Contribution of the fibres to the load bearing capacity of a bar- and fibre-reinforced concrete beam

DEPARTMENT OF CIVIL ENGINEERING

Report 5-78-9

Contribution of the fibres to

the load bearing capacity

of a bar- and

fibre-reinforced concrete beam

by

Dr. Ing. H.W. Reinhardt

STEVIN LABORATORY

CONCRETE STRUCTURES

^ ^ ^ v^A "/ 5 ^^ S^-y<P^(4 Bibliotheek

^ afd. Civiele Techniek T.H. Stevinweg 1 - Delft.

Report 5-78-9

D e l f t , J u l y 1978

*f O

Contribution of the fibres to the load bearing capacity of a bar-and fibre-reinforced concrete beam

by

Dr.-Ing. H.W. Reinhardt

^V^

Contribution of the fibres to the load bearing capacity of a bar-and fibre-reinforced concrete beam

by H.W. Reinhardt

1. INTRODUCTION

In a concrete beam which is subjected to pure bending it is assumed that

- cross sections remain plain (Bernoulli's law) - strain distribution is linear (Navier's law)

These two assumptions are correct in the uncracked stage, but become approximations in the cracked stage. Because of the benefit of simplicity they will be used for both stages.

The stress distribution depends on the mechanical proper-ties of the materials in which the beam consists. In the elastic uncracked stage, the elastic moduli of the mate-rials are the determining quantities whereas in the cracked stage the bond properties of the reinforcing steel, the pull-out behaviour of the fibres and the shape of the stress-strain diagram of the concrete play the important roles. In the following, three cases will be considered: the

linear elastic cracked stage, the nonlinear elastic cracked stage emd the nonlineea" elastic plastic cracked stage.

THE LINEAR ELASTIC CRACKED STAGE

Prom tensile tests / l / it is known that the stress-strain diagram of a fibre reinforced concrete can be given sche-matically like fig. 1. It shows a linearly increasing

Fig. 1. Schematic o'-£-diagram of fibre reinforced con-crete in tension.

branch and a horizontal branch, similar to a mild steel with a pronounced yield point. The "yield point" here is

the cracking stress and the horizontal line is due to the pull-out resistance of the fibres.

Using such a relation for the concrete in the tensile zone of a beam, the contribution of the fibres can be described or can be determined from test data.

Por thatJ the following strain and stress distribution are used, fig. 2.

Pig, 2, Strain and stress distribution in a cross

section.

According to fig, 2 the upper strain £, , leads to a

compressive concrete stress c^ whereas the lower strain

leads to a tensile stress C . in the concrete. Because

ct

of the relation of fig. 1 the tensile stress block is

taken constant over the whole tensile zone, so neglecting

the real behaviour around the neutral axis.

The contribution of the fibres can be concentrated in a

tensile force T . The force in the bar reinforcement is T .

c s

The compressive force is C.

The following expressions eire got

C = -è- b X4,E^

_{* ' c}

₍₁₎

assuming a constant modulus of elasticity of concrete E ,

G

and

T^ = b (h - x) . C^^

s s s s

(2)

(3)

- 4

The steel strain £ and the height of the compressive

zone T( are derived from fig. 2 and become resp.

(1 -ÏÏ )(^.-^,) + £)

I,

'r^^

(4a)

(4h)

where £^ is a negative and ^^Is a positive strain.

In the cross section the equilibrium of forces must be valid,

Thus

C + T + T„ = 0

c s

(5)

With eqs. (l) through (4) and after some rearrangement eq. (5)

becomes

~ y^

'r^z -^

^^ ^^n^'r^ht ^ Vx[(i-U^.-^0-^J="

The reinforcement ratio can be expressed in

s - ^

(7)

Using this relation, eq, (6) becomes

2

J_ — 1 1 — r

1 '•z

>- TTzh

- ? ^ 4 O - M H - O ^ ^ , ]

-

o

(2)

Assumed, that in a test program the quantities E , E , f , c, h

are given and the top amd bottom strains £, and Sj^are measured,

then the contribution «T* , on the fibre concrete can be

deter-ct

mined,

Another relation, which must also hold true, is the

equilibrium of moments in a cross section, i.e, the

external bending moment must be equal to the internal

moment :

M

ex

M. .

xnt

(9)

Making use of the notation in fig, 2, the internal moment

in respect to the compressive force C is given by

M

int

^c

(^-h-x

- | ) -hT^ ( h - c - | ) (10)

U

After inserting eqs. (2) through (4) and rearranging eq. (IO)

one gets with eq. (9) the expression for the moment

equili-brium:

? -

\i

^r^i

t'i

£ ,-£.

?f4^'-r)(vO-,][(-t]-i 1^1

If in a test program the values for M , B , b, c, h , P are

ex s J

given and if £ and &^ are measured, the contribution of the

fibres e' . can be calculated.

ct

do

In fact, eqs, (8) and (II) are two independent relations which

can be used for determining the fibre contribution. S o , one

given or measured quantity could be omitted,

Such a case is the following where the surface strains are

not measured, but, instead of them, the mid span deflection

is determined in the experiment. Then both eqs, (8) and (11)

are needed for the evaluation,

6

-Let us consider a pure bending zone with length -d and

mid span deflection f (fig, 3 ) , The known relation between

f , ^j and

circle with

r a d i u s r

Pig, 3, Deflection in a pure bending zone.

the radius of the circle r is

r..4f

₍₁₂₎

which becomes for 4/- ^< t

(13)

The curvature ^ is then

3£ = a —-1-

₍₁₄₎

In terms of strains.the curvature is

je = <f^-<£,

(15)

and the strain difference becomes

If in eq. (8) the strain difference is replaced by eq. (16) it becomes

2- 5£ ^

5 £,

[iu'c) ^

* 2, ]=o (17)

X

A similajT manipulation of eq. (ll) leads to the new expression

( « )

Prom this set of second order algebraic equations,it is

pos-ti

sible to calculate ^ . and £ if the quanties M , E , E , b, c*c ' ex c s c, hf f are given and the curvature ?€, (or the mid span de-flection) is meastired,

The way to solve this set of equations is enclosed in the appendix,

THE NONLINEAR ELASTIC CRACKED STAGE

Up till now the concrete (in compression) and the steel were assiimed to show a linear el astic behaviour. In fact, that is only true till approximately a third of the concrete strength, B§-ond that limit the c-£ line is curved. In fig. 4 the strain and stress distribution are given in the nonlinear cracked sta-te, The only difference to fig. 2 is that the distance e, of the compressive force is no longer j x but some function of x depending on the shape of the <ƒ-<£-diagram of concrete.

^

Pig. 4» NonlineajT elastic cracked stage.

The first term in eq. (8) which represents the compressive

force of the concrete becomes

^,- ^x

'.-U^>)

(19)

where ^(^,) stands for an integration factor.

So, the whole eq.(8) gets the form

^^'0-['-^J«;. ^?e[;^6=-o.^,7.o

f -£.

(20)

^(^€,j can be taken from graphs or formulae / 2 / , it is for

instance ^E in the linear case and -r E for a parabolic

C 3 C£=f,

<r - ê - l i n e .

Bibliotheek

afd. Civiele Techniek T.H. Stevinweg 1 - Delft

Eq.(ll) for the moment equilibrium has also to be ajusted

to the new center of gravity of the compressive stresses.

That can be done by two coefficients ^^ and -h^ which

de-pend also on the shape of the «r-£-line, Eq,(ll) becomes

then

i * ^''^]^f'4-t^'•-'^''•li('-^ri^ftJ'

- !"

U^

.

(21)

^1 is illustrated by fig, 5, and ^ = ^ ~ k » Por the

li-near

center

of gravity

Pig, 5» Compressive stresses in the beam and stress center

of gravity,

case 4, =3 -r and -4

^ 3 -I

for a parabolic shape i^s -g and -^^ =

/

The same processing can be applied on eqs. (1?) and (18),

where the curvature «. is introduced inst ead of the individual

strains.

-

10

-4.

THE NONLINEAR ELASTIC PLASTIC CRACKED STAGE

If the stress in the reinforcing steel reaches the yield point, the elastic stage goes over into the plastic stage. All equations can thei?. be maintained if the modulus of

elasticity of steel E^ is correctly interpreted. Pig. 6 illustrates that E is a constant value up to the yield point and starts to be a variable beyond this point.

Pig. 6. Definition of E in the post yield stage, s

That means that E depends on «•, or £, ^respectively^ and-the type of steel has to be kept in mind.

STATIC BEiroiNG TESTS

M a t e r i a l s

Static bending tests were caxvled out on beams with

150 X 100 mm in cross—section and with a free span of

2000 mm. The beams were loaded with two symmetric loads.

In the pure bending zone the top and bottom strains were

measured by means of electrical extensometers.

The beams were made from steel fibre reinforced concrete

and wete additionally reinforced by continuous bars. The

types of fibres used are shown in fig. 7 together with

the corresponding

1,4 V.

25

_-A

J^

0,4

mm

0,9 •/•

1.5

J: -T

M^

25

T

30

- ^

0"

0,4

mm

1.5 7,

L l l .

42

50

^ 9S

0,8

mm

4

Pig. 7. Types of fibres used.

percentages by volume. The steel quality of the bars was a

FeSSoo with diameters of 4. ^ or 10 mm, resulting in

reinforcement ratios of 0^11%, 0,75^, and 2,09^ resp.

12

-The concrete mix consisted of 400 kg/m type A Portland cement, round sand and gravel aggregate with a maximum size of 16 mm. The v/ater cement ratio was 0,48, The cylinder strength of the various batches fluctuated between 41 and 45 N/mm ,

The modulus of elasticity of the three types of concrete was practically the same as can be seen from fig. 8, The tests were done on cylinders with a height of 400 ram and a diameter

of 150 mm, 50 Nlmm iO 30 20 10 0 -0 I 2 3 4 S S C 'U

Pig, 8. stress-strain—curves of the concretes used,

5,2, Contribution of the fibres

Eq. (8) was used for the analysis of the test results.

Prom this the contribution of the fibres could be calculated taking E = 2,G.10^ N/mm and E = 2,1.10^ N/mm , and c and h

c s as measured an the beams,

Prom the results^ it could be clearly distincted between the uncracked state and the cracked state. In table 1, the average results of the two conditions are listed dependent on the percentage of bar reinforcement and the tsrpe of fibres. The individual results are given in the appendix,

1

Type of fibres : Q

0,17

straight i 0,75

i 2,09

.. 0,17

hooked 0,75

' 2,09

paddled

0,17

0.75

2,09

ö'ct N/mm

uncracked cracked

i

1,12

1,15

1,22

0,87

0,85

0,67

0,76

2,20

2,54

2,86

2,99

1.84

1,41

2,90

2,25

1,47

Table 1, Contribution of the fibres,

In the iincracked state^ the calculated contribution of the fibres is mucdi less than in the cracked state. This difference can be interpreted by the fact that the strain in the outmost fibre does only reach 0,25?Sowhat means that the stress distri-bution as assumed in fig, 2 is not quite correct, A trian-gular stress distribution in the tensile zone would be a better approach,

V/hen cracks are formed and the tensile strain reaches values up to 2%e , the assumed stress distribution certainly does much

closer correspond with the reality. The calculated contribution is equal to the uniaxial tensile strength of the concrete with. straight fibres and reaches about 8 5 ^ of the uniaxial tensile strength of the concrete with hooked and paddled fibres.

14

-At this moment, no comments will be given on the in-fluence of the percentage of the bar reinforcement. A good discussion of the results can only take place when

some possible scatter in the material properties have been considered.

SUMMARY AND CONCLUSION

Based on a linear strain distribution and on results from ^ensiie testi^ a stress distribution in bar and fibre reinforced concrete beams subjected to pure ben-ding is discussed. In this model the contribution of the fibres is schematized by a rectangle. The theory was applied to experimental results, and it turned out that the calcïulated contribution of the fibres reaches 85 till 100^ of the uniaxial tensile strength of the fibre reinforced concrete,

It can be concluded that this simple model describes the real behaviour with sufficient accuracy, at least in the

cracked elastic stage.

REFERENCES

/ I / Shah, S.P. et al. Complete stress-strain curves for steel fibre reinforced concrete in uniaxial tension and compression. Intern. Symp. on Testing and Test Methods of Pibre Cement Composites, Sheffield 1978.

/ 2 / Leonhardt, P., Monni g, E. Vorlesungen iiber Massiv-bau. Vol. 1, Springer Berlin - New York 1973.

16

-APPENDIX

Al, Determination of e and o* , from -st only

The two relevant equations are

'^-^) ^ - ^,J -

(AI)

Isolate ö', in eq. ( A 1 )

Replace ^ . in eq. (A2) by right hand side of eq. (A3)

^ U"!

Rearrange eq. ( A 4 ) making use of k..=5— /(^

'^ 1 1 1 .

0

,' < r ^ t.'.^ U

Simpli-fy the equation and get

t # ^ ' ' " ^ ^ ^ ' ^^^sh> -^i^^L-'-^^]^,

-^ ?6-^3e

^-c

i i ^ '

1 - ^ . 0

2. r _J ) (

Prom this cubic equation £. can be calculated for a

measured *€ , All other quantities like

rC^^t

•= , E , h, c,

* 3

M must be given,

Generally there aj&three roots of eq. ( A 6 ) , one of which

is the right one. That c, must be put in eq, (A3) in order

to get the contribution of the fibres c'.,

A2, Individual results,

£ upper concrete strain in %o

£ steel bar strain in ^o

^ . contribution of the fibres as calculated by eq, (8) in

2

N/mm

A

e percenteige of b a r r e i n f o r c e m e n t ('s* = TÏT )

Hooked fibres

Straight fibres

? =

- ^ ,

0,14

0,33

0,64

0,89

0,17^

h

0,18

0,58

1,53

2,66

S^t

1,12

1,93

2,38

(2,28)

S"

- ^

0,03

0,11

0,12

0,16

0,38

0.56

0.72

0,82

0,17fo

^ :

0,02

0,15

0,15

0,25

0,55

0,97

1,57

2,06

<^ct

0,66

0.76

0.97

1.03

2.75

3,25

3,14

2.83

18

-^ =

0,13%

-^, h

^ct

0,09 0,07 1.15

0.42 0,53 2,44

0,95 1,61 2,64

e

= 2,09^

- ^

0,10

0,34

0,57

1,72

^s

0,06

0,26

0,47

1,94

<^ct

1,22

2,60

3,60

2,39

Paddled fibres

f

= 0,17^

- 2 , 0,04 0,08 0,19 0,38 0,52 0,69 0,87 ^s 0,06 0,14 0,31 0,56 1,03 1,48 2,11 ^ct 0,32 0,53 1,15 2,69 2,63 3,07 3,22

(O

=. 0,75^

-^, 0,08 0,37 0,59 0,89 1.25 ^r 0,09 0.51 0,96 1,65 2,61 ^ t 0,54 1,79 1,94 1,82 ( 1 . 1 3 )

f = 2,09%

-•?. ^s ^ t

0.34 0,35 0,85

0,96 1,09 1,16

2,32 2,73 1,65

- 2 , 0,22 0,34 0,53 0,85 1.13 ^ 5 0,19 0,43 0,79 1,41 2,07 ^ c t 2,19 1,89 2,16 2,61 2,39

ƒ - 2,09%

- ^ ; 0,10 0,19 0,31 0,38 0,64 1.15 ^s 0,08 0,19 0,27 0,36 0,68 1,33 ^ c t 0,83 0,69 1.73 1,53 1,39 1,21