# Experimental study on fiber-reinforced concrete panels subjected to shear , Literature Survey - Tesi

Anteprima

### ESTRATTO DOCUMENTO

2. LITERATURE SURVEY

Moreover, using the Mohr’s Circle, it is possible to define the slip strain into a

orthogonal components relative to the reference system:

(2 )

= − ∙ 9:

2 (2.42)

(2 )

= ∙ 9:

2 (2.43)

)

= ∙ <=9(2 (2.44)

where: Tensile strain in the reinforcement along the x-direction;

Tensile strain in the reinforcement along the y-direction;;

Shear strain in the reinforcement;

Following Figure 2.20 shows a reinforced concrete element that is experiencing

deformation composed of both continuum straining and discontinuous slip along crack

surfaces.

Figure 2.20: Compatibility Conditions: (a) Deformations due to average (smeared)

constitutive response; (b) Deformations due to local rigid body slip along crack; (c)

Combined deformations

The element can also be subjected to elastic or plastic offsets: the elastic strain offset

## Iε K

c

0 include effects due to thermal and mechanical expansion, and due to the shrinkage.

41

2. LITERATURE SURVEY

e

dε f

0

On other hand, plastic offset is due to cyclic loading conditions or loading into

post-peak levels.

So the total (apparent) strains are the sum of the continuum stress-induced strains, the

shear-slip strains and the elastic and plastic offset strains.

So, the compatibility conditions became: g

## I K I K I K I K

= + + + d f

c (2.45)

This thesis takes the elastic and plastic offset to zero, and the lag in the rotation of the

principal stresses in the continuum, in relation to the rotation of the apparent principal

strains is defined as: ∆ = −

[ i (2.46)

Moreover, the reinforcement is considered perfectly bonded to the concrete and this

makes it possible to calculate the average strain in reinforcement from total strains as:

+ −

= + ∙ <=92V + sin 2V + c

; ;

2 2 2 (2.47)

## S S

α ε

c

5 H

where is the angle of the orientation of the reinforcement, is the initial prestrain of

## R

the reinforcement; at the crack location, the local stresses and strains in the

reinforcement must increase, to compensate the local reduction in the concrete average

tensile stress.

So the local strain in the reinforcement must be represented by:

= + ∆ ∙ <=9

1 1 N (2.48)

## S S S

And is also possible estimate the average crack along the x- and y-direction.

The compressive response of cracked reinforced concrete is characterized by significant

degrees of softening given by the effects of transverse cracking, as shown by Vecchio

and Collins (1986). 42

2. LITERATURE SURVEY

0

The principal compressive stress in concrete , is a function of the principal

compressive strain (as normal) and also of the coexisting principal tensile strain where

the factor that express this reduction is:

1

k = ≤ 1.0

l 1 + m ∙ m (2.49)

l ⁄

C ε ε

o 0 0

where the factor is a function of the ratio as:

(− ⁄

m = 0.35 − 0.28) c.q

l (2.50)

while the MCFT proposed the following formula:

( ⁄

m = 0.27 − 0.37)

l c (2.51)

## C

## H

On other hand, the factor accounts for the slippage on the cracks; if slip is taken into

account, the rate of compression softening due to transverse cracking must be reduced

to compensate the strains obtained.

Slip on the cracks introduces softness into the concrete response, and less stiffness

degradation can be attributed to the tensile strain effects.

Experience on panel tests show that the experimental panels with equal reinforcement

= ρ )

(ρ were subjected to proportional loading, and the shear failure was prior to the

yielding in the reinforcement.

These panels have no crack direction during the course of tests, and the response of

these panels was stiffer and stronger than that for the test panels at all.

C = 0.55,

## H

Examining conditions, it was found that using improving large improvement

was obtained: this is the value adopted for the DSFM.

If slip on the cracks is not taken into account in these compatibility relations, as done

C = 1.0

## H

from the MCFT formulation, then is used.

After cracking, concrete can continue carry tensile stresses as a result of two

independent mechanism: tension softening and tension stiffening. 43

2. LITERATURE SURVEY

Tension softening refers to the fracture-associated mechanism, particularly significant in

concrete structures containing reinforcement or not, and it is calculated as:

( )

− 1

= %1 − )

$ ' ( )

T − (2.52)

T 1

ε G

tH v

where the terminal strain is calculated from the fracture energy parameter .

Tension stiffening stresses come from the interaction between the reinforcement and the

concrete where the load is transmitted between cracks from the reinforcement to the

concrete via bond stresses, making significant levels of average tensile stress in the

concrete.

As previously seen in the MCFT paragraph, the concrete tension stiffening is modeled

as: '

## T

=

w 1 + ∙

2m (2.53)

## T

C = 200

t

MCFT considers a parameter for small elements or for elements containing a

C = 500

t

spaced mesh of reinforcement, and for large-scale elements.

Benz (1999) shows that the degree of tension stiffening is more dependent on other

ρ d

y

factors, such as the reinforcement ratio and the rebar diameter , formulating an

C = 2.2 m

t

improved relationship and a tension stiffening coefficient as where:

## N

1 4 ;

= M ∙ |<=9 |

## N

{ (2.54)

## S

w S

## ;O

Following Figure 2.21 can explain the behavior of the different situations, and as

y

0

shown, is limited to the amount that can be transmitted across the cracks, as

discussed. 44

2. LITERATURE SURVEY

Figure 2.21: Constitutive Relations: (a) Compressive Softening Model; (b) Tension

Softening Model; (c) Tension Stiffening Model; (d) Reinforcing Steel Response

So the tensile stress in the concrete is valued as the larger of the two values:

)

= { }( ,

$ w

$ (2.55)

and a trilinear stress-strain relation is used to model the response of the reinforcement in

tension or compression:

= 0 < < (2.56)

= < < U (2.57)

( )

= + − < <

U U U • (2.58)

and = 0 > • (2.59)

where: Yield Strength;

Modulus of elasticity of steel bars;

U Strain Hardening modulus; 45

2. LITERATURE SURVEY

Yield Strain;

U Strain at the beginning of the hardening;

• Ultimate strain.

2.4.3 Slip Model

δ

## H

The crack slip is continuously under study in the all literature, and it is found be a

a

05

function of the acting shear stress , of the crack width w, aggregate size and

'

0 00

concrete compressive cylinder strength (or cube strength ).

Walraven (1981) proposed the follow formulation:

;

b =

$ (0.234E

1.8E + − 0.20) ∙

\c.q \c.‚c‚ (2.60)

δ

ƒ

## H

and once the slip has been found, it is possible to determine the crack slip shear

γ ƒ

## H

strain as previously viewed: b

=

$ 9 (2.61)

There are two problems arising when the shear-slip models are implemented into

analytical formulations: firstly, although they correctly recognize that some initial slip

is required before the gap between opposite crack surfaces is closed and traction is

developed, including the initial slip component in the analysis procedures proves to be

numerically problematic. Secondly, supposing that the element is locally unreinforced,

the equilibrium equations resulted in zero shear stresses on the crack surface: no

account is made of the shear stresses that arise from the aggregate interlock or other

mechanism.

A supplemental approach is to relate the changes in direction of principal stresses to the

changes in the direction of the apparent principal strains; this happens in panels with

reinforcement crossing the cracks in skew angles, where the stress field rotation tends to

lag behind the strain field rotation. 46

2. LITERATURE SURVEY

This lag is established soon after first cracking and remains relatively constant in the

earlier stages of loading until one of the reinforcement components begin to yield.

θ 50

Relative to the initial crack direction , the rotation in the apparent principal strains

∆θ Z is determined as: ∆ = −

[ [ ; (2.62)

∆θ „

And the change in inclination of principal stress direction can be found as

(∆ ) |∆ |

∆ = − >

… …

i [ [ (2.63)

for |∆ |

∆ = ∆ ≤ …

i [ [ (2.64)

for

θ ‡ is 5° for biaxially reinforced elements, 7.5° for uniaxially

Where the constant lag

reinforced elements and 10° for reinforced elements.

2.4.4 Conclusions

DSFM is considerable as an extension of MCFT where compatibility, equilibrium and

constitutive response are formuled in terms of average stresses and strains, with a

particular attention to the compression softening and tension stiffening mechanism.

Another important aspect is the consideration about the local conditions at the crack

location, where disturbances in the stress fields that can influence the behavior are

found; the analysis from the slip formulation is in middle between fixed crack models

and rotating crack models, capturing the strength of each one.

The DSFM allows for a gradual and progressive reorientation of concrete principal

stress and crack directions.

In a few words, the improvements from the MCFT are reasonable as:

- The inclination of principal stresses and of principal strains are no longer

necessarily equal;

- The degree of softening of the compression response is reduced and more

consistent with that reported in other theory; 47

2. LITERATURE SURVEY

- Behavior and failure conditions influenced by crack shear slip are better

represented;

- The difficult check on the crack shear stress is eliminated;

- Softening behavior is reduced on compression response.

2.5 Disturbed Stress Field Model for reinforced concrete: Implementation

– Frank J. Vecchio

This second part of the thesis, is formed to implement the formulations of the DSFM

into a nonlinear FE algorithm.

All the procedure is based on a load secant stiffness approach, were the crack slip

displacement are treated as an offset strains.

Notice that DSFM was developed from Vecchio (2000) to describe the behavior of

cracked concrete, and to enhance the MCFT formulation, the goal was to address the

small accuracy where the element is loaded under certain conditions, in particular for

element (such as beams) with no shear reinforcement.

Particular attention is given to the creation of new equilibrium, compatibility and

constitutive equation; for the concrete component, the compatibility equation is, as

shown in the previous DSFM Formulation: g

## I K I K I K I K

= + + + d f

c (2.65)

where:

I K Total strain condition;

I K Elastic train of the concrete, due to the stress;

I K Average strain due to rigid slip along the crack surfaces;

## I K

c Elastic offset strains due to thermal expansion or shrinkage;

g

d f Plastic offset strains due to mechanism related to loading history or

damaging. 48

2. LITERATURE SURVEY

Thus, once times that the elastic principal strains in the concrete is obtained, all others

component of the equation (2.65), can be determined.

With the following equation (2.66), knowing the stress acting at any point, the resulting

total strain will be calculated as follows:

## IJK ILKI K IJ K

= − c (2.66)

where:

ILK Composite material stiffness matrice;

## IJ K

c Element pseudo-prestress due to elastic and plastic strain offsets in

concrete and reinforcement;

As shown in following Figure 2.22, the Mohr’s Circle can be used to determine

whatever is needed: (a) (b)

Figure 2.22: Reinforced Concrete Element: (a) Element properties and applied

Stresses; (b)Total Average Strain Condition in Element.

Cracked concrete is treated as an orthotropic material with principal axes corresponding

to the direction of the average principal stresses.

For each reinforcement component, a stiffness matrix is evaluates, and transformed to

the global reference system and then summed. 49

2. LITERATURE SURVEY

2.5.1 Shear Slip Model

Various alternatives were considered for modeling the crack slip.

One was to create a constitutive model to relate the amount of shear slip along the crack

to the magnitude of the shear stress acting on the crack.

Another was to fix the degree of “lag” between the rotation of the stress field in the

concrete and that of the strain field: a hybrid approach was taken using both.

Using the two formulations of Walraven (1981) and Maekawa (1991), the theory define

the slip displacement along the crack in two methods, that have few differences between

them: the first, with rotation lag, provide very slightly increased deformations at early

stages and, the second (with stress-based formulation),gives a reduced strength and

stiffness at the intermediate stage of loading, as shown in Figure 2.23

Figure 2.23: Comparison of experimental and Theoretical Response.

As shown in previous Figure 2.23, MCFT also gives a good representation of response

but the deformation at the ultimate load was significantly underestimate.

The MCFT also predicts failure by the concrete stress attaining its peak strength while

DSFM, on other hand, indicates behavior governed by excessive slip along the crack

surface leading to a concrete shear failure.

The transverse reinforcement ratio is another important point that MCFT and DSFM

show little differences: at low reinforcement ratios, shear strength predicted by the

DSFM are significantly higher as shown in the following Figure 2.24. 50

2. LITERATURE SURVEY

Figure 2.24: Predicted Response of Hypothetical Panel.

More important differences between MCFT and DSFM are shown in the following

Figure 2.25, there describes the corresponding orientation of stress and strain fields: at

low transverse reinforcement levels, the DSFM predicts angles of inclination for these

fields 8° higher than those predicted by MCFT.

Figure 2.25: Predicted Response of Hypothetical Panel.

These differences become important where the flow of force from the load application

point to the support may, or may not, form a direct strut. 51

2. LITERATURE SURVEY

The consideration of crack shear slip in the compatibility equations of this theory results

in reduced element stiffness so, less stiffness degradation need to be considered in the

softening of the concrete in compression.

So DSFM approach is more rational and more consistent with the degree of

compression softening.

2.5.2 Conclusions

The crack slip formulations permit the divergence of principal stress and principal strain

directions in the concrete.

In elements containing a little or no transverse reinforcement, the rotation of concrete

stress field is retarded by up 10°, resulting in reduced shear capacities.

2.6 Variable Engagement Model – S.J. Foster – Y. L. Voo and K. T. Chong

This thesis is prepared to work on a finite element model that developed for the analysis

of fiber reinforced concrete plane stress members failing by fracture. The constitutive

law is done where the behavior of a fiber composite is obtained by integration of its

parts (fibers and concrete matrix) over the cracked surface. This model is generally

applicable to any type of steel fiber-cement based matrix with any combination of fibers

in the mix, in any ratios. In this model it is used local and non-local modeling because

the finite element formulation is shown to be capable of modeling the girder, with good

accuracy for the global load versus displacement history and is shown to correctly

capture the localized shear failure mechanism.

In this thesis, a model is developed for the analysis of fiber reinforced concrete failing

in tension, based on integrating the behavior of individual fibers crossing a cracking

plane. The model, named the variable engagement model is used for the constitutive

relationship for finite element modeling of steel fiber-reinforced concrete structures. 52

2. LITERATURE SURVEY

2.6.1 Introduction

In this paper, a model is developed to study the behavior of a single fiber introduced in

the concrete matrix to study the behavior. In this document, the following assumptions

are done:

1. The behavior of a fiber reinforced composite may be obtained by a summation

of the individual components. The effects of each individual fiber can be

summed over the failure surface and added to the concrete-matrix component to

yield the overall behavior of the composite;

2. The geometric centers of fibers are uniformly distributed in space and all fibers

have an equal probability of being oriented in any direction;

3. All fibers are pullout from the side of the crack with the shorter embedded

length while the longer side of the fiber remains rigidly embedded in the matrix;

4. Displacements due to elastic strains in the fibers are small, relative to

displacements resulting from slip between the fibers and the matrix;

5. The bending stiffness of a fiber is small and energy expended by bending of

fibers can be neglected.

The slip between fibers and matrix must occur before the anchorage is engaged, and this

happens only after that the adhesion between the fibers and the matrix is broken. The

crack opening displacement (COD) for which the fiber becomes effectively engaged in

## E

ˆ

the tension carrying mechanism, is termed the engagement length and denoted as .

Assuming the engagement-length versus fiber-slip relationship can be described using a

continuous function, the boundary conditions dictate that:

## E

ˆ

θ

1. for a fiber angle of = 0 the result obtained is = 0 and

‰

θ

2. the function is asymptotic at = .

One such a function is E = V θ

ˆ (2.67) 53

2. LITERATURE SURVEY

where:

E Š COD at the point of the engagement of the fiber;

V Material parameter (has a units of length);

θ Fiber angle with respect to the crack.

The fiber is taken to be effectively engaged at the point corresponding to 50 percent of

the peak load to don’t have problems on the plateau of load-COD curves while

E ‹

Œ

Š . With this condition, the force in a single fiber, is obtained as:

determining =0 E < E and w >

‹ •

ˆ (2.68)

− E

( )

‹ = Ž P : E < E < •

•

Œ w ˆ $ (2.69)

where: Diameter of the fiber;

• $ Initial length of the embedment of the fiber;

## P

w Shear stress between fiber and matrix;

E Crack opening displacement;

(• − E

$ ) Remaining portion of the embedded fiber.

One point is to determine the angle in which a fiber become embedded and become

θ 1;T this

active to transmit the load across the crack in the concrete: the VEM terms

angle. < ≥

θ θ θ θ

1;T 1;T

When the fiber is not engaged, while the fiber is involved in the

tension load.

θ 1;T is a function of the current COD (w), as shown in the following Formula

The angle

(2.70): ‘

= • ’

\

1;T $ (2.70)

Considering that the maximum possible fiber slip engagement is: 54

2. LITERATURE SURVEY

•

Œ

E= 2 (2.71)

The Formula (2.71) become: •

Œ

= tan • –

\

θ 2

<“: (2.72)

2.6.2 Stress-COD Model Excluding Fiber Fracture

For fibers randomly orientated in three dimensions, Aveston and Kelly (1973) show that

⁄

2

Œ Œ

the number of fibers crossing a plane of unit area is where is the volumetric

fraction of fibers placed in the concrete.

•

Œ Œ

For fibers of length and diameter crossing cracking plane with the fiber pulling out

from the side with the smaller embedded length, Marti et al. (1999) noted that for w = 0

•

Œ

the average length of embedment is /4 and that the number of bonded fibers decreases

linearly with increasing COD.

Watching many substitution in the previous formulas, the result obtained to have the

tensile stress is, rewriting the previous formula (2.70):

## P

J = — • ˜

w

Œ Œ Œ Œ (2.73)

where:

J Tensile stress of the fiber;

—

Œ Global orientation factor;

P w Shear stress between fiber and matrix;

Œ Diameter of the fiber.

The orientation factor can be determined using probability theory and is affected of the

shape of the specimen which the orientation is considered, and it can be described as: 55

2. LITERATURE SURVEY

›

1

— = M š

Œ ;

™ (2.74)

## ;O

where:

™ Number of fibers crossing a plane of unit area;

— : TU

; Local orientation factor for the fiber. ⁄

• 2

Œ and, considering the probability

The average value is considered between zero and

that any fibers can be oriented in any directions, this average of the local orientation

factor for the embedded fibers is: 1 E

— = −

$œˆ 2 • (2.75)

Œ

Assuming that there is a uniform distribution of fibers, the actual proportion of fibers

## E

embedded across the crack, for a given is shown in the following equation

## E

= 1−2 •

Œ (2.76)

The following formulation assume that all the fibers are pulled out from the concrete

matrix and there is no fiber fracture. Thus the length of the fiber is less that the critical

• (obtained by setting the total force in the fiber due to bond stresses):

length ⁄

tan (E V ) 2E

\

— = •1 − –

Œ Ž • (2.77)

Œ

And the length of the fiber is so determined: J

Œ Œ•

• < • =

Œ 2 P (2.78)

w 56

2. LITERATURE SURVEY

2.6.3 Stress-COD Model Including Fiber Fracture

If there is constant bond shear stress along the fiber length, any fiber will fracture if

## J

Œ Œ•

• = + E

$ ˆ

4 P (2.79)

w

Experimental studies are done, and this model shows to have a good correlation with the

experimental data.

2.6.4 Conclusions

A constitutive model for analysis of steel-fiber reinforced concrete members is

presented. The model is based on the VEM with the composite behavior of a fiber

composite obtained by integration of its parts over the cracked surface (fibers and

concrete matrix). The formulation is made generally applicable to any type of steel

fiber-cement based matrix with any combination of fibers in the mix in any ratios. The

model was demonstrated for a RPC girder failing in shear using local and non-local

models. It was shown that the FE formulation is capable of modeling the RPC girder

with a good correlation observed compared to experimental load-displacement curves

and with the local shear failure mechanism.

2.7 Diverse Embedment Model (DEM) for steel fiber-reinforced concrete

in tension: model development – Seong-Cheol Lee, Jae-Yeol Cho and

Frank J. Vecchio

This Theory studies the beneficial aspect of adding discrete fibers in a concrete matrix,

giving ductility and making this matrix a non-brittle material.

In Figure 2.26 there are compared different types of behavior of Concrete subject to

tensile stress, and there are differences between the different types of concrete: 57

2. LITERATURE SURVEY

in Normal Concrete (NC) the tensile stress decreases rapidly after cracking while in

Fiber Reinforced Concrete (FRC), fibers bridging the crack improve the behavior and

contribute to the load-carrying mechanism.

Figure 2.26: Tensile behavior of FRC.

Several researches have been done to develop analytical models for the tensile behavior

of FRC, such as Marti et al, that derived a relationship between crack width and tensile

stress of FRC members: it was shown that tensile stress provided by fibers,

decreases with an increase in crack width.

Foster found that also the random distribution of fibers in the concrete matrix, was

important to evaluate the amount of fibers engaged: the fibers were taken into account

to carry the load if they have an inclination angle to the crack normal direction of less

⁄

π 3

than .

2.7.1 Theory Assumption

This Theory used an effective engagement concept, where fibers have an inclination

angle less than the critical value, increased with an increase in the crack width.

A constant bond stress between the steel fibers and the concrete matrix was assumed

(the model is so suitable for end-hooked fiber types). 58

2. LITERATURE SURVEY

As seen in the VEM (Variable Engagement Model) an effective engagement is used

where the fibers have an inclination angle less than the critical value, increased with an

increase in the crack width.

Fiber slip, as in the VEM model, occurs only on the side where the shorter part is

embedded, even though the crack width should equal the sum of the fiber slips from

both sides of the crack.

Obviously, the slip from the longer part of the fiber embedded may not be negligible, in

particular when the embedded lengths of the fiber at the crack on either side are

relatively similar.

In this thesis, the pullout characteristics of fibers and their potential orientation are taken

into account, so the tensile characteristic of the FRC can be correctly predicted.

2.7.2 Pullout behavior of single fiber

As previously mentioned, the tensile behavior of FRC is quite different from that of NC.

The tension model of FRC is derived analyzing two different conditions of the

embedment of fibers:

- Only one side of the fiber is embedded in the concrete matrix, while the other

side is free or fully fixed;

- Both sides of the fiber are embedded in the concrete matrix.

2.7.2.1 Pullout behavior of single straight fiber embedded on one side

While the concrete matrix can’t sustain tensile load over a certain amount, cracks occur.

When a crack forms in FRC, the tensile stresses took by fibers bridging cracks are

transferred back to the concrete matrix through bond behavior between both fibers and

concrete matrix as shown in the following Figure 2.27: 59

2. LITERATURE SURVEY

Figure 2.27: Free diagram for infinitesimal element with single straight fiber.

The pullout behavior of a single straight fiber is calculated as following equation (2.80),

from many researchers: 9 1 1

= P Ž • + –

w Œ

} (2.80)

Œ Œ

The pullout behavior of a single straight fiber embedded only on one side, must be

considered to occur in two stages because this fiber length is relatively short, compared

to its transfer length: firstly, when the slip is relatively small at a crack, the transfer

length is less than the embedded length of the fiber. Thus the fiber slips over only part

of its embedded length, as shown in following Figure 2.28:

Figure 2.28: Pullout behavior of a single straight fiber embedded on one side.

Secondly, when slip at crack is relatively large and the fiber slips extend over the entire

embedded length and slip at the end of the fiber occurs.

More important is when the fiber stress at a crack increase linearly up to a peak, where

the bond stress reach the full bond strength and then decrease linearly because there is a

reduction of the embedded length of the fiber in the matrix. 60

2. LITERATURE SURVEY

It can be assumed that the slip can considered constant along the fiber, and that

elongation of the fiber can be considered pointless: to be clear, the pullout behavior of a

single straight fiber can be considered as a rigid body translation (so the deformation

due to elastic elongation of the fiber can be neglected, as discussed previously in the

## VEM).

2.7.2.2 Pullout behavior of single straight fiber embedded on both sides

In this case, the crack width is composed of the sum of slips from both sides of the

cracked concrete: the slip on the shorter embedded part is bigger than of the longer

embedded one but, considering the body translation and a bilinear stress-slip

relationship between fiber and concrete matrix, an equilibrium equation can be derived:

(• )

Ž − 9 — 9 = Ž B• − • − 9 C — 9

Œ $ Už1T Œ Už1T Œ Œ $ …žNŸ Œ …žNŸ

9 9 (2.81)

Už1T …žNŸ

≤

For

where:

9

Œ Slip corresponding to the full bond strength;

• $ Fiber embedment length on shorter side;

•

Œ Fiber length;

—

Œ bond modulus, which is slope for elastic behavior in bond stress-slip

relationship for fiber, of which inclination angle is 0 degrees;

Œ Fiber diameter;

9

…žNŸ Slip at crack for longer embedded part of fiber;

9

…žNŸ Slip at crack for the shorter embedded part of fiber.

9 ,

Œ

When the shorter part has a slip bigger than the bond behavior of the longer part can

be considered an unloading mechanism on the bond stress-slip while the bond stress of

the shorter embedded part reached the bond strength. 61

2. LITERATURE SURVEY

Figure 2.29: Pullout Fiber stress at crack: crack width response.

The previous Figure 2.29 shows how the embedment of the fiber influence the stress in

the concrete matrix: as the fiber embedded lengths to each side approach one-half the

29

Œ

fiber length, the crack width for the peak fiber stress at a crack increase to .

It is also important consider if the fiber is normal to the crack surface or if it is inclined,

to consider how it is engaged to transfer the tensile stresses: this thesis shows that for

crimped steel fibers the inclination fibers with angle less than 30 degrees, the frictional

pullout behavior strength is slightly affected.

There are some contradistinction about the pullout strength of inclined fibers, but in this

theory the bond strength is considered constant, regardless of the variation of the fiber

inclination angle.

2.7.2.3 Pullout behavior of single end-hooked fiber embedded on both sides

This kind of fiber, as the previous, benefits of mechanical anchorage provided by the

end-hook: after that the slip amount exceeds the length of the end hook, the tensile force

due to the mechanical anchorage becomes zero because it is seen a deterioration of the

concrete matrix near the end hooks and a straightening of the hook.

Once that the shorter embedded part of a fiber reaches its pullout strength, it has a post-

peak behavior for the mechanical anchorage effect, whereas the longer part has a partial

62

2. LITERATURE SURVEY

unloading for frictional bond and mechanical anchorage.

There are three possible cases that must be considered, to calculate the stress at a crack

in a end-hooked steel fiber:

1. The end hook in the sorter embedded part of the fiber remains embedded;

2. The end-hook is pulled out;

3. The end hook in the shorter embedded part of the fiber was not originally fully

embedded.

In the case 1, it is assumed that:

(• )

Ž − 9 P + ‹ =

Œ $ Už1T Už1T ˆU, Už1T

Ž B• − • − 9 C P ‹

Œ Œ $ …žNŸ …žNŸ ˆU,…žNŸ (2.82)

⁄

• − 9 B• − • C 2

$ Už1T Œ ;

≥

For

where:

Œ Fiber diameter;

• $ Fiber embedment length on shorter side;

9 Už1T Slip at crack for shorter embedded part of fiber;

P Už1T Frictional bond stress for shorter embedded part of fiber;

‹

ˆU, Už1T Tensile forces due to mechanical anchorage of shorter embedded part of

end-hooked fiber.

“Long” is for the longer side of the embedded fiber.

So the stress of the fiber in the location of the crack is as the total amount of mechanical

anchorage and frictional bond stress along the fibers, as follows:

(• )

4P − • ‹

Už1T $ Už1T ˆU, Už1T

J = +

Œ, 1 Ž (2.83)

Œ Œ 63

2. LITERATURE SURVEY

where:

## J

Œ, 1 Fiber stress at crack with given fiber inclination angle and embedment

length;

P Už1T Frictional bond stress for shorter embedded part of fiber, respectively;

• Už1T Slip at crack for shorter embedded part of fiber;

‹

ˆU, Už1T Tensile forces due to mechanical anchorage of longer and shorter

embedded part of end-hooked fiber, respectively;

⁄

9 9 <=9

Už1T ˆU

If is less than , the fiber is at the maximum stress experienced at the

crack; it is obvious that, if the maximum experienced stress is more than the fiber

strength, the fiber has been broken.

In case 2, the deterioration of the concrete matrix near the fiber is considered possible if

⁄ ⁄

• B• − • C 2 9 B• − • C 2

$ Œ ; Už1T Œ ;

is between + and .

Considering that the anchorage resistance of the longer embedded part is bigger than the

shorter one, it can be assumed that the slip from the shorter embedded side is the

dominant part that govern the crack width. So this case 3 can be postulated as if the

fiber stress at crack is calculated from the pullout behavior of a straight fiber embedded

on only one side.

2.7.3 Consideration of member dimension and fiber embedded length

The general assumption is that the fiber inclination angle and the fiber embedment

length are randomly in the concrete matrix.

This thesis consider a steel fiber randomly oriented in a 3D infinite element, illustrated

as a sphere, as the following figure Figure 2.30 shows.

The probable density of a fiber inclination angle is expressed with a sine function and

the fiber stress t crack can be obtained as:

⁄

‰ (• )

J = J , sin

Œ, 1,G Œ, 1 $ (2.84)

c 64

2. LITERATURE SURVEY

Figure 2.30: Probability of fiber inclination angle using a sphere representation.

Also the behavior of fiber orientation in 2D member is subject of numerous study,

because when the fresh mixture of concrete is placed in forms and the dimension of the

member element is relatively small and the finishing of exposed surface are still present,

all this can influence the disposition in two dimension of the fibers.

In this case, the fiber orientation can be affected by three cases:

1. The fiber orientation is affected both by the long and the short embedded part of

the fiber;

2. The fiber orientation is affected only by the long embedded part of the fiber;

3. The fiber orientation is not affected.

Moreover, to define the tensile stress on a crack surface of a unit area, the number of

fiber that cross the crack surface should be known: this number is usually mentioned by

## V

Œ

employing a fiber orientation factor as: Œ

™ = V

Œ Œ (2.85)

Œ 65

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher ale.baselli di informazioni apprese con la frequenza delle lezioni di Tecnica delle costruzioni e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Brescia - Unibs o del prof Minelli Fausto.

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