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

Tesi di laurea "Experimental study on fiber-reinforced concrete panels subjected to shear", in Tecnica delle costruzioni, Corso di laurea magistrale in ingegneria edile-architettura dell'università di Brescia.
Tesi sperimentale sul comportamento di SFRC e MSNFRC.
Literature Survey.

Materia di Tecnica delle costruzioni relatore Prof. F. Minelli

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

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)

## 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 − )

$(2.55) and a trilinear stress-strain relation is used to model the response of the reinforcement in tension or compression: = 0 < < (2.56) = < < U (2.57) ( ) = + − < < U U U • (2.58) and = 0 > • (2.59) where: Yield Strength; Modulus of elasticity of steel bars; U Strain Hardening modulus; 45 2. LITERATURE SURVEY Yield Strain; U Strain at the beginning of the hardening; • Ultimate strain. 2.4.3 Slip Model δ ## H The crack slip is continuously under study in the all literature, and it is found be a a 05 function of the acting shear stress , of the crack width w, aggregate size and ' 0 00 concrete compressive cylinder strength (or cube strength ). Walraven (1981) proposed the follow formulation: ; b =$ (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

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

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

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

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DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria edile-architettura (a ciclo unico di durata quinquennale)
SSD:
Università: Brescia - Unibs
A.A.: 2013-2014

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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