý = R1/2 (4.8)

would be a more appropriate formula for the estimation of the

bearing capacity of the concrete block. The ratio R is the

footing to the loading area ratio a/a1 and in this case it

should be related to the reduced dimensions (or effective block

size), a2 for a2<a. Therefore,

fb/f' = (a2 /a1)1/2

(4.9)

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105 CHAPTER 4

Substituting Eq. 4.9 into 4.6,

(fb/f ') 2= [16z2 -F` +S (z, -z 1+2z2) -F31 /3Pu -ai +1 (4.10)c

since Pu = ai"fb and let r= fb/fý, therefore,

r2 =[16z2'F2+8

(z3-z1+2z2) -F3]/3a12-fb

r3 -r-[16z2

-F2+8 (z J-z 1+2z2) -F J,/3a1

By solving the cubic equation (Eq. 4.11) the

strength to cylinder splitting strength can be

+ fb/fý

f' =O (4.11)c

ratio of bearing

obtained.

For blocks with height to width ratio less than 0.75, the

calculation is based on the assumption that the tension zone

ends at the bottom of the specimen. Moreover, for blocks with

eccentric loading, the near edge side has a similar amount of

confinement as the corresponding concentrically loaded block

with a width equal to twice as the edge distance. Therefore,

by using a= 2Wa, Eq. 4.11 can also be used for the estimation

of the bearing strength in eccentrically loaded concrete

blocks.

4.6 COMPARISON WITH TEST RESULT

4.6.1 PLAIN CONCRETE BLOCKS

All the test results available from various sources

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106 CHAPTER 4

E32,49,62,653 are analysed in the light of the proposed method

and also compared with the method given by Meyerhof [62],

Hawkins 132], Shelson (76], Kriz (49], Muguruma (62],

Niyogi (65] and Jensen 136]. The results are presented in

table 4.4 for all concentrically loaded specimens and table 4.5

for all eccentrically loaded specimen. For easy comparison,

ratios of the values of bearing capacity obtained from the

experiment tothe

corresponding values obtainedby

calculation

are tabulated together with the mean and standard deviation for

each group of data with the same height to width ratio.

Fig. 4.1O-18 are plotted with the ratios of bearing to cylinder

strength obtained from experiment to those from the calculation

with different formulae. They can be read in conjunction with

table 4.4 and 4.5, so as to gain an idea of how the data are

distributed about the diagonal line, on which experimental

values would be equal to the calculated values of bearing

strength. Furthermore, an overall mean and standard deviation

are shown at the end of each table. In the calculation with

Hawkins's and Meyerhof's formulae the values of m is assumed to

be 27.5 degs. which is suggested by Hawkins (14). The

internal angle of friction, w is assumed to be 37 degs. as is

recommendated in the literature (6,7,36] when calculated with

Jensen's model.

It can be seen from fig. 4.10 and table 4.4d that

Heyerhof's equation can give a fairly good value of the mean,

0.98 for all the specimens with height to width ratio greater

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107 CHAPTER 4

than i but values are rather scattered around the diagonal line

with a large value of standard deviation of 0.24. However, the

deviation is reasonable for each group of specimens with the

same height to width ratio (table 4.4), for instance, it has a

standard deviation of only 0.04 and 0.08 for the specimens with

H/a =2 using Muguruma's and Niyogi's data respectively.

Meyerhof tends to over-estimate the strength of high blocks and

under-estimate for lower ones. This phenomenon becomes obvious

for a block with height less than its width (table 4.4b and

fig. 4.10). It has an overall mean for all the specimens of

1.224 and standard deviation of 0.406. Meyerhof has considered

the effect of height on the bearing capacity of the concrete

block but gives the wrong trend. With his method of

calculation

bearing

strengthdecrease

withthe decreasing

height of the blocks but in fact, bearing strength is increased

with decreasing height except for very short blocks H/a<0.33.

Generally, Meyerhof's formula is not accurate for the

estimation of bearing strength of the concrete block especially

with specimens which have their height less than the width.

Hawkins gives better result in comparison with those

calculated by Meyerhof. Although the mean for all specimens

with H/a>1 of 0.888 is not as good as the 0.98 given by

Meyerhof, the standard deviation is much smaller at 0.101 for

specimens with H/a>1 and 0.158 for all specimens. Moreover,

the mean for all specimens is good with only 5.4%

under-estimated. With Hawkins' formula the results are similar

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108 CHAPTER 4

to those obtained from Meyerhof's formula, in that there is a

trend of increasing under-estimation as the height to width

ratio decreases. However, with H/a<0.33 a sudden decrease in

bearing strength is recorded with Niyogi's data (mean test to

calculated bearing strength equal to 0.81). This is probably

due to the restraining force, which has increased to its

maximum as the height decreases. Further reduction in height

of the specimens may result in the weakening of the splitting

strength of the specimen as the cross-sectional area is

decreasing with height. Hawkins equation represents an

improvement in the estimation of the bearing strength of

concrete blocks in comparison with Meyerhof's estimation. It

is an acceptable means of estimation, but it should be borne in

mind

that there is a certain percentage of over-estimation.

Shelson, Kriz, Muguruma and Niyogi developed empirical

formulae. Amongst these four formulae, Shelson gives the best

result with an overall mean of: 1.005 and a standard deviation

of 0.242, although there is obvious scatter on the graph

(fig. 4.12). The result is improved somewhat if only blocks

with H/a>1 are considered: the formula has a slightly

over-estimating mean of 0.958 and smaller deviation of 0.208.

Kriz's empirical formula gives the safest estimate which has

over 207. of under-estimation for specimens with H/a>1 and 257.

for all specimens. His result is even more scattered than

those with Shelson's formula; it has an overall standard

deviation of 0.387 and 0.325 for all the specimens with H/a>1.

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109 CHAPTER 4

Muguruma's formula is the only empirical formula which

considers the effect of height on the bearing capacity of the

concrete blocks. This gives reasonably good results

(mean = 0.915, deviation = 0.121), for specimens with H/a>1.

However, his formula fails to gives a reasonable estimate for

specimens with H/a<1, and his estimation of bearing capacity

becomes negative, for a block with H/a = 0.33 (table 4.4b),

which is not acceptable. Fig. 4.14 shows that his equation

over-estimates the strength of nearly all the specimens

particularly for specimens with H/a<1. The overall mean and

standard deviation are 0.751 and 0.368 respectively.

Niyogi's formula is another one which over-estimates the

bearing capacity of the concrete block with a large standard

deviation. He seriously over-estimates for concrete blocks

with H/a? 1, mean = 0.889 and with a good standard deviation,

0.244. The mean is improved at, 0.946 when all specimens,

including those with H/a<1, are considered but the deviation

becomes worse at, 0.329. In general, Muguruma's formula is

good for the estimation of the bearing capacity of concrete

blocks with H/a>1 but it cannot be used for blocks with H/a<1.

Shelson gives a reasonably good estimate for all specimens

including those with H/a<1. Kriz's formula is the safest

estimator even for specimens with H/a<1 and it can be used as

an upper bound bearing capacity.

Jensen's model is based on the equilibrium of internal

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110 CHAPTER 4

energy and external work. With his method of calculation, he

seriously over-estimates the bearing strength of the concrete

blocks with H/a>1 and with a large standard deviation

(mean = 0.643, deviation = 0.205). Fig. 4.16 shows that he has

more or less over-estimated the strength of all the specimen

with H/a>1 and under-estimated all those with H/a<1. In

general, his formula gives scattered results with an overall

mean and standard deviation are 0.884 and 0.374 respectively.

The proposed model gives excellent results for all the

specimens with H/a>1; it has a mean of 1.005, which represents

an under-estimate of only 0.5% and a small deviation of 0.098

(see fig. 4.17). However, if specimens with H/a<1 are also

considered, the result is not so good. The overall mean and

standard deviation are 1.072 and 0.181 respectively. Moreover,

the under-estimate seems to increase with the decrease in H/a

ratio. Fig. 4.9 plots the mean of the ratio of bearing capacity

obtained in the tests to values obtained by the proposed model

for each group of data against its corresponding height to

width ratio. The number beside each symbol is the number of

blocks involved in the relevant group. This shows that the

under-estimate increases exponentially as H/a decreases.

Regression analysis leads to the following equation:

fb Ifb=0.657 e

1.15H/a+0.9 (4.12)

(test) (cal.)

The increase in bearing strength for shorter blocks is

believed to come from three sources:

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111 CHAPTER 4

<1> As the blocks become shorter, the end of the tension zone

is disturbed and the transverse tensile stress will not be

zero at the bottom of the block. These restraining forces

will increase the bearing strength of the concrete block.

<2> The vertical compressive stress at the base of the block

will not be distributed linearly. It will change to a

triangular or even trapezoidal distribution of stresses.

<3> With the rearrangement of vertical compressive stress at

the base of the block, the horizontal restraining force

which had been ignored in high blocks becomes dominant and

this will increase the bearing strength of the concrete

block.

The proposed model can be improved for shorter blocks by

solving the above three problems. However, due to the

complexity of stress in the tension zone and the variable

source of the restraining force, it is quite difficult to

analyse systemtically. However, the proposed model can still

be improved by means of Eq. 4.12. Using this equation to modify

the bearing capacity calculated by the proposed model, the

result will be improved considerably.

This modified proposed model, gives a mean and standard

deviation of 0.984 and 0.085 for all the specimens with H/a>1

respectively. Moreover, it improves the overall mean and

standard deviation from 1.072 to 1.010 and 4.191 to 0.137

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112

i. 6e

®4® --- Nt.yogt. ' a data

Nuguruna' a data

1.5 --- Krtz'a data

® --- Present data

!. 4

1.30

L

O

.m.ä.2

m

V

1.0

0.9

0.8

0 0.5 1.0 1.5 2.0 2.5 3.0

Hei.gth to width ratu4 H/a.

Fi.g. 4.9 Effect of height on the bearing capact. yof concrete bLocks.(Calculated by the proposed model)

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114

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C. Lou lot od bear t rng as rouge h ratio

F ig. 4.1 0. Expt. ve ca Lou Let ed beer I ng strength.

(Meyer-hof 'a formu La)

8/8/2019 uk_bl_ethos_373477


izo

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8/8/2019 uk_bl_ethos_373477


I1I

FIg. 4.1 2. Expt. ve ca lcu Lat ed boor t ng et rongt h.

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I , Lz

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8/8/2019 uk_bl_ethos_373477


ýý3

F19.4.1 4. Expt. vs ca lcu Lat ed bearing st rangt h.

(Mugurums 's formu La)

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8/8/2019 uk_bl_ethos_373477


c114

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8/8/2019 uk_bl_ethos_373477


1-ý5

Ftg. 4.1 6. Expt. ve cs Lcu let ed beer t ng at r®ngt h.

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8/8/2019 uk_bl_ethos_373477


%2b

CGLou L. s. d bow- t ng "I rer4t hr -as c

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(Proposed model)

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(Modified proposed modal)

8/8/2019 uk_bl_ethos_373477


128 CHAPTER 4

respectively. Fig. 4.18 shows that after multiplication by the

factor in Eq. 4.12, most of the data are congested around the

diagonal line.

Furthermore, table 4.5 tabulates the estimation of bearing

capacity (test/cal. ) with different formulae and models for all

the eccentric loaded specimens available C30,49]. It shows

that the proposed model also produces good results for

eccentrically loaded blocks. It has the best mean and standard

deviation of 0.945 and 0.140 respectively when compared with

estimations from other formulae. Therefore, it is concluded

that the proposed model is a sound method of estimation of

bearing capacity of concrete blocks both under concentric and

eccentric loading.

4.6.2 REINFORCED CONCRETE BLOCKS

Table 4.6 tabulates the result of all the reinforced

concrete blocks tested with the calculated values using

Kriz's C49] and Al-Nijjam C63] formula. It seems that both

Kriz and Al-Nijjam have over-estimated the bearing capacity of

the reinforced concrete blocks by around 277. and 7%

respectively. They both have similar standard deviations of

around 0.130 and 0.129 respectively. However, over-estimation

of the strength of the reinforced concrete block may not be due

to the theory itself, it is possibly due to the ineffectiveness

8/8/2019 uk_bl_ethos_373477


1'29

Spec. Na Ast r ft Kr. z Eq. 2.54 L-Ntjjan Eq. 2.5 Proposed Eq.4. i2

no. (on. (ma.e(NIan 1 IN/nn) tb/to GA8Ycot tb/to C88YCoL tb/to cesyco,,

R1/i 180 226 45.3 3.37 1.80 0.72 1.59 0.82 1.22 1.07

Ri/2 180 628 46.6 3.63 2.28 0.55 1.89 0.66 1.14 1.10

R2/i 180 402 46.0 3.70 2.04 0.65 1.61 0.83 1.21 1.10

R2/2 180 402 46.0 3.70 2.04 0.62 1.44 0.88 1.21 1.04

R3/i 180 622 43.4 3.50 2.35 0.74 1.77 0.99 1.21 1.45

R3/2 180 603 43.4 3.50 2.33 0.63 1.76 0.83 1.21 1.21

R4/i 180 402 47.0 3.45_ 2.02 0.70-1.61

0.88 1.19 1.19

R4/2 180 402 47.0 3.45 2.02 0.68 1.61 0.86 1.19 1.16

R5/i 180 301 45.9 3.77 1.90 0.73 1.47 0.95 1.21 1.15

R5/2 180 301 45.9 3.77 1.90 0.72 1.39 0.99 1.21 1.13

R6/i 130 301 39.5 3.29 1.84 0.72 1.50 0.88 1.08 1.22

R6/2 80 301 39.5 3.29 1.56 0.83 1.50 0.87 0.97 1.34R7/i 280 301 45.3 3.36 - 2.22 0.73 1.47 1.11 1.48 1.10

R7/2 230 301 45.3 3.36 2.08 0.75 1.47 1.06 1.33 1.17

R8/i 180 - 463 3.29 1.02 1.15 1.14 1.03 1.27 0.99

R8/2 330 301 463 3.29 2.32 0.75 1.47 i. 18 1.60 1.07

mean 0.729 0.926

standard devi. att. on 0.130 0.129

Table 4.6 fb/fc caLcuLated by various researchersfor reinforced concrete blocks.

I \

O 40 1LÖ

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8/8/2019 uk_bl_ethos_373477


130 CHAPTER 4

of some forms of reinforcement. For instance, A1-Nijjam's

formula successfully estimates the bearing strength of block

R3/1 which may be said to have effective reinforcement, with a

test to calculated bearing capacity of 0.99 which is a good

result. On the other hand, a test to calculated bearing

capacity ratio of 0.66 is obtained for block R1/2, which is

consisered to have ineffective reinforcement. Therefore, at

this stage it cannot be concluded whether Kriz's or Al-Nijjam

model can be used generally as a means of calculation for

reinforced concrete blocks.

Table 4.6 also tabulates the bearing capacity estimated by

the modified proposed model as if they were plain concrete

blocks..

The ratio of the test to calculated values of bearing

capacity is also tabulsted on the table. This can give an

indication of the effectiveness of each form of reinforcement,

and it is found that there are mostly 10 to 20% increases in

bearing strength due to the presence of reinforcement. The

most extra-ordinary one is block R3/1, which has more than 457.

increase in strength with the presence of reinforcement.

Fig. 4.19gives a plot of

thy?percentage increase in strength

with reinforcement for all the specimens in the present

investigation.

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131 CHAPTER 5

5 LITERATURE REVIEW ON THE SHEAR STRENGTH OF REINFORCED

CONCRETE DEEP BEAMS

5.1 INTRODUCTION

Nowadays, short span reinforced concrete panels are often

used as structural members in civil engineering works and are

referred to as deep beams. They can be defined as beams whose

depth is of the same order of magnitude as span. The stresses

in a deep beam differ radically from stresses predicted by

ordinary beam theory because simple beam theory does not

account for the vertical normal stresses induced by the applied

loads and supports nor for the shearing deformations. The

shear strength of a deep beam is significantly greater than

predicted using expresions derived from simple beams. As

increasingly design is carried out on a basis of ultimate

strength, there is a need to understand the ultimate behaviour

and strength of deep beams.

The investigation of the stresses and ultimate strength of

deep beam is not a new subject. Different researchers have

different techniques of investigation and have arrived at

various results. In this chapter, an overall review of

previous research on deep beams is presented.

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132

5.2 ELASTIC SOLUTION

CHAPTER 5

Dischinger [22,1932] was the first researcher looking into

the stress distribution in deep beams with periodic loading

represented by Fourier series and he constructed a stress

function to satisfy the boundary condition. His results were

later published graphically in a pamphlet by the Portland

Cement Association [72]. In 1951, Chow 118] undertook the

investigation of a single span deep beam by superimposing two

stress functions. He used the stress function of the

continuous deep beam that satisfied all but one of the boundary

conditions and then constructed another stress function by the

principle of virtual work to elimate the normal stress left at

the vertical edges from the first stress function. He solved

thedifferential

equationby the

method offinite differences

and presented the results in graphical and tubular form for

direct use in design. However, Chow could not produce accurate

results at all cross-sections of the beam. Uhlman [80,1952]

was another researcher of that period to investigate a single

span deep beam. He solved the governing differential equations

using Richardson's method of successiveapproximations

and

computed the stress trajectories for a number of loading cases.

His solution can give some guidelines for the design of

reinforced concrete deep beams with discussion of the effect of

the presence of web openings. In 1954, Caswell [16] made use

of photoelastic analysis to investigate the stress distribution

in a simply supported deep beams. In comparing the fringe

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133 CHAPTER 5

'patterns from photoelastic analysis with theoretical estimates,

a close estimate of stress in deep beams subjected to simple

loading was obtained. Kaar [37,38] performed experiments on

small deep beams of aluminium and steel in 1957. He recognized

the gradual departure from the simple linear flexure

relationship to a highly non-linear stress state in a very deep

beam. Archer and Kitchen [4,1956] constructed a stress

function directly by means of virtual work. Their solution of

bending stress agreed well with the results of Chow, but their

shear stresses were not in good agreement. Later, Archer and

Kitchen summerized the solutions of eight loading cases on deep

beams with span/depth ratio equal to 1 together with correction

factors for deep beams with span/depth ratios equal to 1.5 and

2.0 respectively. They presented these in tabular form for the

purpose of design in the later paper [3]. Geer [25,1960] also

used the method of finite differences to solve the differential

equation but with a much finer computational grid than Chow.

He discovered that the greatest tensile stress occurred not at

the mid-span but near the face of the support. In 1961, Saad

and Hendry [74,75] reported the results of a series of

photoelastic tests on simply supported deep beams with either

central load or gravitational load. In the latter case, they

used a large centrifuge to simulate the gravitational loading.

They summarized the results in figures showing the magnitude

and direction of the principal stress. Holmes and Mason [34]

made use of virtual work to present a method in solving the

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134 CHAPTER 5

problems of a single span deep beam supported by a parabolic

shearing forces applied at the vertical edges. Since, this

loading condition does not create a high bearing pressure

around the supports, their results did not differ from the

results obtained by shallow beam theory by as much as might be

expected for beams of deep proportions. In 1973, Bhatt E11]

generated a general procedure for solving continuous deep beams

in statically indeterminate supports. The magnitudes of the

reactions were determined by imposing the condition of

displacements at the support. His solution comprised a power

series and a series of hyperbolic functions. However, his

results of stress distribution for deep beams with three spans

differed little from those obtained by elementary beam theory.

Recently, (1983), Barry and Ainso EiO] applied the multiple

Fourier technique to a simply supported deep beams under

uniform loading at either top edge or bottom edge. They

superimposed three stress functions in order to have all

boundary conditions safisfied. Contour maps of the stress

field were presented and illustrated the apearance of regions

of pure tension and pure compression in the stress fields.

Previous elastic analyses fall within the three

categories: (1) Fourier series technique, (2) Method of finite

differences and (3) Photoelastic technique. However, with the

invention of computer, the above mentioned elastic analyses

have lost some value and have been replaced by the powerful

method of finite element analysis (FEM), fig. 5.1 shows the

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135

a

Z

C

a

2

c

Ne

äZ

O

v

1

N!

1

äC

.4

U

T1

N

x

4r*

inö

O

a.ý

vL

yCCDL0'4-

t

3

to

to

0Ca

W

.oCO

a.ý

.DL

to

'O

0(0wLa.U,

U,DD

LL.

ýý ýý

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I

CL

13(0

N

.Ca.1

3

to

EO

LO4-

N

Z

CL

0

to

to

OCO

x

WLL

.aC0

y7

LyCo

'O

to(AwL

U,

N

0,

lt.

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137 CHAPTER 5

horizontal stress distributions at the centre of a simply

supported beam with span/depth ratio from 0.5 to 4.0, analysis

by FEM. It illustrates clearly how the stress dirtribution

deviates from elementary beam theory as the span/depth ratio

decreases. Fig. 5.2 is also the result of FEM analysis; it

gives the vertical and horizontal stress distribution at

various sections along a beam (fig. 5.2) with a span/depth ratio

equal to one, together with the stress contours of the

principal tension and compression stresses (fig. 5.2b).

5.3 EXPERIMENTAL ANALYSIS OF REINFORCED CONCRETE DEEP BEAMS

The gradual introduction of limit state analysis into most

national codes implies that elastic analysis is no longer of

prime importance in design. It only gives information of

stress distributions in reinforced concrete deep beams up to

the onset of cracking and also it sheds no light on the modes

of failure which are important in design. For the above

reasons, extensive experiments have been carried on reinforced

concrete deep beams for the past two decades. Some of these

tests are summarized in this section.

Paiva and Siess [69,1965] were early researchers

investigating the shear strength and behaviour of moderately

(span/depth ratios from 2 to 6) deep reinforced concrete deep

beams. The main variables involved in the study were the

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138 CHAPTER 5

amount of main and web reinforcement, concrete strength and the

span to depth ratios. All the beams were 711 mm long and

supported on a span of 610 mm. The depth was varied from 178

to 330 mm with three effective span/depth ratios of 4,3 and 2

respectively. The widths of the beams were varied in such a

way to give a constant cross-sectional area. The bearing plate

at load points and reactions were 101 mm long and it was loaded

at the third points. Fig. 5.3 indicates the dimensions of the

specimens and types of reinforcement used. From the results of

the tests, they concluded that an increase in the concretehas no effect on the beam failing in flexure but increasesthe shear strength

strength n of the beam particularly at low span/depth ratios.

The presence of web reinforcement had no effect on the cracking

load for the shear capacity of a beam with shear span to

effective depth ratio (x/d) greater than 3. However, for x/d

smaller than this value, there is a large increase in shear

capacity beyond the cracking load. Web reinforcement tended to

reduce the deflection at ultimate load.

Leonhardt [53,54,19667 reported the tests of five simply

supported deep beams with span/depth ratio of 0.9. They were

1600 mm square and 100 mm thick. The bearing length was

1600 mm, leaving a span of 1440 mm. All specimens were tested

under uniformly distributed top load; dimensions and

reinforcement details are shown in fig. 5.4. In one of these

beams (WT4), the bearing area was increased by a transverse

strip up to a height of 600 mm from the bottom. It was found

from the tests that tensilestresses

in themain reinforcement

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1

i78 t330

Ij

178

33C

1

º-1 iui t- w

610 --ý

133

1

178 t330

1

sý lUl t-

'1ITrr1 rTrr

11111 IIIII

iiltI 11111JIILL

__J11LL---+t O

j

178

33C1

i7 610 rf---- 610

F..g. 5.3. Test specl. mens of Patva and St.ess (1965)

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140 CHAPTER 5

do not decrease towards the support as in ordinary slender

beams butremains constant almost to the supports.

Inorder

to

provide anchorage of the main reinforcement, Leonhardt

suggested that long horizontal hook loops should be used in the

main reinforcement at the supports. He stressed that bent up

bars from the main reinforcement mainly received compressive

stresses and therefore serve no purpose. On the other hand,

the weakening of the main reinforcement through bent up rods

reduced the fracture load. In 1968, "Ramakrishnan and

Ananthanarayana (73] presented experiments on 26 single span

rectangular deep beams having span/depth ratios of 1 to 2.

They were tested under concentrated (at a single point and two

points) and distributed loads. All specimens have a constant

span of 686 mm and height varied from 381 to 762 mm. Plain

mild steel round bars were used as reinforcement in all beams.

Details of the dimensions and reinforcement of the beams are

shown in fig. 5.5. Different modes of shear failure were

reported in their investigation. They were classified as

follows;

ti) Diagonal tension failure - characterized by a clean

and sudden fracture. For concentrated loads, it

occurs along a line joining the support and the

loading point. For uniformly distributed loads, it

is along a line joining either support with the

nearest third span point.

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141 CHAPTER 5

(2) Diagonal compression failure - involved with the

formation of two cracks. The first one, developed

nearly along the line joining the support and load

points. The second one is formed parallel to the

first one but closer to the support. Failure of the

beam was by the destruction of the portion of

concrete between these two cracks.

(3) Splitting of the compression zone - characterized by

the clear vertical fracture of the compression zone

at the top of the inclined crack.

(4) Flexure-shear failure - Although the tensile steel

of the beam had considerably yielded but before the

beam could fail in flexure, suddenly diagonal tension

cracks developed and caused the collapse of the beam.

Kong et al 139-47] carried out major tests on reinforced

concrete deep beams. This included experiments with normal

weight and lightweight concrete and with different forms of web

reinforcement. In 1970 and 1972, Kong et al [29,44] presented

an experimental study on the effectiveness of different web

reinforcement in 45 normal weight concrete deep beams. The

types of web reinforcement used are shown in fig. 5.6. Each of

the specimens had an overall length, 915 mm, 762 mm span and

76 mm width. They were tested with span/depth ratios varying

from I to 3 and- loaded with two point loads at their third

span. The bearing length of both the support or loading point

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1412

I81

to

762

1

T81

to762

1

I81

to

762

I

I81

to762

I

Rg. 5.5 Test spect.mens of Ramakrt shnan G Anathanarayana (1968).

76.2

T254

to

762

1

Sen. es 4

C------ýC------ý

Sert. 7

Seri. es 2

Seri. es 5

Seri. es 3

Sert. es 6

Seri. S&0

813 11

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143 CHAPTER 5

was 76 mm. Anchorage of the main reinforcement was provided by

steel blocks at the ends of the beams. It wasfound that the

strength, deflection and crack widths were dependent on the

depth (L/H) and clear span to depth (Xc/H) ratios. For low L/H

and Xc/H ratios, only horizontal web reinforcement near the

bottom of the specimen was effective while for higher L/H (>3)

and Xc /H (>0.7) vertical web reinforcement was more preferable

than others. Inclined web reinforcement was the best in

controlling crack widths and deflection, and could also

increase the ultimate strength of the beams. However, they

pointed out that inclined web reinforcement may be uneconomical

in construction.

Fongand

Robins (42]reported

testson simply supported

lightweight concrete deep beams with dimensions and types of

reinforcement similar to those with normal weight concrete

beams [39,44] except deformed bars were used for reinforcement.

They arrived with similar conclusions as for normal weight

concrete deep beams in that inclined web reinforcement was the

most effective form of reinforcement in controlling crack

widths and deflection and produced higher strength as well. It

was found that the formulae for normal weight concrete deep

beams were not necessarily suitable for lightweight concrete

beams.

Further investigation on lightweight concrete. deep beams

has been carried out by Kong and Singh (47]. Results were

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144 CHAPTER 5

obtained for 45 rectangular deep beams with constant depth,

508 mm, but varying span from 508 to1524

mm.They

weretested

with L/H ratios ranged from 1 to 3 and Xc/H ratios from 0.23 to

0.7. Different types and amounts of web

used as shown in fig. 5.7. It was reported

critically affected the crack and ultimate

Inclined web reinforcement was the most

reinforcement for all ranges of Xc/H rat

reinforcement were

that the X /H ratioc

load of the beams.

effective form of

io tested. For low

Xe/H ratio, the next most effective reinforcement was

horizontal reinforcement placed close to the bottom of the beam

but for higher Xc/H ratios it was vertical web reinforcement.

Generally, The de Paiva and Siess's formula could be used to

estimate

the ultimate strength of lightweight concrete deep

beams, but was not so accurate as for normal weight concrete

beams.

Reinforced concrete deep beams subjected to repeated

loadings were also tested by Kong et al [48]. All the beams

tested were 76 mm thickness, 1524 mm span and had two different

heights of 508 and 762 mm (span/depth ratios were 3 and 2

respectively). They were loaded with 3 clear shear span/depth

ratios (Xc/H = 0.25,0.4 and 0.55). It was found that inclined

diagonal web reinforcement was still the most effective in

controlling crack width, reducing deflection and increase shear

strength of the beam. de Paiva and Siess design formula was

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05

T0.8

I

º{ f-76.2

50 to 152 -. -i Series iB

Sert. es IA

Series 2A

Sert, es 4A

Sert.es 2B

Sert. es 4B

Sen. es 5A

Series 6A

Ft.g. 5.7

Series 5B

f

Serves iC

Series 2C

Serves 4C

Serves 5C

Test spemmens of Kong (1972)

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146 CHAPTER 5

more accurate for deep beams with a repeated loading history.

Repeated load (within the tested range, mean level of load and

number of cycles) had no overall effect on the ultimate shear

strength of the reinforced concrete deep beams.

Amongst all the deep beams tested by Kong 139-47] four

principal modes of failure were reported. They are as follows

(1) Splitting of the beam into two by a diagonal crack.

(2) Crushing of the concrete between two diagonal cracks.

(3) Penetration of a diagonal crack into the concrete

compression zone near a bearing plate and failure of the beam

resulting from by the crushing of the concrete in the reduced

compression zone. (4) Crushing of the concrete at a load

bearing block (true crushing failure mode).

Al-Najjim (63] reported tests of 24 simple supported

beams, of which 6 had no web reinforcement, and the others were

reinforced with different forms of reinforcement; horizontal,

vertical, orthogonal and inclined web reinforcement. Amongst

the tested beams, 8 had vertical stiffening ribs to prevent

failure due to bearing at the supports. In fact, with the

exception of the specimens with stiffened ribs, most of them

failed by crushing of concrete above the supports; However, he

arrived a conclusion similar to Kong et al 147] in that for

small clear span/depth ratios, horizontal and diagonal web

reinforcement were more effective while vertical web

reinforcement was suitable for beams with larger clear

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147

span/depth ratios.

CHAPTER 5

Smith and Vantsiotis E77] presented tests on 52 deep

reinforced concrete beams under two point loading. All the

beams tested had a constant depth of 356 mm, 102 mm thickness

and varying lengths from 1420 to 2080 mm. They were tested

with spans varying from 813 to 1470 mm in 4 clear span/depth

ratios (Xc/H) of 0.77,1.01,1.34 and 2.01 respectively. Five

of the beams had no web reinforcement, the rest of them having

horizontal and vertical web reinforcement percentages ranging

from 0.23 to 0.91% and from 0.31 to 1.25% respectively. The

forms of reinforcement used are shown in fig. 5.9. It was

reported that failure of the beams was caused by crushing of

concrete in the reduced compression zone at the head of the

inclined crack or by fracture of concrete along the inclined

crack. Beams with web reinforcement has smaller cracks and

less damage at failure. Increasing the ratio Xc/H could

increase the deflection and reduce the ultimate load of the

beam. It was also found that the effectiveness of the vertical

reinforcementdiminished

andthe influence

ofhorizontal

web

reinforcement increased as Xc/H decreased. Concrete strength

played an important parts in the ultimate shear strength of

deep beams exceptionally at low clear shear span/depth ratios.

Besser (12,13] carried out tests on 7 deep reinforced

concrete panels with depth ranging from 720 to 2880 mm

corresponding to span/depth ratios from 0.25 to 1.0. All the

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vart, esP-

v-t. es

508

or762T

J@f'{. 6S A

6mmnomi,aL dta. att. rrupeHort. zontat spacing 152 mm.Vertt. cat spaci. ng 76 mm.

6 am nom.aL di.a. st t rrupa.Hor. zontal, spacing 152 nn.Vertical, spacing 38 ma & 108 mm.

Serves B

6 mmnom.oL di.a. tncU. ned stirrup.

at 45 dogs. to hort.zontaL. at 76 mmspaCi. g hori.zontaLLU. -

Series C

Ft.g. 5.8 Test spect. mens by Kong (1974).

0

Lai

LAj

LU

148

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149 CHAPTER 5

specimens had span 720 mm, overall length 872 mm and 72 mm

thickness, the depth/thickness ratios varied from 10 to 40.

They were reinforced with four 10 mm diameter bars as main

reinforcement which were anchored by external steel blocks at

the ends of the panel. Web reinforcement was orthogonal

arranged with 5.3 mm plain mild steel bars. Detail of the

dimension and reinforcement of the specimen is shown in

fig. 5.9. It was reportedthat the

diagonal crackloads

were

increased as span/depth ratio decreased from 1 to 1/3 and

thereafter they were unaffected by the depth of the panel. For

a specimen with span/depth ratio equal to 1, failure was by

shearing along the line joining the load and support points.

Bearing failure at the support was dominant in the specimens

with span/depth ratios ranging from 0.28 to 0.67. However,

failure by buckling of the specimen was reported with

span/depth ratios equal to 0.25 (height/thickness ratio equal

to 40).

5.4 RECOMMENDATIONS FOR THE DESIGN OF REINFORCED CONCRETE

DEEP BEAMS

A brief summary of various documents providing guidance

for the design of reinforced concrete deep beams follows;

5.4.1 PORTLAND CEMENT ASSOCIATION C72,1946]

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150 CHAPTER 5

This is a design method based on the elastic analysis and

not on the results of ultimate load tests. It is applicable to

reinforced concrete deep beams with the span/depth ratio (L/H)

not exceeding 1.25 for simply supported beams and 2.5 for

continuous beams. Design charts are used as a method of design

and it is based on two parameters: the height to span ratios

(p=H/L) and ther ratios of the support width to span (E=a1/L).

They can be calculated according to the loading and supporting

conditions as follows;

r of/L continuous beams

-{ 0.5 U. D. L. (simple beam) (5. la)I

a1/2L point load (simple beam)

f H/L continuous beamp=i (5. ib)

I HI2L simple beam

After obtaining the values of e and p from eq. 5.1, tensile

force T in terms of total load can be found from a design

chart. Thus, the area of the main longitudinal steel can be

calculated by

Ast = T/fs (5.2)

where f5 is the allowable working stress in steel. It was

suggested that the main longitudinal reinforcement should be

placed as close as possible to the lower edge of the beam.

Shear stress in the beam should be limited by

v= 8V/7b-d < (1+5H/L)v /3 (5.3)c

where v= shear stress of the beam

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151 CHAPTER 5

V= applied shear force

vc = allowable shear stress for slender beams.

5.4.2 DE PAIVA AND SIESS 169.19651

Previously, Laupa 151] had derived an expression for the

shear strength of reinforced concrete beams;

v= V/b-H = 200 + 0.188f' + 21300Pt (5.4)

where v= nominal shear strength, psi.V= shear force, lb.

f' = cylinder compressive strength of concrete, psi.

H= depth of the beam, inches

b= thickness of the beam, inches.

and

Pt = AS (1 + sing)/b-H (5.5)

in which As = Total sectional area of steel crossing a vertical

section between the load point and support.m= Angle of inclination of reinforcement to the axis

of the beam and should not be greater than62.7 degress.

Using De Paiva's experimental data to calculate shear

strength by Eq. 5.4, a linear relationship was found between

experimental and calculated values. It varied with the clear

shear span/depth ratio in such a way that

V(expt.)/V(cal. ) = O. 8(1-0.6Xc/H) (5.6)

and is valid for Xc/H between 0 and 1. Therefore, an

expression for computing the ultimate load of reinforced

concrete deep beams was obtained as follows.

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152 CHAPTER 5

Pu = 2-b-H-v = 1.6(200+0.188f'+21300Pt)(1-O. 6Xc/H) (5.7)

5.4.3 RAMAKRISHNAN AND ANANTHANARAYANA [73]

According to the results of their experiments, it was

found that most beams failed in a diagonal tension mode. This

mode of failure was similar to that of a cylinder splitting

test, and therefore, equations were developed to predict the

ultimate shear strength of reinforced concrete deep beams based

on the equations for the evaluation of splitting strength of

concrete. The splitting strength of the concrete ft can be

expressed as;

ft= F/K-A (5.8)

where F= Maximum splitting force.

A= Area resisting the splitting force.

K=1.57 for cylinder splitting test.

Consider an eccentric single point load acting at the top

of the beam as shown in fig. 5. lOa and resolve it according to

the figure. It can be seen that the splitting force of the

failure strut will be equal to

F= Pu-cos4/sin (e++) (5.9)

where 4>e

The area resisting the splitting force is

A= b"H-cosec+ (5.10)

By substituting Eq. 5.8-10, the ultimate load of the reinforced

concrete deep beams failed in diagonal tension will be

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153 CHAPTER 5

pu = K(1 + tane-cat+) -ft"b-H (5.11)

For a central concentrated load e=+, thus the ultimate diagonal

tension failureload will be

Pu = 2K-ftb-H (5.12)

For a two-point loaded deep beam as shown in fig. 5.10b, the

splitting force, F is

F=P -cosecs/2 (5.13)u

and the area resisting the splitting force is

A= b-H-cosece (5.14)

Therefore, the ultimate load for a deep beam with two-point

load is

Pý = 2-b "H-K-ft (5.15)

For uniformly distributed load on the top of the beam

(fig. 5.10c), the ultimate load can be found by considering it

as a superimposition of a series of point loads and integrating

it throughout the span of the beam. The splitting force

reached a maximum when the diagonal crack plan was defined by

tans = 3H/L (5.16)

and the total ultimate load Pu on the beam is given by

Pu = 2K-ft "b -H (5.17)

The value of K can lie between 1.0 and 1.57 depending on

the method used for accessing the tensile splitting strength of

concrete. Generally, a value of 1.12 is a 'reasonable lower

bound for all the tested beams. Moreover, during the

derivation of the above expression, the effect of web

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154 CHAPTER 5

reinforcement has been

authorhad little

or

conclusion can be made

used for beams with web

tool to estimate the cr,

deep beam.

neglected and tests carried out by the

no web reinforcement.Therefore,

no

as to whether these equations can be

reinforcement. However, it provided a

acking strength of a reinforced concrete

5.4.4 COMITE' EUROPEEN DU BETON - FIP 1171

CEB-FIP recommended that beams with span-depth ratios less

than 2 and 2.5 for simply supported beams and continuous beams

respectively should be designed as deep beams. In this section

only simply supported deep beams with top load are discussed.

The area of main reinforcement should be calculated from the

largest bending moment in the span, using the lever arm, z

defined as follows:

f O. 2(L + 2H) I< L/H <2

z=i (5.18)10. bL L/H <1

The main reinforcement should be extended without reduction

from one support to the others and anchored with a force equal

to 0.8 times the maximum force calculated. It should also be

distributed uniformly over a depth 0.25H-0.45L (<0.2L),

measured from the lower face of the deep beam. In order to

facilitate anchorage at the support and limit the development

of cracks and crack width, small diameter bars should be

employed. Anchorage by means of vertical hooks is not

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155 CHAPTER 5

recommended as it tends to promote cracking in the anchorage

Zone.

For top loaded deep beams, CEB-FIP proposes the use of

orthogonal reinforcement in the web, consisting of vertical

stirrup and horizontal bars on both faces of the beam. The

area of the reinforcement is given by 0.0025b. s for a smooth

round bar or O. 002b. s for a high-bond bar, where b is the

thickness of the beam and s is the spacing between bars. It is

interest to note that most of the CEB-FIP recommendations are

based on the findings of Leonhardt and Walther (53,54].

5.4.5 ACI COMMITTEE 318

The proposed revision of ACI 318-63: Building Code

Requirements for Reinforced Concrete E2] has a section of

recommendations on the design of deep beams and they will be

discussed briefly here.

The revised ACI Code is applicable to mambers with clear

span to effective depth ratio (Lo/d) less than 5 and loaded at

top or compression face when designed for shear. The shear

strength of the reinforced concrete deep beams, v is believed

to be composed of the nominal shear strength provided by

concrete, vc, and the nominal shear strength provided by shear

reinforcement, vs, so that

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156 CHAPTER 5

v=v+v (5.19)c5

Regardless of the amount of web reinforcement, the nominal

shear strength, v is limited by the following expressions

v= k- v (5.20)u

ra-f,

0.5Lo/d <2

and vu j(5.21)

l2(1O+Lo/d)f'0.5/3 2< Lo/d <5

where k= the capacity reduction factor and is taken as 0.85

The nominal shear stress, vc carried by concrete is calculated

by

vc = (3.5-2.5M/V-d) (1.9f'0.5+2504p"V-d/M)

< 2.5 (1.9fß "'+2 500p -V -d/M) (5.22)

< 6f* 0.5c

where M, V = the design bending moment and shear force at the

critical setion respectively.

p= the ratio of main reinforcement As to the area bxd

of the concrete section.fc = compressive cylinder strength, psi.

The remaining shear stress is carried by the web reinforcement

and it can be calculated by

v= V-Vsc

=fy-Awv(1+La/d)/(12Sv-b)

+fy-Awh(11-Lo/d)/12Sh-b)

(5.23)

where A= area of vertical shear reinforcement within awv

distance, Sv

Awh = area of horizontal shear reinforcement within a

distance Sh.

fy = specified yield strength of reinforcement, psi.

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157 CHAPTER 5

However, irrespective of the values of shear stress, the

cross-sectional areas of vertical and horizontal reinforcement

should not be less than 0.15%B. L and O. 25%b. d respectively.

5.4.6 KONG [40.45]

Kong has proposed a design formula to calculate the

ultimate shear strength of reinforced concrete deep beams for

both normal and lightweight concrete. The formula can be used

for beams with span/depth ratio not greater than 3 if the clear

span/depth ratio does not depart widely from the range 0.23 to

0.7. The formula is as follows:

Pu = 2[C1(1-0.35Xc/H)ft-b-H

+ C2 EA -y"sin2a/H] (5.24a)5

where Pu = ultimate load of the deep beam.

C1 = coefficient equal to 1.4 for normal weight concrete

and 1.0 for lightweight concrete.

C2 = coefficient equal to 130 and 300 N/mm2

for plain

round bars and deformed bars respectively.ft = cylinder splitting strength, N/mm2.

b= thickness of the beam, mm.H= overall depth of beam, mm.y= depth of bar, measured from top of beam to the point

where it interests the line joining the inside. edgeof the bearing blocks at the support to the outsideedge of that at the loading point.

a= angles between bars and the line described above.

n= numbers of bars, including the main reinforcementthat cross the line between support and loadingblock.

Recommendations have also been made for the design of flexural

reinforcement. Longitudinal main reinforcement should be added

so that the bending moment will not exceed.

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158 CHAPTER 5

M<O. 6As-fy-H/ym or O. 6As-fy-L/rm (5.24b)

Since theamount of

flexuralreinforcement required

in deep

beams is small and main reinforcement can also act as web

reinforcement, although the above formula gives conservative

results, it does not lead to wasteful use of reinforcement.

5.4.7 CIRIA GUIDE 2 E68.1977]

This is the most comprehensive set of rules and

recommendations available for the design of deep flexural

members. It can be used for beams with span/depth ratios less

than 2 for single span or less than 2.5 for continuous

supports. A brief summary of the design method is listed

below.

(a) Design for flexure

The area of main reinforcement can be calculated by

AS = M/O. 87fy-z (5.25)

where M= Design bending moment.fy = yield strength of the reinforcement.

z= lever arm at which the-reinforcement acts and is

given by

4.2L + 0.4H single span L/H <2

z=1 (5.26)L O. 2L + (). 3H continuous L/H < 2.5

For a simply supported single span deep beam, the main

reinforcement should be distributed uniformly over a depth of

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159 CHAPTER 5

O. 2H at the bottom of the beam. Reinforcement is not curtailed

in the span and must be anchored to develop 80% of maximum

ultimate force beyond the face of the support. Twenty percent

of bars should anchor beyond a point 0.21 from support.

(b) Design for shear

For top loaded beams, tl

defined as either the clear

contributes more than 50% of

distributed load. In the case

has more than 50% of the total

clear span should be taken as

Xe. The shear strength for top

he effective clear span Xe is

shear span for a load which

shear or 4.25L for uniformly

of more than one load but none

shear, the weighted average of

the effective clear shear span,

loaded reinforced concrete deep

beams can be estimated by

Vu = rl"b-H(1-0.35Xe/H)-fcu+r2-E100A5-y-sin2oa/H (5.27)

< 1.3b"H-ri -fcu

f 0.44 for normal weight concretewhere ri =j

1.0.32 for lightweight concrete

i1.95 N/mm2 for deformed bars

r2 =11

0.85 N/mm2 for plain bars

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160

(c) Bearing capacity

CHAPTER 5

CIRIA Guide limited the maximum bearing stress to O. 4f'

and the bearing length is regarded as the lesser of the actual

bearing length al and O. 2Lc.

5.4.8 AL-NAJJIM (63)

Al-Naijim proposed a structural model of failure of

reinforced concrete deep beams with and without web

reinforcement. There are a numbers of assumptions and they are

listed below:

(1) Steel is assumed to be properly anchored so as to

developa

tieand strut action.

(2) Steel is assumed to be perfectly plastic and has a

yield stress, fy in tension.

(3) Steel is assumed to carry only uni-axial stress alongthe original bar direction.

(4) The size of the compressive strut is determined eitherby the yield resistance of the tension steel or by

local conditiona at supports.

(5) The struts between loads and suports are deflected bythe presence of web reinforcement. For uniform websteel, the deflected strut is in parabolic form.

(6) The force in the main steel decreases towards the

supports and this is due to the presence of vertical

stirrups.

Al-Najjim 163] presented his model in many loading conditions

(single point, two point and uniformly distributed load) and

with different types of web reinforcement. To be brief, only

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161 CHAPTER 5

beams with uniformly distributed orthogonal web steel and under

two-point load are discussed here. Fig. 5.12shows

the

dimenmsions of the beam, notation and the forces to maintain

equilibrium. There are two modes of failure as follows:

(A) Failure of the tie

By taking moments at the intersecting

of the strut with the direction of the

considering the horizontal equilibrium of

the following equatons are obtained.

Vu = (Ts -z + 0.9d-Zw-H - AT

point of the centre

load (point A) and

one of the struts,

5)/X (5.28)

and yo = (TS + 0.9d-NW - ATS)/(b-ft2-f (5.29)

where z= d-y0/2 & ZW = 0.45d - ya/2 (5.30)

and Hw = Asw/b-Sh

V=A /b -Sw sv v

Ts = Tensile force of the main reinforcement.

ATs = Loss of tensile force towards the support due to the

presence of vertical reinforcement.y0 = Depth of the compressive zone at the top of the beam.

Combining Eqs. 5.28-30, the shear strength of the deep beam with

the yielding of the tie is

Vu = Ts (d-K')/X + 0.9d-Hw(O. 45d-K')/X (5.31)

where K' _ (TS 0.9d-Hw-ATs)/2b-p2-fc

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162

(B) Failure of the strut

CHAPTER 5

Consider the equilibrium at one of the support.

Vert.: Vu = wo-b-p`-f'-sine = K-b-p2-fýsin`s (5.32)

Hor.: TS-ATs = wo-b -pL-f' -cose = K-b -p2-fc "sin(2e)/2 (5.33)

since singe =[1

- (1-sin2(2e)]°5

(5.34)

By substituting Eqs. 5.32,5.33 into 5.34, the magnitude of the

tensile force in the main reinforcement Ts' can be found by

solving the equation below.

A-T '2 + H-T +C=0 (5.35)ss

where A=0.81d2 /X2 +1

B=4.73Hw "d3/X2 - 0.9K-d "b -p2-fim/X - 2d -Ts

C= ATs` + 0.16d4. Hty2/X2 - 4.41d2. Hw"K-b"p2"f*/X

After getting the values of TS', the shear strength of the beam

can be found by substituting back into Eq. 5.28, thus

Vu ={Ts'-z

+ 0.9Hw-Zw"d}/X (5.313)

It is noted that the value of ATs (decrease in tension force of

the main reinforcement from mid-span to the support) can be

determined by assuming that both main and web reinforcement

yield, taking moments at A (fig. 5.12) and considering all

forces from mid-span to the support. Integrating it along the

main reinforcement and ATs can be found-by solving the equation

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720

to2880

Fi.g. 5.10 Test specu,mens of Besser (1984).

T1

1Ft.g. 5. i2 Structural, model, proposed by Kong (1972)

.Vu vu

TIw

1

Z

1

fFý'S

a -j X I-"

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CA

Pu

/

fý

H

fIý.

P. cot$

cote+coto

n /n n /e

Pu Cate

cot9+coto

coto. aec8cote+cot

PUCoto

Pu

F . g. 5. ii Effect of Load Ln deep beam by Rama rL shnanand Ananthanarayana (1968)

.

seoo. cot 0Pýcot0+cote

2=Coto

PU cos¢ P corecos(et0

cos(et

PUcosecs

2=cosece

2 Pd2

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165

below.

CHAPTER 5

6T52+( 2b -d -p2 "f '-TS 4.9d -HW

)6Ts-b

-/12-fim W-X2 =0(5.37)

5.5 SUMMARY

(1) Early research on deep beams concentrated on finding the

distribution of stresses within the structures and falls

into three categories (1) Fourier series technique,

(2) method of finite differences and (3) photoelastic

technique.

(2) With the invention of the computer, the finite element

method of analysis has been introduced and virtually

replaced allthe

above mentionedtechniques.

(3) The introduction of limit state design and the non-linear

behaviour of concrete in the presence of cracks has

limited the value of elastic analysis. This has lead to

extensive experimental analysis on reinforced concrete

deep beams during the past two decades.

(4) Paiva and Siess E69] concluded from their experiments that

web reinforcement had no effect on cracking strength of deep

beams with large clear span/depth ratios ( >3 ) but

increases strength with smaller clear span/depth ratios.

Concrete strength increases the shear capacity slightly

with low span/depth ratio but flexural strength is

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166

unaffected.

(5) Leonhardt found that tensile

reinforcement do not decrease ti

ordinary slender beams. He also

of the main reinforcement is very

owing to bent-up bars may lead to

and tie system of the beam.

CHAPTER 5

stresses in the main

awards the suport as in

stressed that anchorage

important, any weakening

the failure of the strut

(6) Ramakrishnan and Ananthanaarayana (73] observed four

different modes of failure and they were (1) diagonal

tension failure, (2) diagonal compression failure,

(3) splitting of the compression zone and

(4) flexural-shear failure. However, the diagonal tension

mode was the most common in their tests and they

" constructed a failure model able to estimate the ultimate

failure load as

Pu = 2K"ft.

b-H (5.12)

(7) Kong E39-47] found that the shear strength, deflection and

crack widths were dependent on the clear span/depth

ratios. He also pointed out that inclined web

reinforcement was the most effective form of reinforcement

and horizontal web reinforcement was more effective than

vertical for low span/depth ratios. Four different modes

of failure were reported; (1) splitting along the diagonal

crack, (2) crushing of concrete along the diagonal cracks,

(3) crushing of concrete in the compression zone and

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167 CHAPTER 5

(4) crushing of concrete in the bearing zone. Based on

the experimental data, empirical formula was drawn up for

the estimation of ultimate loads for reinforced concrete

deep beams.

Pu = 2[C1(1-0.35Xc/H)ft-b-H+C2EA5-y-sin2a/H, (5.24a)

(8) Al-Najjim 163] arrived at the conclusion that inclined and

horizontal web reinforcement were more suitable for beams

with small clear span/depth ratios. Structural models of

the failure mechanism were proposed based on (1) failure

of the strut and tie system in which its shear strength

can be found by Eq. 5.31, and (2) compression failure of

the strut where its shear strength can be found by Eq. 5.36

after obtained the value of Ts' by Eq. 5.35. The predicted

ultimate shear strength will be the lesser of those

obtained by the above two methods.

(9) Smith and Vantsiottis [77] arrived at conclusions similar

to Kong's suggestion. They added that concrete strength

can affect the ultimate shear strength of deep beams

especially at low clear span/depth ratios.

(10) With the exception of one specimen (L/H=1), which. failed

by shearing along the diagonal crack, all the others

(L/H<1) tested by Besser C12,13] were dominated by

crushing of concrete above the supports. Buckling of the

specimen was only observed with specimen having L/H=4.25

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168 CHAPTER 5

and height to thickness ratio equal to 40.

(11) Recommendations for the design of reinforced concretre

deep beams by Portland Cement Association C72] were based

on elastic analysis. They involved the estimation of the

amount of tensile steel required with the help of design

charts and limited the shear stress by Eq. 5.3.

v= 8V/7b-d < (1+5H/L)vc/3 (5.3)

(12) CEB-FIP recommended that the area of main reinforcement

should be calculated from the largest bending moment in

the span with lever arm, z defined by Eq. 5.18. Shear is

controlled by orthogonal web reinforcement with

cross-sectional area equal to 0.0025b"s and 0.002b-s for

plain and deformed bars respectively.

(13) ACI Code limited the maximum shear stress by Eq. 5.21 and

evaluated the shear stress taken up by concrete to be

calculated by Eq. 5.22. The contribution of shear by

reinforcement can be obtained by Eq. 5.23.

(14) CIRIA Guide is the most comprehensive set of rules for the

design of deep beams. The area of main reinforcement can

be calculated by

A= M/0.87fsy

where z is the lever arm and is given by Eq. 5.26. Shear

force is limited by Eq. 5.27 and bearing and bearing stress

is not recommended to exceed 0.4f'.c

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169 CHAPTER 6

6 EXPERIMENTAL INVESTIGATION OF THE SHEAR STRENGTH OF

REINFORCED CONCRETE DEEP BEAM

6.1 INTRODUCTION

In order to cater for large spans in buildings, new

structural systems consisting of frames and deep beams have

evolved. Because of their proportions in depth and span, the

strength of deep beams is usually controlled by shear, rather

than flexure, provided that normal amounts of longitudinal

reinforcement are used. On the other hand, the deep beam's

shear strength is significantly greater than that predicted

using the expression for slender beams, because of its special

capacity to redistribute internal force before failure.

Investigation has been made of the shear strength of deep

beams with different span/depth ratios (0.7 to 1.1) with a

uniformly distributed load on top. In particular, for beams

with low span/depth ratios, bearing failure usually takes place

around the supports. Special forms of reinforcement

(chapter 3) are put into this region to prevent it from failing

in this mode.

6.2 DESCRIPTION OF TEST SPECIMENS

The testspecimens consisted of 6 beams with different

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170 CHAPTER 6

spans varying from 700 to 1100 mm. All had the same dimensions

1260x

1000x

100mm

except for a step of 5 mm which varied in

length according to the span of the beam tfig. 6.1), leaving a

central section over which uniformly distributed load was to be

applied.

All beams were reinforced with a similar amount of steel.

Main longitudinal reinforcement consisted of S plain mild steel

bars with 10 mm nominal diameter. This reinforcement was

placed in 4 layers, consisting of 4 closed stirrups, at 50 mm

spacing. The web reinforcement was provided by an orthogonal

arrangement of bars on both faces of the beam. They were 6 mm

diameter plain mild steel bars, with 100 mm centre to centre

spacing. Vertical reinforcement above each support was four

plain mild steel stirrups, 6 mm in diameter at 66.7 mm spacing.

Beams D82 to DB6 had additional stirrups in the bearing zone

above the supports of the beams so as to resist bearing

stresses. They were plain mild steel bars, 6 mm nominal

diameter and were placed alternatively around the horizontal

reinforcement (fig. 6.1). In order to avoid the anchorage

problems, particularly for main reinforcement, all stirrups

were welded to form a closed link.

6.3 MATERIAL AND MIX DETAIL

Basically similar materials were used as for the bearing

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'7'd

Cb 43' in 90

to

ýF 5a

C0mCl

OA.

mC

toEOCD

.0

ccm

A.ti-0

0tocuM

OJJ

'O

CmEm

ULO

C

mL

CO

a0

.C(0

v

L

CoEO

wC7

ouN

EC

t0

C60,

U..

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172 CHAPTER 6

capacity blocks. Ordinary Portland cement confirming to

British Specification was used throughout. Coarse aggregate

was North Notts quartzite gravel with maximum size of 10 mm,

'irregular' shape and 'smooth' surface texture as classified by

BS 812. Air-dried sand from the same quarry as the coarse

aggregate was used and it was classified as zone 3 according to

BS 882. The grading curves for the fine and coarse aggregatea

are shown in fig. 3.2.

Reinforcement was plain round mild steel bars and a

typical stress-strain curve and strength properties are shown

in fig. 3.3.

The concrete mix used was identical for all the six

specimens, in order to obtain similar strengths of concrete.

The mix proportions by weight were 1: 1.96 : 2.03, with a

water/cement ratio of 0.54. It was designed to give a more

workable mix so that concrete can get through the congested

steel in the bearing zone. Workability tests gave average

values of 120 mm slump, V-B time less than 1 sec. and

compacting factor of 0.95.

6.4 CASTING AND CURING

The reinforcing cage was prepared as shown in fig. 6.1 and

6.2, placed in position on the mould and adjusted to give the

designed cover for the reinforcement. An oiled steel mould

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X73

FL, . 6.2 Typ.cal, cage ret of orcement for deep beams.

FLg. 6.3 Deep beams cost hors.zont aLLy on a vibration t abL e.

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174 CHAPTER 6

1260 x 1000 x 100 was used throughout. The depressions of 5 mm

on each side of the beam were provided by attaching a5 mm

thick plastic strip on the mould. In order to avoid the

variation of strength of concrete with height of the

specimen 1631, all the test specimens were cast horizontally

(fig. 6.3).

For each mix, all the constituent materials were weighed

inthe

required proportionsbefore being fed into

a mixer of

350 kN dry weight capacity. The materials were turned over for

about 15 secs. before the addition of the required quantity of

water. The materials were mixed for 3 minutes in order to

ensure a unform workable mix. It was then poured into the

mould and compacted in two layers on a vibrating table.

Control specimens consisted of three 100 mm cubes, three

15C) mm cubes, eight 300 x 150 mm cylinders and two

100 x 10-0 x 500 mm prisms. They were cast together for each

mix and also compacted on a vibrating table. They were

stripped from the moulds and placed in the curing room at

20 degs. C., relatively humidity of 95-100 percent, 24 hours

after casting. The test specimens were covered with damp

hessian for 3 days, watered constantly and then transferred to

the curing room.

Compressive strengths were obtained by three 100 and

150 mm cubes and four 100 x 150 mm cylinders. Tensile strength

was assessed from four 100 x 150 mm cylinders by the splitting

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175 CHAPTER 6

cylinder test and from one 100 x 100 x 500 mm prism by the

modulus of rupture test. The remaining prism was used to

obtain the Young's Modulus and Poisson's ratio of the concrete.

Similar procedures were applied for casting and curing of the

tests and control specimens so as to have a better indication

of the strength of concrete in the test specimen. The control

tests were made at the time when relevant beams were tested and

were according to E+S 1881. The properties of each specimen are

listed in table 6.1.

6.5 INSTRUMENTS AUD TEST PROCEDURE

Strains on the surface of the concrete were measured by

Demecgauges with

100mm gauge

lengthexcept near

theedge

where space is not available for a 100 mm gauge length, a 5() mm

Demec gauge was used. Strains were measured on twelve

particular sections of the beam (fig. 6.4). Section 1 was a

vertical section along the centre line of the support, and

measurements were made at the level of the reinforcement both

vertically and horizontally soas

toobtain

thevertical

and

horizontal strains. Sections 2 and -3 were vertical sections

similar to section 1 except that they were 200 mm from

section 1 and of the centre of the beam respectively.

Sections 4 and 5 were horizontal sections 20 mm below the top

and 1() mm above the bottom of the beam respectively. They were

chosen to measure the horizontal strains at the top and bottom

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7'C

Con reseLve Tenst. Le Young's Poast.onSpec.

Span SPA

htou(00) fo_Uso) fo ft(oU.. ) tt(pm. ) Modulus ratt. o

Pu

no.

(an. )

dept

ratio N/mn22

N/ ne N/mat N/ mat N/mm22

Ii'/nn IN

Present tnveatt. gatton

081 1000 1.0 67.5 62.7 46.9 3.57 4.06 34.2 0.149 1415

D82 1100 1.1 70.1 58.8 45.4 3.36 3.98 31.3 0.150 1400

083 1000 1.0 74.6 66.2 50.9 3.57 4.06 34.9 0.169 1700

084 900 0.9 69.9 59.4 50.2 4.03 4.53 34.9 0.175 1960

D85 800 0.8 71.4 62.8 47.5 3.82 4.24 35.4 0.173 1975

086 700 0.7 68.1 65.6 49.8 3.77 5.62 32.5 0.158 1980

Notes For aLL the apeci. raen$ a- 1260mA b- i00na h- i000ma.

TabLe 6A Concrete propret t es of the test beams (OBI-0B6).

span/320

Sectton4

ý-"200 neý

I\//

ioo

\On

cu cn '6

c

0IUm m\

ý\ N)l mN

cn tnl wN\\I

nýSeýýi

ýý

r\ \\0or%

eö

Section 5/ T

tospan

FLg. 6.4 Sect Lons on the surface of the beam

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177

Spri

test

specLnen

; eal bean. ng)Late

F , g. 6.5 Loadingg mechant sm of tests wt, h deep beams.

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1 6wýr

ýý, r`yKyýý

yt J, ti'f

9

CLr" i+ y`s. if" [ '. "ariý 71k' ' i/

Lý

.}

ý_

Oi .T "i

..ký

ý ý

i O

ý_ .`c, t}h'$...ý

". ; iº...wig

Iz: CA

m

r. SaW, I

r

wI

K 'J 1L

lot

c

.{ý

Ö

O

Xi ý"

CDIon

F t..

ýý

`Ya

8/8/2019 uk_bl_ethos_373477


179

of the specimens.

CHAPTER 6

It was pointed out by Ramakrishnan [73] and Al-Nijjam 163]

that failures of deep beams were in fact failures of the strut

and tie system. Ramkrishnan had suggested that for uniformly

distributed loaded beams, the strut lies on the line between

the support and a third span point on the top of the beam.

Strain measurements on sections 6-12 were specially designed to

investigate this. Sections 6-9were perpendicular

to the

direction of this compressive strut and strains were measured

along the direction of the strut so as to give an idea of the

distribution of compressive strain. Sections 6,7,8 and 9

were at. 100,200,400 and 800 mm respectively from the centre

of the support. Sections 10-12 were lying parallel to the

direction of the strut. Section 11 was passed through the

centre of the support while section 10 was 100 mm inside and

section 12 was 50 mm outside the centre of the support.

Strains were measured both along and across the strut so as to

give compressive and tensile strains at different positions.

Strain of the steel was measured by electrical resistance

gauges and recorded by a data-logger. Strain in the main steel

was measured in three positions; along the centre line, along

the centre of the compressive strut and at the end of the main

reinforcement. Strain in the web reinforcement was measured

along the centre-line of the compressive strut. For those

beams with bearing steel, strain was also measured at the

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180 CHAPTER 6

appropriate positions. Detailed positions of the strain

measurements are shown individually in chapter 7.

When a specimen was ready for test, it was taken out from

the curing room together with the control specimens. A thin

coat of white emulsion was applied. Demec points were fixed

into positions. The specimen was then carefully placed on to

the testing rig. Two bearing plates, each 140 x 100 x 50 mm,

were placed in an appropriate position to give the correct span

of the beam. A roller and a half roller were put underneath

the bearing plates and they were supported by two I-section

beams placed on the floor. A spreader beam was put on the top

of the specimen to obtain an uniformly distributed load. In

order to have a better contact between steel and concrete,

plaster of paris was applied onthe

surfacesbetween the

spreader beam and the specimen and on the two bearing plates

and concrete. The mechanism of loading is shown in figs. 6.5-6.

The beam was then checked for position, vertically

verified and dial gauges were placed and adjusted both under

the base and at the back of the beam for measuring deflection

and horizontal movement respectively. After taking the initial

readings of all the gauges, load was applied in constant

increments of 100 kN. At each stage of loading, gauge readings

were recorded, cracks were marked on the surface and the load

at which it was observed was noted at the end of the crack.

The widths of the cracks were measured by a hand microscope

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lei CHAPTER 6

with a magnification of 40 and graduation in the eyepiece scale

of the microscope corresponded to a crack width of 0.02 mm.

This procedure was repeated until the specimen failed, it was

then removed from the test rig and photographed to record the

final crack pattern. The control specimens were tested on the

same day.

6.6 BEHAVIOUR DURING TESTS

The appearance of the test specimens and their crack

patterns after failure are shown in figs. 6.7-12. The numbers

shown at the ends of the cracks should be multiplied by 10 in

order to obtain the load in kN. Fig. 6.13 shows the development

of crack width withload for three types

of crack;flexural,

shear and bearing.

In beam DB1, fig. 6.7, the first crack to appear was a

flexural crack around the middle bottom section of the beam at

400 M. As load increased to 500 kN, more flexural cracks were

observed in the bottom of the beam. They extended upwards, to

a height of 500 mm above the bottom edge of the specimen, and

cracks widened gradually as load was increased. Cracks in the

middle of the beam extended more quickly than those near the

supports. At 900 kN, a diagonal crack was observed above the

right hand support extending at an angle of 68 degs. which

agrees well with the direction of the imaginary compressive

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182 CHAPTER 6

strut with the horizontal. It extended its length and widened

rapidly, finally coming to a point 200 mm below the top of the

specimen at 1300 M. At 1000 kN, a similar diagonal crack was

formed above the left hand support. As load was further

increased to 1300 kN, vertical cracks (bearing cracks) were

formed above the two supports. Their length extended slowly

but the crack width increased even more rapidly than the

diagonal cracks. Finally, failure of the beam took placeas

the vertical cracks widened so much that pieces of concrete

fell away from the right hand support which could not substain

more load. The beam failed with an audible report and this was

considered as a bearing failure.

Beam DB2 behaved similarly to DB1 in the first stages of

loading. A flexural crack formed at around 500 kN at the

bottom of the beam and extended to 550 mm above the bottom of

the specimen. At 900 kN, a vertical bearing crack was found at

the right hand support and this increased its width vigorously

as shown in fig. 6.13. Shear cracking was found at 1000 kN.

This formed at the end of a flexural crack near the support and

began to extend its length at an angle of 70 degs., which is

again in the direction of the predicted compressive strut, and

stopped when it reached a height of 600 mm above the support.

In the presence of bearing steel, less vertical cracks were

found above the support, as shown in fig. 6.8. However, the

support in this specimen was too near to the edge of the beam

and bearing cracks near the edge were uncontrolled. These

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183 CHAPTER 6

bearing cracks were widening so quickly that at 1400 kN, the

bearing zone is totally distorted and affecting the stability

of the beam. Beam DH2 had not actually failed but it slid off

the testing rig and test was abandoned.

Beam DB3 was identical to DB1 except that reinforcement

was added in beam DB3 near the support in the bearing zone. It

behaved rather similarly to DB1 with the formation of flexural

cracks at 400 kN. Again, these extended upward to a height of

700 mm above the bottom edge of the specimen and widened

gradually as the load was increased. The formation of flexural

and shear cracks was observed at loads of 400 kN and 900 kN

respectively, as in beam DB1. The shear cracking was inclined

at an angle of 70 degs. with the horizontal and agreed well

with the direction of the predicted compressive strut by

Ramakrishnan (73]. It extended almost to the full height of

the specimen. However, the behaviour of the bearing cracks was

the major difference between DB1 and DB3, (fig. 6.9). Beam DB3

had more controlled bearing cracks. They occurred at more or

less the same load as in beam DB1 but crack widths remained

approximately constant (0.04 mm) in beam DB3 whereas extensive

widening up to 0.27 mm at 13DÜkN took place in beam DB1

(fig. 6.13). Moreover, the bearing cracks in beam DB3 remained

short and concentrated in the region of 100 mm above the

support while those in beam DB1 were more widely spread to

200 mm above the support. This shows the effectiveness of the

form of bearing steel used. With this reinforcement, the

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184 CHAPTER 6

bearing zone above the support was held as one unit and beam

DB3 was prevented from failing in bearing, (fig. 6.9b). Loads

could then be increased beyond 1415 kN and finally, failure of

this specimen was by shearing of the concrete along the crack

in the centre of the compressive strut, with the buckling of

vertical steel above the support, together with the dowel

failure of main reinforcement at 1700 kN.

Despite differences in span, beams DB4, DB5 and DB6

behaved very similarly to each other. In general, the first

crack which appeared was a flexural crack in the bottom middle

section of the beam at around 500 to 600 kN. The length and

width of these flexural cracks decreased with the span of the

beam, (fig. 6.10-13). Shear cracks appeared at 700 to 1000 kN

and they were inclined approximately in the direction of the

suggested inclined compressive strut, rising to a height of

900 mm above the bottom of the specimen. As load was

increased, the crack width increased gradually. Around 1200 to

1600 kN, bearing cracks emerged but crack widths did not

increase with load and remained at 0.02 mm until the beam

failed. At later stages of the test, around 1800 to 1900 kN, a

new vertical crack was formed. This was vertically above the

inner edge of the support, originating at a height 600 mm above

the support and extending in both directions until the beams

failed by vertically shearing of the two concrete blocks on

either side of the crack, as shown in fig. 6.10-12. Beams DB4,

DB5and

DB6 failedwith similar

loadsof

1960,1975and

1980 kN

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185 CHAPTER 6

respectively. With the exception of beams D81 and DB2, all

specimensfailed

quite gently with a reasonably audible report.

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I8&

iý Lam.2.-

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DB1Vk load

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90

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Twi

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of beam DB3.

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DB... :;[sov t, load i6o

100

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1,10 9000

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Ft,g. 6. i0 Crack pattern of beam DB4.

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191

Ft,g. 6.12 Crack pattern of beam DB6.

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19-2

in

lfm

Im

Z 12ý

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of C c3 c1 C5co, / o-j, 01 PI 02 03 cd 05 C6 c7 oh nyCrso kwt ds h (mm. ) Cr. o k width (mm. )

Fig. 6.13. Load ega i nst crack width in beams DB 1-OB6

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193 CHAPTER 7

7 EXPERIMENTAL RESULTS - SHEAR STRENGTH OF REINFORCED

CONCRETE DEEP BEAMS

7.1 INTRODUCTION

In this chapter, the strains on the surface of the

concrete and in the reinforcement measured during the tests and

the deflections of the specimens will be shown graphically and

discussed in detail. At the end of the chapter, a proposed

model of failure mechanism is drawn up and predicted failure

loads both by the proposed model and formulae from other

researchers are compared with the experimental failure loads.

7.2 STRAIN DISTRIBUTION ON CONCRETE SURFACE

Strain distributions for every measuring section on the

concrete surface (fig. 6.4) are shown in figs. B. 1-18.

Experimental strain distributions are plotted together with the

theoretical values obtained by FEM analysis. Details of their

behaviour will be discussed in later paragraphs.

Fig. 8.1 shows the vertical strain distribution in

section I (fig. 6.4), above the support, for all the six tested

beams. In general, the strain distributions are similar in all

the tested beams. Vertical compressive strain increases

gradually as the support is approached and have a maximum at a

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194 CHAPTER 7

point 100 mm above the bearing plate. Below this maximum

point, the vertical compressive strain decreases slightly.

Strains obtained by FEM analysis are plotted in dotted lines on

the same figure (fig. B. 1). It can be seen that the

experimental and theoretical values agree well for small loads

(less than BOO kN) on the upper section of the beams. At

higher loads, the experimental strain is larger than predicted

by FEM analysis particularly in the regions near the supports.

This is probably due to the formation of a wedge in the

concrete above the bearing plate and large compressive strains

correspond to the action of this wedge of concrete an the

concrete block.

Figs. B. 2 and B. 3 show the vertical strain distributions in

sections 2 and 3 respectively. Strain distributions are

similarin these two sections and with all the test beams, but

they are different from the distribution in section 1. In

sections 2 and 3, the vertical compressive strain has a maximum

at the top and decreases gradually towards the depth of the

beam. At a position 100-200 mm from the bottom of the beam,

the vertical strain becomes zero, and their vertical tensile

strain emerges and increases in magnitude towards the bottom of

the beam. Occasionally, maximum vertical tensile strains were

recorded at 100 mm above the bottom of the beam (DB3 and DB6).

Analysis from FEM shows similar results, a maximum vertical

compressive strain occurs at the top of the beam, it is

constant over the top 200 mm and then decreases more or less

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195 CHAPTER 7

linearly with the depth of the beam. In section 2, the

vertical compressive strain decreases more rapidly at the

position 200 mm from the bottom until a point 100 mm from the

bottom of the beam. Thereafter, the vertical compressive

stress becomes constant. In section 3, the position at which

the compressive strain becomes constant depends on the span of

the beam. Larger spans have a larger constant region which

varies from 250 to 100 mm from the bottom of the beam.

The horizontal strain distributions in sections 1-3 are

shown in figs. B. 4-6 respectively. The horizontal strain

distribution in section 1 (fig. 8.4) is very similar to the

transverse strain distribution along the loading line of the

bearing concrete blocks which has been discussed in chapter 4

(fig. A. 27-34). It has a high compression region close to the

bearingplate and

thena

tensionzone with maximum

tensile

strain occurring at 100 mm from the loaded surface. This

agrees well with the assumption that maximum tensile strain

occurs at a distance al from the loading surface. The tensile

strain gradually diminishes and comes to zero at 900 mm from

the bearing plate. Theoretical analyses with FEM have

tensile strain is not so high as the experimental values.. This is again due

identical results except that the maximum A to the formation of

cracks and a wedge in the bearing zone. It can be seen from

fig. B. 5 that at low load (less than 800 kN) the horizontal

strains in section 2 are very small and approach zero for the

region 200 mm above the bottom and 100 mm below the top of the

beam. For the section below this region, tensile strains were

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196 CHAPTER 7

recorded and for the section above, compressive strains

measured. This is due to the flexural behaviour of the beams.

However, at higher loads intercept the strain-measuring line

creating large strains which correspond to large strain in the

middle section section of the beam. In section 3, large

tensile strains were recorded at the bottom of the beam for

loads greater than 800 kN. The presence of these tensile

strains is due to the formation of flexural cracks at the

bottom of the beams. Therefore the magnitude of these tensile

strain and the affected region is dependent on the span of the

beam. At the top of the beam, small compressive strains were

recorded. In general, theoretical and experimental horizontal

strain distributions in section 1-3 agree well for the region

which is not affected by cracks.

Fig. 8.7 shows the distribution of horizontal strain across

the top of the beams (section 4). It can be seen that

compressive strain was found in all six beams between the two

supports. There is a parabolic distribution of compressive

strain with a maximum at the centre of the beam. Slight

tensile strains were found over the supports. However

analytical results from FEM produce a flatter distribution of

horizontal compressive strains between the supports and a rapid

increase in tensile strain over the supports. In most beams

experimental compressive strains between the supports are

larger than the predicted values while the experimental tensile

strains overthe

supports are smallerthan the

corresponding

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197

theoretical values.

CHAPTER 7

Horizontal strain at the bottom of the beam (section 5) is

shown in fig. 8.8. At small loads (less than 400 kN),

experimental tensile strain between the supports has a

magnitude similar to the analytical values by FEM. At higher

load (greater than 800 kN), the experimental strain

distribution fluctuates due to the existence of flexural

cracks.Therefore,

comparison cannotbe

madebetween

experimental and analytical values at higher loads. However,

rapid changes in strain around the support are recorded by FEM

analysis which seem to be unreasonable in practice.

It was suggested by Ramaskrishnam 1737 that failure of

deep beams was in fact failure of a strut and tie system.

Sections 6-12 were specially designed to investigate the

behaviour of this compressive strut. Sections 6-9 are

perpendicular to this strut, transverse strains being measured

at positions 100,200,400 and 800 mm from the support

(fig. 6.4). Fig. B. 9-12 show the compressive strain across this

compressive strut in section 6-9 respectively. It can be seen

that the magnitude of the compressive strain falls dramatically

at sections further away from the support. In section 6,

maximum compressive strain is found in the centre of the strut

and gradually falls to zero at 150 mm from the centre. In

beams DD4-6, a sudden decrease in compressive strain is found

at 50 mm to the left of the strut; this is believed to be

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198 CHAPTER 7

affected by the presence of shear cracks. Section 7 has a

similar distribution of strains as in section 6 but with

smaller magnitude. Again the maximum strain is found at the

centre of the strut and becomes zero at 100 mm to the left of

the strut. Similar to section 6, a sharp decrease in

compressive strain, affected by shear cracking, is found at

100 mm to the left of the strut in beams DB3-5. It is

interesting to note that to the right of the shear crack high

compressive strains are found but little or no strain was

recorded to the left of it. Therefore, the shear crack is

actually the dividing line of the compressive strut with the

beam. Sections 8 and 9 have similar magnitudes of compressive

strain, these are much less than in sections 6 and 7. The

distribution of strain is comparatively uniform and

occasionally, depressions are present due to the presence of

shear cracks.

Figs. 6.13-15 show the distributions of longitudinal strain

in sections 10-12 respectively (fig. 6.4). In section 10

uniform compressive strain was recorded along the section.

Compression is small in magnitude, around 700 micro-strain,

even at 1600 M. Sections 11-12 have a distribution of strains

similar to section 1 but with larger magnitudes (fig. 8.1). An

increase of compressive strain occurs as the support is

approached and there is a maximum at 100 mm from the support.

Below this point compressive strains begin to decrease.

Transversestrain

distributions insections 10-12 are shown

in

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199 CHAPTER 7

figs. B. 16-18. Compressive strains are found at the top for all

three sections. These compressive strains decrease slowly and

become zero at 900 mm from support. Transverse tensile strains

begin to emerge below this point. Rapid increases in tensile

strain are recorded in section 10 with loads greater than

800 kN. It increases linearly to a point 300 mm from the

support and then decreases. This rapid increase in strain

actually indicates a rotational movement of the concrete block

from the shear crack. The decrease of transverse strain below

300 mm from the support is due to the presence of four main

steel bars of larger diameter which help to hold back the

concrete block. Only small strains were recorded in

sections 11-12 showing that the concrete block is not rotating

along these sections but rotating in section 10.

7.3 STRAIN DISTRIBUTION IN REINFORCEMENTS

The distributions of strain in main and web steel are

shown in figs. B. 19-24. The strains in the bearing steel are

givenby figs. B. 25-29.

On each figure, reinforcement details

are drawn; the position and direction of every strain gauge is

marked on the reinforcement and numbered beside it. It is

drawn together with the variation of strain in each gauge with

loads.

Gauges 1-4 were mounted on the reinforcement in the centre

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200 CHAPTER 7

of the beam. As load is increased, strain increased linearly.

At around 400 to 1000 kN, depending on the span of the beam,

strain increased at a faster rate. This indicates the flexural

cracking of the beam at that load. In the presence of flexural

cracks, of course, strain increases more rapidly remains at

this rate until failure of the beam. It is noted that none of

the gauges shows yielding of the main reinforcement in this

position. Gauges 5-8 were placed along the direction of the

imaginary compressive strut. They again increase in strain at

two rates, before and after the formation of shear cracks at

800-1000 kN. With the exception of beams DB1 and Db2, which

did not fail in shear, all gauges show yielding of the

reinforcement in this position. In most beams, except DB2, the

upper main steel exhibits larger strains. With the exception

of beam D82, gauges 9-12 are placed on the main reinforcement

above the outer edge of the bearing plate. These four gauges

have similar strains and they all have demonstrated an enhanced

rate of increase in strain with loads. In beam DB2,

gauges 9-12 were actually in the main reinforcement above the

bearing plate. They had different characteristics to those in

beams DB3-6. Gauge 11 recorded the highest strain, it yielded

at an early stage (400 kN) but gauge 10 is virtually

unstrained. This outstanding characteristic is believed to

stem from the small edge distance from the support. Gauge 34

is at the end of the bottom main reinforcement, it is designed

to monitor the anchorage problem of the main reinforcement.

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201 CHAPTER 7

Strain in this gauge is low, therefore, it is accepted that

there is good anchorage of the main reinforcement.

Gauges 13-17 were in the horizontal web reinforcement in the

direction of the suggested compressive strut. It is found that

with the exception of gauge 13, all gauges show tensile strain

in the reinforcement. Compressive strain in gauge 13 shows a

constant increase in strain with load while all the others show

two stages of increase before and after the formation of shear

cracking. In beam DB2, gauges 14-17 are virtually unstrained

as DB2 has not failed in shear and no shear crack is found. In

beams DB1 and DB3-6, only small strains are found in

gauges 14-17 at loads below 700-900 kN, at higher loads, rapid

increase in strain is taking place. Generally, web steel near

the bottom has larger strain and yield occurs before the

specimen fails. This suggests that shear cracking originates

at a point above the main reinforcement but below the web

steel. Gauges 18-20 were installed in the vertical web steel

and again along the direction of the imaginary shear crack.

Compression found in this reinforcement and higher compression

was recorded at the reinforcement near the bearing plate.

Bearing steel in the form of small interlocking stirrups

is inserted in beams DB2-6 to prevent bearing failure above the

supports (figs. B. 25-29). Gauges 21-26 are placed on the centre

stirrup above the support while gauges 27-29 and 30-32 are on

either side of the central stirrups. In beam DB2, only small

strains are recorded in the gauges, especially for

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203 CHAPTER 7

beams DB3-6 and no buckling was observed for these depth to

thickness ratios.

7.5 SHEAR TRANSFER BY AGGREGATE INTERLOCK AND DOWEL ACTION

From the behaviour in failure of deep beams in this

investigation, it is believed that the shear strength of deep

beams is determined by aggregate interlock and dowel action. A

review of past researches in this area is discussed in the

following paragraphs.

Shear forces can be transmitted across a crack by the

interaction between two rough surfaces of the crack (aggregate

interlock) or by the dowel action of tensile reinforcement

(dowel action). However, aggregate interlock and dowel action

are interdependent and not easy to separate. During the

initial stage of crack formation, aggregate interlock plays an

important parts in shear resistance. As external shear force

is increased the diagonal crack is widened and lengthened by

the rotation and shear displacement of the beam. When the

crack meets tensile reinforcement, part of the shear resistance

is taken up by the dowel action of the reinforcement. This

leads to splitting of the concrete at the level of the

reinforcement and increases the crack width at a higher rate.

Therefore, the shear force taken up by aggregate interlock is

further reduced. Houde and Mirza 1353 stated that for a beam

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204 CHAPTER 7

without web reinforcement and after cracking, shear carried by

aggregate interlock, concrete in the compressive zone and dowel

action of the main steel was 50,30 and 20% respectively. They

are distributed as shown in fig. 7.1, [79].

Some attempts had been made to separate the action of

aggregate interlock and dowel action and then investigate the

effect of them with various parameters. Houde and Mirza (35]

eliminateddowel

action withthe

absence of reinforcement

across the crack. A tensile crack was introduced by applying

direct tensile force on either ends of the block and a

predetermined crack width was maintained by restraining the

specimen as shown in fig. 7.2. Shear force was applied across

the crack and shear displacement was measured. It was found

that the shear stress carried by aggregate interlock ranged

from 0.5 to 1.2 N/mm`. The magnitude of the stress was mainly

dependent on the crack width and was proportional to the square

root of the cylinder strength of concrete. However, it was

independent of the size of aggregate used in the concrete.

Twenty seven tests of aggregate interlock was carried out by

Paulay and Loebar (70] with constant and variable crack widths

under constant or variable restraining forces. The test

specimen is shown in fig. 7.3. They arrived at similar

conclusions to Houde et al in that the aggregate size, shape,

hardness and cement mortor had no noticeable effect an the

shear carried by aggregate interlock. The largest single

factor affecting shear resistance was the width of a crack

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X05

Cracked

regt, on

Conpresai. on zone Shear stress &atri. butt on

Ln the conpreee. on zone

x (30%)

Shear etreae by

aggregate interlock( 50% )

Shear stress by

dowel. oct i, on( 20%)

F .,g. 7.1 0' st rt. but L. n of shear stress along a shear crack.

P

L

___=_ý

ý_ __^ 11II

ý_====ýJRet.nforcenent to prevent

fLexurol. & dkagonot crocks

AdJustobLe

et.de roLLer

Crack formed

by Cens. on

III

Crock open. ng

II I

c=--== J3

P

mLLd steel

g"Load cell,

Ft,g. 7.2 Aggregate LnterLock test by Houde and Mt, za.

dt.o. bars

TensLon crackalong shear plane

3-3/8' dLa.

etLrrupeShear surface

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206 CHAPTER 7

across which shear stress were to be transfered. Tests with

restrainingforces

showedthat the force

actingat right angles

to the shear plane required to maintain constant crack width

was considerable. Typical results are shown in fig. 7.4.

Logarithmic regression analysis was performed and the following

relationship obtained.

fL = 0.473 fa1.03 (psi) (7.1)

It was suggested that for design purposes the mean shear stress

could be approximated by a straight line corresponding with a

coefficient of friction of p=1.7, while p=1.4 was recommended

by the ACI Building Code C1]. Dowel action of the

reinforcement can be eliminated by placing the reinforcement

within an oversize duct and a shear force is applied on either

side of the crack. This method of testing the effect of

aggregate interlock was employed by Millard and

Johnson 160,617. They found that the shear stiffness across

the crack and the ultimate shear stress both decreased as the

initial crack width was increased. The shear stiffness also

diminished with increasing shear displacement which was

associated with crack-widening regardless of the size of the

initial crack width. Crack-widening was sensitive to the

stiffness normal to the crack plane. Furthermore, it was

believed that shear is resisted by a combination of crushing

and sliding that cannot be represented by a conventional

friction model.

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1200

1000

a

"

8000

cU

600mc

4000

0

ý. 200

ýqx

ti7%

xA i7

x

if Se

LfL- 0.473r

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rL - 0.582 ,

0 200 400 600 800

fL Hestroi. rn.ng stress (p«.

Ft g. 7.4 Shear vs restratning stress by Paulay.

I. -

dId 1

V

1 'f-

IL. Plo-v - Hi 1'l-VN

FLEXURE SHEAR KINKINGV. 211

34ý V-A,; /j3 V-A, coed

Ft.g. 7.5 The mechancsmof dowel, action.

------------ k

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1 Load ceLL1

-1

Loadý-------- Load

cell f ceLL T

V-ýd

"N/: \

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208 CHAPTER 7

Dowel action is believed to be come from three sources:

(1) the flexure ofthe

reinforcement,(2) the shear across the bars, and(3) the kinking of the reinforcement.

These shear mechanisms are illustrated in fig. 7.5 associated

with the shear strength expressed in terms of the diameter of

the bar and its strength. However, when the dowels are large

the shear capacity of a dowel is determined by the strength of

the surrounding concrete rather than the yield strength of the

reinforcement. Thirty push-off type specimens were tested by

Pauley 161] with varied surface preparation from smooth to

keyed surface and three different amounts of reinforcement

across the shearing surfaces. It was found that the dowel

force is proportional to the total steel area (square of the

diameter of the reinforcement). This infers that shear and

kinking are the principal mechanisms of dowel action as the

dowel force produced by flexure of the reinforcement is

proportional to the cube of the diameter of the reinforcement.

Thirty two beam-end specimens shown in fig. 7.6 were tested by

Houde and Mirza (35] to determine the ultimate dowel strength

under dowel acting alone or dowel action combined with

predetermined pull-out forces. Dowel cracking loads were found

to be directly dependent on the beam width and the concrete

tensile strength. The bar size and the embedment length did

not have any influence on the dowel cracking load. This

contradict with Pauley's finding in which dowel force is

proportional to the total area of steel. The pull-out farces

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209 CHAPTER 7

had no effect on the dowel capacity which did not exceed 207. of

the shear capacity of the beam. The dowel cracking load may be

expressed by

Df = 40-b-fc'1/3

(7.2)

Similar specimens were used by Millard and Johnson C60] in

aggregate interlock tests, except that this time no duct was

surrounding the reinforcement. The shearing surfaces were

smoothed by casting each specimen in two stages and separating

them with two layers of thin polythene sheeting. The

experimental results show that increasing the diameter of the

reinforcement resulted in a higher shear stiffness and ultimate

stress. It also increased the tendency for the smooth crack to

widen. An increase in the strength of the concrete had only a

small effect on the behaviour but an increase in axial force inO\

the reinforcement resulted in a lower shear stiffness and

ultimate shear stress together with an increased tendency for

crack widening. This is due to some localized damage and

softening of the concrete by the axial tension force.

However, interaction between aggregate interlock and dowel

action makes the combined effect in reinforced concrete more

complex than if considered separately. For instance, in dowel

action tests, elimination of aggregate interlock by

artificially smooth cracks also suppressed the tendency for

crack faces to override which causes widening of the crack,

increase in axial tension force in the reinforcement and

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210 CHAPTER 7

reduction in the shear stiffness of the dowel action. In

aggregateinterlock tests,

elimination ofdowel

actionby

means

of an oversize duct also removed the local bond between

reinforcement and concrete. This could lower the axial

stiffness and thus underestimate the shear stiffness provided

by aggregate interlock. Also the internal crack widths are

dissimilar for two specimens with the same surface crack width

but with bonded and unbonded reinforcement. Crack width is a

prime factor of aggregate interlock; therefore these two

specimens cannot be expected to have similar shear stiffness.

Hofbeck et al and Mattock [33,57] have presented some

tests on the combined effect of aggregate interlock and dowel

action. Their test apecimens included orthogonal, inclined or

parallel reinforcement of initially cracked or uncracked

concrete along the shear plane in fig. 7.7. It is found that

the spacing and diameter of the reinforcement did not affect

the linear relationship between p. fy and the ultimate shear

force Vu. Reinforcement with high yield strength and a small

yield plateau give higher shear resistance. A structural model

was constructed according to the observed mode of failure shown

in fig. 7.8, with the following assumptions:

(1) The stress in the reinforecment, fs

component of relative displacement

sides of the shear plane in

reinforcement.

(2) The relative displacement at ultim

is proportional to the

of concrete on the two

the direction of the

ate, Su is constant and

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V

ord in no.nd size ofLupe

ShearPlane

stLmips1.5ms dLa.

V3/e di.a.a i. rupivanes n

ShearPLone

LoP%.

V

yvv

Ft,g. 7.7 Push-off test by Mattock and Hofbeck.

A F, IFT

i"ia ý_' '1 (xy

.. ctyýy I Iý 0

N. yNý%II

Y/ ro Cr" rt

of

1F

I F.

Ft.g. 7.8 Shear model. of tnt. tt oL uncracked spect. men by Mattock.

Yppv

k

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212 CHAPTER 7

equal to that necessary to produce a strain cy in the

reinforcement in the shear plane.

During the tests on initially uncracked specimens, diagonal

cracks at short intervals were formed across the shear plane.

Movement of the two halves of the specimen relative to one

another occurs by rotation of the concrete strut between the

diagonal cracks. Therefore, the strain Es in reinforcement at

angle e to the shear plane is

Es = C-Su*Cos (9Ü+a-e) (7.3)

where C= constant and when e+90, cs=may

therefore E=E seca"cos(90+m-e) (7.4)sy

stress fs can be expressed as:

r -fy O<0<2m-90

fs =j fy. secm. cos(90+(K-e) 2a-90<e<90 (7.5)

tC fy 90<e<180

Total steel force perpendicular and parallel to the crack are

respectively

F= AS f -sin 2e/Sb (7.6)

Fv = AS -fs -sin(2(3) (7.7)

At failure, the direct stress ox acting across the shear

plane as a result of the stresses in the reinforcement and any

externally direct stress aex is

oaa = F/b-d + aex (7. A)

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213 CHAPTER 7

where b= width of the shear planed= length of the shear plane.

Thus Txy can be found by the Mohr circle (fig. 7.8) with a given

compressive and tensile strength of concrete. The ultimate

shear force can be estimated as

V=F+ K-T (7.9)uv xy

where K may be taken as 0.84

For initially cracked specimens, movement of the two

halves of the specimen was along the crack (shear plane). The

faces of the crack was rough, and hence when slip occurs, the

crack faces were forced to separate. The relative displacement

äu which takes place is assumed to be in a direction at angle v

to the crack,

where 'r" = arctan}t (7.10)

= coef. of friction between crack faces,

taken as 0.8 ()f=38.7)

Thus using the foregoing assumptions, the strain at ultimate Es

in reinforcement at an angle e to the crack is given by

Es = C2-Su-cos(e+'") (7.11)

similarlythe

stress at ultimate ofthe

reinforcement at angle

e to the crack is

I -fy

O<e<90-2w

fs =i -fy.

cosecw-cos(e+w) 90-2v<e<90 (7.12)

L fy 90<e<18O

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214 CHAPTER 7

It has been proposed C58] that for the case of

reinforcement crossing a crack at right angle to the ultimate

shear transfer strength may, for the purpose of design, be

taken as

ýu = 200 + O. ep-fy < 0.3fc' c7. i3)

The first term in Eq. 7.13 represent the shear transfer by dowel

action of the reinforcement. But this gives the lower bound

value for design purpose. Since, shear force along the crack

is resisted by dowel force perpendicular to the reinforcement,

therefore shear force is proportional to sine times the dowel

force perpendicular to the reinforcemment (i. e. )

Vu =k -D f' sine (7.14)

Dowel force is produced by steel stress in the direction

perpendicular to the crack, thus dowel force is proportional to

sine times steel stress (i. e. )

Df = k"sine-fs (7.15)

Combining Eqs. 7.14 and 7.15,

Vu =k "sin2e -f (7.16)

Therefore, Eq. 7.13 is modified to give the mean value of shear

strength as

Vu = 404b-h-sin2e + 0. OF +F (7.17)

where F= A5-h-fs-sinze

& Fv = -As-h-fs-sin(2e)

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215 CHAPTER 7

7.6 PROPOSED FAILURE MECHANISM OF REINFORCED CONCRETE DEEP

BEAMS

Consider a reinforced concrete deep beam with orthogonal

reinforcement subjected to two point loading as -shown in

fig. 7.1Oa. It had been observed from experiments that the

shear crack was usually formed along the line joining the inner

edge of the support and the outer edge of the loading plate.

Equilibriumof

the forcesalong

thecrack were maintained

by

the shear in the compression zone, aggregate interlock,

stresses of any reinforcement across the crack and the tensile

strength of concrete. Various forces acting along the crack

are shown by the free body diagram in fig. 7.1Ob. For the

purpose of analysis, certain assumptions have been made and

they are as follows:

(1) A crack is formed along the line joining the inner edge ofthe supporting plate and the outer edge of the loading

plate.

(2) Movement of the concrete block is by the rotation about the

centre of forces in the compression zone (point A) andmovement of each point along the crack is perpendicular tothe direction of the crack.

(3) Strain in the reinforcement across the crack, cs is

proportional to the displacement of the concrete, Su along

the direction of the reinforcement.

(4) Ultimate displacement between two concrete surface, Su is

equal to that neccessary to produce yielding of the

reinforcement.

(5) A reinforced concrete deep beam is said to fail when the

ultimate displacement, Su has taken place at the

bottom-most of the main reinforcement.

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>i(o

vu vu11

6ual. nO

ducoseeb

ca

ca

Yu

I+1

Ar

lotto+ta,X

rL61-1 ro,

t

rLh I_.

Yu

(ä

-1 ý-, I-

I dh

d

FIg. 7. i0 Proposed faLture mechanism.

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217 CHAPTER 7

As shown in fig. 7.10b, the shear crack is formed at an

angle e with the horizontal and according to the

assumptions 3-5 above, we may write

Esh = k-Su-sine (7.18)

if when e=90 degs., Esh-Eyh Eyh=k-Su

thus Esh = Eyh-sine (7.19a)

similarly, Esv = Eyv'Cosa (7.19b)

andfsh = fyh'sine (7.20a)

fsv = fyv'Cosa (7.20b)

where Esh'Esv = Strain in the horizontal and vertical

reinforcement respectively.Eyh, Eyv = Yield strain in the horizontal and vertical

reinforcement respectively.fsh'fsv = Stress in the horizontal and vertical

reinforcement respectively.fyh, fyv = Yield stress in the horizontal and vertical

reinforcement respectively.

Assuming a triangular distribution of stresses in the

reinforcement resulting from the rotation of the concrete block

about the centre of compression in the compression zone

(point A). Tensile force in the reinforcement can be

calculated as

Th = Ash 'f sh (yh-d ') / (d-d ')

= Ash . fyh -sine (yh-d') / (d-d') (7.21a)

Tv = Asv -fsv

(H-Xv. tane-d ') / (d-d' )

= Asv-fyv-cose(H-Xvtane-d')/(d-d') (7.21b)

where Th, Tv = Tensile force in the horizontal and vertical

reinforcement respectively.d' = Depth of the centre of compression in the

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218 CHAPTER 7

compression zone (point A) from the top of the

beam.

yh = Depth of the horizontal reinforcement from the

top of the beam.

X= Distance of the vertical reinforcement from thev

inner edge of the support.H= Total height of the beam.

Stresses due to the reinforcement perpendicular and parallel to

the crack are

fLh = Th-sine/(b-Sh. cosece)

Ash-fyh-sin3a yh-d'

(7.22a)

b- Sh d -d '

fph = Th -cose/ (b -Sh' coseas)

Ash"fyh-sin2e"cose yh-d'

(7.22b)

b"Sh

d-d'

fLv = Tv"sine/(b"Sv-secs)

Asv

-fyv-cos'

e H-Xv

-tans-d'

(7.22c)

b -S d-d'v

fpv=

Tv-sine/(b"Sh-Seca)

Asv

-fyv-cos2e-sine

H-Xv

-tans-d'

(7.22d)

b -S d-d'v

where fLh, fph = Stresses perpendicular and parallel to the

crack due to horizontal reinforcementrespectively.

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219 CHAPTER 7

fLv'fpv = Stresses perpendicular and parallel to the

crack due to vertical reinforcement

respectively.

As shown in figs. 6.7-12, flexural cracks extended to a depth of

0.7H from the top. It is reasonable to assume that the maximum

tensile stress in the concrete is at the tip of the crack.

Below this point, the tensile stress is zero and a triangular

distribution of tensile stress is assumed above it. Therefore,

the tensile stress in the concrete in the direction

perpendicular to the crack can be written as

r ft -y/ (D. 7H-d' )y<0.7H

ft' =1 (7.23)

L0y>0.7H

where ft' = Tensile stress of concrete perpendicular to the

direction of the crack.ft = Tensile strength of concrete.

y= Distance from the top of the beam.

Stresses perpendicular to the direction of the crack should be

taken as the maximum of (fLh+fLv) and ft (i. e. )

fL = max( fLh+fLv, ft ) (7.24)

Two modes of failure of reinforced concrete deep beams are

considered. They are (1) splitting failure along the crack and

(2)shear

failurealong

the crack.

MODE 1- Splitting failure

Consider the rotation of the concrete block about point A

(fig. 7.1Ob) and failure of the beam is due to the splitting of

the concrete block. By taking moments about point A, thus

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220 CHAPTER 7

rfL'b -cosec

2a- (y-d ')

.Vui =I d(y) (7.25)

1Q X-d'-cote

where Xc = Clear shear span,

Vu1 = Ultimate shear capacity of the reinforced concrete

deep beam estimated by failure mode 1.

MODE 2 -- Shear failure

Consider the forces acting along the direction of the

crack. They include the components of forces of the

reinforcement, shear stress in the compression zone, aggregate

interlocking stress and dowel action of the reinforcement. By

considering the equilibrium of these forces, the shear strength

of the reinforced concrete deep beam can be found.

where vc = Shear stress in the compressive zone,

fa = Shear stress due to aggregate interlock effect,

fd = Shear stress due to dowel action of the

reinforcement,Vu2 = Ultimate shear capacity of the reinforced concrete

deep beam estimated by failure mode 2.

However, estimation of vc, fa and fd is necessary before the

shear capacity of a deep beam can be found.

Shear stress in the compressive zone can be estimated by

the recommendation of ACI-318 (2), Eq. 5.22.

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22L CHAPTER 7

vom' _ (3.5-2.5M/V -d) (I. 9&/fý'

+ 2500pß "V -d/M)

< 2.5 (1.9ý + 2500ß-V -d/M) (5.22)

< 61fc'

Aggregate interlocking stress is dependent on the restraining

stress (stress perpendicular to the direction of the crack).

It was suggested by Paulay and Loeber 170] that

fL = 0.473fa'

1'03

(psi) (7.1)

Fenwick and Paulay [24] stated that the aggregate interlocking

stress was also proportional to the square root of the cylinder

strength of concrete

fa' = k-, /fr, (7.27)

where k= proportional constant.

Eq. 7.1 may be modified without much loss of accuracy to

fL = 0.582 fa (7.28)

rtSince Paulay and Leber [701 used an average concrete cube

strength of 5300 psi. in their experiment, therefore, the

appropriate proportional constant should be taken as k=0.165

Thus, the aggregate interlocking stress can be estimated as

fa' = O. 165&/fC:

# - (fLh+fLv) /O. 582 (7.29)

It is recognized that the aggregate interlock effect can only

happen in the region where the crack has formed, otherwise only

shear stress of the concrete is effective. Therefore, the sum

of the shear stress in the concrete and the aggregate

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222 CHAPTER 7

interlocking stress should be equal to the maximum of the

aggregate interlocking stress in Eq. 7.29 or the concrete shear

stress in Eq. 5.22 in any position of the crack, (i. e. )

fa+vc = max( fa', vc ) (7.30)

Stresses due to dowel action are considered to be

proportional to the square of the sine of the angle between the

reinforcement and the crack (Eq. 7.16). A proportional constant

of 0.45 is suggested by Kong's formula (Eq. 5.23). He used a

constant of 130N/mm2 for plain steel bars with yield stress of

296N/mm. Therefore, dowel stress, fd can be estimated as2

fd = 0.45(fh. sin2e + fý. cos20) (7.31)

where fh = Stress of the horizontal reinforcement,

fv = Stress of the vertical reinforcement,

fd = Dowel stress of the reinforcement.

Substituting Eqs. 7.22,30 and 31 into Eq. 7.26, the shear

capacity of the reinforced concrete deep beam with failure

mode 2 can be found.

The shear capacity of the reinforced concrete deep beam

shouldbe the

minimum ofthe

values obtainedby Eq. 7.25

and

7.26. In the case of deep beams with uniformly distributed

load, the shear crack is assumed to form along the line joining

the inner edge of the support and the third span on the top of

the beam.

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223 CHAPTER 7

7.7 COMPARISON WITH TEST RESULTS

Test results frort: various sources (44,42,47,73,69] are

analysed with the proposed method and compared with methods

given by Paiva and Siess 169], Ramakrishnan and

Ananthanarayana (73], Kong (45] and Al-Najjim (63]. The

effectiveness of different design guides; CP11O (19],

CIRIA 168] and ACI-318 C1] are also discussed. The results are

presented in table 7.1 as ratios of the ultimate load obtained

during tests to the values given by different design guides and

formulae. Figures with experimental ultimate loads varies the

calculated values are given in figs. 7.11-18 together with a

diagonal line of 45 degrees which represents the calculated

values equal to the experimental ones.

Among the design guides, CP110 gives the most conservative

results. It gives an average safety factor of 6.5 and a high

standard deviation of 4.4. Generally, beams with small shear

span to depth ratio have a higher factor of safety. Fig. 7.11

shows that none of the beams was over-estimated by British Code

CP110 and most of its estimates lie within a 50kN range even

for those with 600kN capacity. The CIRIA design guide gives

much better results than CP110, in fact it is the best among

the three codes: CP11O, CIRIA and ACI-316. It gives an average

value of experimental to calculated ultimate shear strength of

1.706 and a standard deviation of 0.3G. Fig. 7.12 shows that

CIRIA design guide gives more conservative results for higher

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224 CHAPTER 7

strength deep beams and it under-estimates the shear strength

for all the beams except for a few of Ramakrishnan and

Ananthanarayana's single point loaded deep beams which failed

in flexure rather than shear. An average factor of safety of

2.054 and a standard deviation of 0.995 are obtained with the

recommendations for the design of deep beams in the ACI-318

Design Code. It gives slightly more scattered results than

with the CIRIA design guide as shown in fig. 7.13.

Paiva and Siess's empirical formula on average gives only

a slight over-estimation of 2.27. and with a reasonable standard

deviation of 0.215. Generally, this formula can predict Kong's

beams quite well but it over-estimates Ramakrishnan and

Ananthanarayana's beams with concentrated loads by 36%

(standard deviation of 0.068) and under-estimates those with

uniformly distributed load by 117. (standard deviation of

0.316). An average ratio of experimental to calculated

ultimate shear strength of 0.969 and 0.671 (corresponding

standard deviation of 0.103 and 0.127) is obtained with their

beams which failed in shear and flexure respectively.

In the calculation with Ramakrishnan and Ananthanarayana's

formula, k=1.57 is used and it is based on the use of the

cylinder splitting test in the estimation of the tensile

strength of concrete. This formula gives an excellent mean

value of 1.007 in the ratio of experimental to calculated shear

strength of deep beams, but the standard deviation is high at

8/8/2019 uk_bl_ethos_373477


225 CHAPTER 7

0.26. The

advantages,

estimate of

should be n

strength of

formula.

simplicity of this

and therefore, it

the shear strength

oted that the effect

deep beams is na

formula is one of its great

can be used for a primary

of deep beams. However, it

of reinforcement on the shear

t taken into account by this

The formula introduced by Kong and later adopted by CIRIA

with modifications, as a design tool for reinforced concrete

deep beams gives good results as shown in table 7.1 and

fig. 7.16. It has an overall average values of experimental to

calculated shear strength of 1.037 (3.7% under-estimate) and a

modest standard deviaton of 0.208.

Al-Najjim's theory was based on the compression of a

curved strut which was deflected by the present of web

reinforcement inside the beams. As shown in table 7.1,

Al-Najjim's theory gives an average of 38% under-estimation and

a high standard deviation of 0.354. Most of the points in

fig. 7.17 lie in the region where experimental ultimate load is

greater than the estimated one, except for a few of Paiva and

Ramakrishnan's datawhich

falloutside

thisregion.

This

method of assessing the ultimate shear strength is rather

complicated and no particularly good results can be obtained,

therefore, its practical use is of limited value. However, it

has suggested an explanation of the behaviour of reinforced

concrete deep beams after cracking.

8/8/2019 uk_bl_ethos_373477


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8/8/2019 uk_bl_ethos_373477


231

zX

a

c"Lv"

t""

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9

E

L

a"X

4l

Ftg. 7.1 1. Expt. va cal. sheer strength of deep beam.

(CP11O design guide)

0 100 200 300 400 500 600 700 e00 900

C. Lou l. t. d . hNr strength (kN)

8/8/2019 uk_bl_ethos_373477


a37a

Z

cv"

t""

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Y

E

w

Fig. 7.12. Expt. vs os 1. shear st rongt h of deep bs. m.(CIRIA design guide)

o 100 200 300 wo Soo 600 700 ow wo

Cs Lou Lst od wh. r "t r. nge h (kN)

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133

az

rvaýC

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L""iS

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M

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a"X

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(ACI design guide)

0 t00 200 300 400 90D 600 700 no 900

C. Lou L. t. d . hMr wt rongt h (kM)

8/8/2019 uk_bl_ethos_373477


.>-14

z

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(Ps t vs &St ess 's formu Ls)

o too 200 300 400 SIX 600 700 900 900

C. Lou L. t. d sh. w .tr. ngt h (kN)

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236

z

v

c"Lv"

La"

t"

JS

Y

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L"

w

Fig.?. 15. Expt. va ca L. sheer strength of deep beam.

(Rams kr t ahnen 's formu La)

0 $00 200 300 400 500 600 700 e00 900

C. Lout. t. d "hesr "tr. ngth (kW

8/8/2019 uk_bl_ethos_373477


z36

2

C"LY"

L""

t"

J"

p

E

L"

a

F 1! .7.16. Expt. vs os 1. shear strength of deep beam.

(Kong '$ formu La)

0 100 200 300 400 90D 600 700 000 400

C. lou Letal . h. v- .tr. ngs h MM)

8/8/2019 uk_bl_ethos_373477


- 37

z

rp

C

Lp"

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r"

J

p

W

F IS. 7.17. Expt. ve os 1. sheer strength of deep boom.

(A l-Nej J1m 's modal)

0 100 200 300 400 500 600 Too 800 900

C. Loutotod . hNr . tr. ngth (kN)

8/8/2019 uk_bl_ethos_373477


z36

z

rr

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FIg. 7.18. Expt. ve ca L. *hour strength of deep boom.

(Proposed mode L)

0 100 200 300 400 500 600 TOD 800 900

C. Lou L. t. d . h. sr .tr. ngt h (kM)

8/8/2019 uk_bl_ethos_373477


239

900

eoo

7

Z

v6

tvW

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2000 x

X1900

1000

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ADD

1000 1500 2000

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f f Kong's data (1972)

0 o Kong's data (19711

+ + Kong'sdata (19701

X xR saw4 dots,

0

Do

Do

DO..

Do

00

Do

DO

V

0 100 200 300 400 500 600 700 wo 900

Cs Lou L. t. d . h. w "tr. ngth (km)

Fig. 7.19. Expt. vs os 1. *hoar strength of deep boom.

(Proposed dos Igm guIdo)

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240 CHAPTER 7

The proposed method of analysis is based on the

calculation of shear along the crack which involved the shear

in the compression zone, aggregate interlock and dowel action

of the reinforcement. This gives good results of an average

ratio of experimental to calculated ultimate load of 0.966

(3.47 over-estimate) and a standard deviation of 0.181 which is

the best among the different formulae in table 7.1. The

proposed method of analysis seems to give more conservative

results for beams with a large shear span to depth ratio; for

instance, with Kong's normal weight concrete beams, it

over-estimates those with Xc/H=4.23 by 0.8`/, and under-estimates

those with Xc/H=g. 7 by 10'%. This method of analysis fails to

give good results for Ranamkrishnan and Ananthanarayana's beams

in series B and C. This could be due to the premature failure

of the anchorage of the main reinforcement: a large diameter of

main reinforcement and a lack of proper anchorage to the end of

the beam is apparent in these two series of beams.

7.8 RECOMENDATIONS FOR THE DESIGN OF REINFORCED CONCRETE DEEP

BEAMS

As shown in the previous sections, the proposed failure

mechanism for reinforced concrete deep beams gives excellent

estimates of the ultimate strength. By introducing the

appropriate factor of safty, Eqs. 7.25 and 7.26 can be used for

8/8/2019 uk_bl_ethos_373477


241

the design.

CHAPTER 7

Material factors recommended by CP114, are taken as 1.5

for both tensile and compressive strengths of concrete and 1.15

for the yield strength of reinforcement. Estimation of

aggregate interlocking strength taken from Paulay's

experimental datas (703 is bounded within 177., (fig. 7.4).

Therefore, a factor of 1.17 should be added to the aggregate

interlocking strength in Eq. 7.29 to become Eq. 7.32

fa' =[o.

16s. fLh+fLV/o. 5e2}f1.17 (7.32)

With the above-mentioned factors, Eq. 7.25 and 7.26 can be used

for the calculation of the design strength of reinforced

concrete deep beams in splitting and shear failure modes

respectively. Again, the lesser of the values from Eqs. 7.25

and 7.26 will be the design strength of the beam.

The ratios of experimental to calculated design strength

for various reinforced concrete deep beams are tabulated in

table 7.1. Fig. 7.18 is plotted for the experimantal ultimate

strength against the design strength. It can be seen that with

the exception of nine values, all the rest are estimated safely

by the proposeddesign

method.Four

out of the nine

over-estimated beams are in series B and C of Ramakrishnan and

Anathaanrayana's beams. As discussed in the previous section,

their over-estimation is due to the premature failure of the

anchorage of the main reinforcement. The other over-estimated

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242 CHAPTER 7

beams are DB1 and DB2 in the present investigation. They are

over-estimated because of bearing failure at the supports;

crushing of concrete due to the lack of bearing steel and

instability of the beam as a result of the distortion of the

supports due to small edge distances are found in beams DB1 and

DB2 respectively. Generally, the proposed design formula gives

an average factor of safety of 1.375 and a standard deviation

of 0.256, which is better than CIRIA guide with an average of

70.6% over-estimate and a standard deviation of 0.38.

A check on the bearing strength of the supports is also

neccessary, especially for beams with height greater than span.

The bearing strength of the supports can be estimated by

Eq. 4.11 (Chapter 4) and bearing steel in form of closely spaced

interlocking stirrups should be added where appropriate. Edge

distances with 2Wa/ai less than 3.5 should be avoided.

Moreover, it is advised to used small diameters, closely spaced

bar rather than larger diameters widely spaced steel as main or

web reinforcement.

7.9 SUMMARY

(1) A shear crack is formed along the line joining the inner

edge of the support and the outer edge of the loading

plate in the case of deep beams with concentrated loads.

With uniformly distributed load, a shear crack is formed

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243 CHAPTER 7

at an angle of 70 degrees from the inner edge of the

supporting plate.

(2) Aggregate interlock and dowel action play important parts

in the shear strength of reinforced concrete deep beams.

(3) Aggregate interlock and dowel action are interdependent

and not easy to separate them from each other.

(4) Aggregate interlock ismainly

dependenton crack width and

is proportional to the square root of the cylinder

strength of concrete. It is independent of aggregate

size, shape and hardness.

(5) The restraining force required to maintain constant crack

width was found to be a function of aggregate interlock

stress (70]

fL = 0.473 fý1.03 (psi) (7.1)

(6) It was found that shear and kinking are the principal

mechanisms of dowel action. The shear stress provided by

dowel action of light reinforcement is proportional to the

square of the sine of the angle between the direction of

shear crack and the reinforcement times the steel stress.

vu =k "sin2e "fs (7.16)

(7) Based on aggregate interlock and dowel action, the shear

strength of reinforced concrete deep beams can be

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244 CHAPTER 7

estimated by the lesser of the values given by Eq. 7.25 and

7.26.

(8) Among the design guides, CIRIA gives the best estimate of

shear strength of reinforced concrete deep beams with a

safety factor of 1.7 and a standard deviation of 0.38.

(9) The simplicity of Ramakrishnan and Ananthanarayana's

formula with reasonable accuracy (an average of 7%

under-estimation, standard deviation of 0.26) enables its

use as a primary estimate of the shear strength of

reinforced concrete deep beams.

(10) The proposed model gives the best result (3.4%

over-estimation, standard deviation of 0.181). It seems

to gives more conservative results for beams with large

shear span to depth ratio.

(11) By introducing appropriate material factors; 1.5 for

concrete, 1.15 for steel and 1.17 for the aggregate

interlocking strength, Eqs. 7.25 and 7.26 can be used to

calculated the design strength of reinforced concrete deep

beams.

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245 CHAPTER 0

8 CONCLUSIONS AND SUGGESTION FOR FURTHER RESEARCH

9.1 REARING CAPACITY

Hearing capacity is important in many cases, such as in

the anchorage zone of a post-tensioned beam and at the supports

of reinforced concrete deep beams. The prime factor affecting

the bearing strength of concrete blocks is the value R; the

footing to loading area ratio. Most researchers have adopted

the cube root formula, Eq. 4.7 but it has been shown to be to

conservative, especially for those with large value of R. It

is found that the square root formula would be more appropriate

Eq. 4.8.

fb/fc =k -JR (4.8)

Restraint at the base of the concrete block can also

affect the bearing strength. This restraint can be the

frictional force between concrete and steel at the base.

Higher specimens has the loading area further away from the

base and there is a lesser effect but shorter specimens

(H/a<0.5) are shown to have a 30% increase in strength. These

effects have been,

neglected by many researchers except

Muguruma (621, Eq. 2.22. He was able to estimate well for

specimens with H/a>1 but seriously over-estimated those with

H/a<l. Specimens with larger bearing plates are affected more

by the condition of the base, because a larger loading area

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246 CHAPTER 8

generates a deeper tension zone and thus can easily intercept

thebase. The size effect in this investigation is found to be

proportional to

fb/fý = 1.45 ea/80+ 0.9 (4.3)

Closely spaced (30 mm) small diameter (10 mm diameter in

practice) interlocking stirrups provide the most effective form

form of reinforcement in concrete blocks under concentrated

bearing loads. Reinforcement should be maintained to the depth

of the tension zone (0.75a below the loading surface) and

extended at least to the width of and preferably to twice the

width as the loading plate. Edge distances with 2Wa/a1<3.5

should be avoided.

In general, the cracking strength of reinforced concrete

blocks and ultimate strength of plain concrete block can be

estimated by

t 0.47Wa/a1 + 0.55 2Wa/a1<3.5

fb/fý _"1

0.12Wa/a1 + 1.16 (plain)i

(4.4)

2Wa/a1>3.5

L 0.22Wa/a1 + 1.01 (reinft. )

The proposed model of failure

bearing capacity of plain concrete

r3 -r-[1672

-F 2+8(7-, -z

where F1 = FZ+F3 F4

mechanism suggested that the

blocks can be calculated by

1+2z2) -F,,]

13a12. f . (4.11)

z1 = 0.4a1

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247

F2 = ft"(z2-z1)/2

F= ft-(7 t2)/2

F4 =0

andr= fb/f'

z2 = a1

z, = O. 75a

z4 =H

CHAPTER 8

This gives excellent results for all higher specimens

(H/a>1); average under-estimation of 0.5%, standard deviation

of 0.098. For shorter blocks (H/a<1), it shows an

exponentially increasing under-estimation with decreasing ratio

of H/a as in Eq. 4.12.

fb

(test)

/fb

(cal. )= 0.657 e-1.15H/a+a.

9 (4.12)

By multiplying the reciprocal of this factor to the bearing

capacity obtained from Eq. 4.11, the modified model gives good

estimates (average over-estimation of 1.67., standard deviation

of 0.085) for the full range of height of all the specimens.

8.2 DEEP BEAMS

After shear crack is formed along the line joining the

inner edge of the support and the outer edge of the loading

plate. For uniformly distirbuted loaded beams, shear cracks

are formed at an angle of 70 degrees from the inner edge of the

supporting plate. After cracking, the shear strength of

reinforced concrete deep beams is maintained by aggregate

interlock and dowel action along the crack.

8/8/2019 uk_bl_ethos_373477


248 CHAPTER 8

The aggregate interlocking strength is found to be

dependent on the restraining stress provided by the

reinforcement and is proportional to the square root of the

cylinder strength of concrete.

fL=0.473 fai'03(psi.

)

Dowel action is taken to be proportional to the square of the

sine of the angle between the crack and the reinforcement.

V= k"sin2e-f (7.16)us

Based on the above two findings, a failure mechanism for

reinforced concrete deep beams is proposed. The ultimate

splitting and ahear strength for orthogonally reinforced

concrete deep beams can be calculated by Eqs. 7.25 and 7.26

respectively.

I' fL.

b -cosec2o - (y-d ')

Vuý =Id (y) (7.25)

J xc - d'-cots4

rN (fpm fph+vc+fa+fd) -cosecs

Vu2 _'d (y) (7.26)

Jsine

The ultimate strength will be the lesser of the two values

found by Eqs. 7.25 and 7.26. This gives the best result

(average over-estimation of 3.4%, standard deviation of 0.181)

among different formulae by recent researchers.

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249 CHAPTER 8

By introducing certain material factors, Eqs. 7.25 and 7.26

can be used for the design of reinforced concrete deep beams.

The appropriate factor for aggregate interlocking strength is

taken as 1.17 (fig. 7.4, E70]). The material factors, as

recommended by CP11O, are taken as 1.5 for concrete and 1.15

for steel. The design equation has been proved to be very

effective with an average factor of safety of 1.375 and a

standard deviation of 0.256. Only 9 beams out of 152 are

over-estimated in strength by the design formula (table 7.1),

of which 2 are due to premature failure of the bearing at

supports and 4 have anchorage problems with the main

reinforcement.

8.3 SUGGESTIONS FOR FURTHER RESEARCH

The effects of height and base friction on the bearing

capacity of concrete blocks need further investigation. The

mechanisms of aggregate interlock and dowel action are

complicated in nature and need to be studied further. There is

limited knowledge on the effects of openings and wall

connections of deep beams and this subject should be further

investigated.

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------- 1600 kN

------ -------1200 kN

iI-------

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900

800

700

600

500

400

300

200

100

0

900

a

700

600

500

400

300

200

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900

a

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5I

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300

200

100

0

-35M -dom -2500 -2000 -1500 -I -500 0 -3000 -2500 -2000 -1500 -1000 -500 0

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Fig. B. 1. Vert Ica L st re In In sect Ion 1.

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900

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700

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300

200

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800

700

600

500

400

300

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,

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Fig. B. 2. Vsr-t tcsL at re tn to sect ton 2.

8/8/2019 uk_bl_ethos_373477


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E

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------- 1600 kN

---- ------- 1200 kN

------- 800 kN

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am

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Ftg. B. 3. Vert Ioal etreIn In sect I on 3.

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600

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400

300

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FIg. B. 4. HorizontaL strain in section 1.

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600

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400

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Flg. B. 5. Horizontal strain in Boot Ion 2.

8/8/2019 uk_bl_ethos_373477


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8/8/2019 uk_bl_ethos_373477


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Ftg. B. 7. Hortzontml strain to section 4.

Expt. FEM

------- 1600 kN

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-------

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VII

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8/8/2019 uk_bl_ethos_373477


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Flg. 8.8. Horizonts L strain to seat 1on 5.

8/8/2019 uk_bl_ethos_373477


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FIg. B. 9. Transverse strain In section 6.

8/8/2019 uk_bl_ethos_373477


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Ftg. B. 1 0. Tr-ensverss st ra 1nIn sect Ion 7.

8/8/2019 uk_bl_ethos_373477


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Ftg. B. 1 1. Trsnsvers" st r. InIn sect I on 8.

8/8/2019 uk_bl_ethos_373477


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8/8/2019 uk_bl_ethos_373477


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1

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Ftg. B. 13. Longitudinal strain in section 10.

8/8/2019 uk_bl_ethos_373477


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1 J

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0

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Fig. B. 14. Longitudinal strain to section 1 1.

8/8/2019 uk_bl_ethos_373477


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Fig. B. 1S. Longitudln. L strain in sect Ion 12.

8/8/2019 uk_bl_ethos_373477


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Ftg. B. 16. Tr-sneverm® st rte toto sect ton 10.

8/8/2019 uk_bl_ethos_373477


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a See1000150020002510300039004= 450050 5600 0 500100015002000ä00 3000Moo M 150050005600

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332

REFERENCES

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