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

IJK IL K I K K I K

= + MIL ; ; (2.29)

;O 37

2. LITERATURE SURVEY

where: Number of reinforcement components;

IJK Stress matrice;

IL K Concrete Stiffness matrice;

I K Net Strains in the concrete matrice;

IL K ; Reinforcement stiffness matrice;

I K ; Net Strains in the reinforcement matrice.

Equilibrium conditions become quite simple if the panel is in the special case where it is

orthogonally reinforced and the reinforcement is aligned with the reference axes:

J = + ∙ (2.30)

J = + ∙ (2.31)

and P = (2.32)

0 0 0

Concrete stresses , and are easily determined from the principal stresses

using the Mohr’s Circle of stress shown in Figure 2.18.

It is finally necessary to check that the average stresses can be transmitted across the

cracks.

For this reason, the principal tensile stress in concrete is taken to be zero along the crack

0

locations: to transmit the average stress , there are local increases in the

reinforcement stresses. H0Q

These local stresses are denoted as (as shown in Figure 2.19) and the magnitude of

R

0 that can be carried by this mechanism is limited by the capacity of the

reinforcement, which is given by the difference between the average and yield stresses.

38

2. LITERATURE SURVEY

So: N

≤ M B − C ∙ <=9

; S S (2.33)

;O

where:

; Reinforcement ratio;

Average Stress;

S : TU

Yielding Stress for the reinforcement component;

S

N; Angle obtained by the difference between the angle of the orientation of

V ;

the reinforcement (knows as ) and the normal to the crack surface

= − V

N ; (2.34)

S

H0Q

Local reinforcement stresses are determined from local reinforcement strains, and

R

must satisfy the equilibrium conditions: average concrete tensile stresses must be

transmissible across the cracks

N

M B − C <=9 =

; 1 S S (2.35)

;O

05

Shear stresses along the crack surfaces are generated from the local increases in the

reinforcement stresses at crack locations:

N

= M B − C <=9 ∙ sin

; ; 1 N N

S S S S (2.36)

;O

Compatibility conditions are taken into account for a reinforced concrete element

where the straining is obtained as sum of mechanical compliance and smearing of

cracks widths over a finite area.

The slip component is the result of the rigid body movement along the crack interface

and using extensometers of a gauge length sufficient to span several cracks, the

measured strains contain both component of deformation. 39

2. LITERATURE SURVEY

IεK = Wε , ε , γ Y

These measured total (or apparent) strains are denoted as and the

θ Z

apparent inclination of the principal strains is calculated by:

1

= % )

\

[ 2 − (2.37)

while the net (or actual) strains within the continuum are denoted as:

I K= , ,

W Y (2.38)

So the principal strains are determined from the net strains, using the standard

transformations: B + C 1

, = ± ^B − C + _ (2.39)

2 2 θ

So the actual inclination of the principal strains in the continuum and the assumed

θ 0Q

inclination of the principal stresses are compared in:

1

= = % )

\

1 2 − (2.40) θ,

Assuming that cracks are inclined in the directions of the net principal tensile strain

w s,

and spacing and that the slip along the crack

that cracks have an average width

δ

H

surface is of magnitude , the average shear slip can be defined as:

b

= 9 (2.41)

where:

w Average width;

s Spacing;

δ

H Slip along the crack surface; 40

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 th