Journal of Advanced Concrete Technology Vol. 5, No. 2, 247-258, June 2007 / Copyright © 2007 Japan Concrete Institute247 Scientific paper
A Cohesive Interface Model for the Pullout of Inclined Steel Fibers in Cementitious Matrixes
Alessandro P. Fantilli1 and Paolo Vallini2
Received 5 April 2007, accepted 15 May 2007
Abstract
The nonlinear behavior of fractured quasi-brittle materials is conventionally modeled with a fictitious crack model, which relates stress on the crack surfaces to the corresponding crack widths. Its definition for fiber reinforced con-crete is only possible by introducing a cohesive model for the matrix, and by modeling the pullout of randomly oriented fibers. To this aim, a new cohesive interface model, able to predict effectively the pullout respon of inclined fiber, is prented in this paper. Bad on the nonlinear behavior of steel fibers and cementitious matrixes, the propod ap-proach also takes into account the bond-slip relationship between the materials. By means of an iterative procedure, numerical results similar to experimental data can be obtained. In particular, maximum pu
惊蛰节气的习俗llout forces at given inclina-tion angles, as well as the complete pullout load vs. displacement diagrams, can be correctly predicted. Moreover, ac-cording to test results, the propod approach shows, from the first pullout stage, the dependence of the respon both on crushing of cementitious matrix and on yield strength of steel fibers.
上古大神1. Introduction
Fracture energy of Fiber Reinforced Cementitious Com-posites (FRC) can be higher than that relead by tradi-tional cement-bad concretes. Practically, in the cohe-sive relationship, at a given crack width, higher tensile stress can be detected on the crack surfaces of FRC, becau of the bridging effect produced by fibers. To better define this phenomenon, the pullout respon of a single fiber in a cementitious matrix needs to be investi-gated (Hillerborg, 1980).
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Pullout involves fibers inclined with respect to crack surfaces, as they are randomly positioned within the matrix. This has been investigated both theoretically and experimentally (, Shah et al., 2004). In a huge number of tests, fibers made of different materials and shapes have been pulled out from cement-bad or plas-tic specimens. For the sake of simplicity, only pullout of straight steel fibers in a cementitious matrix is consid-ered in the prent paper.
In the first tests by Naaman and Shah (1976), the maximum pullout load of inclined fibers appeared higher than that measured in specimens with aligned fibers (fibers direction orthogonal to crack surfaces). Moreover, if the pullout diagram (that is, load P vs. dis-placement w) of an aligned fiber ends approximately with zero loads, in ca of inclined fibers a significant load persists up to the complete slippage. In other words, due to the nonlinear behavior of materials, the area un-der a P - w curve, usually called pullout work, generally increas with fiber inclination. For the reasons, ac-cording to the results of Leung and Shapiro (1999), in specimens having the same cementitious matrix, the yield strength of steel fibers plays a fundamental role on the crack-bridging efficiency. In fact, at the same incli-nation angle, both the maximum load and the pullout work em to increa with the yield strength of steel fibers (Leung and Shapiro, 1999).
It must be remarked that bond properties and yield strengths are strictly connected to fiber production (e.g. hot rolled or cold drawn, e CEB, 1991). Thus, to model the pullout of inclined fibers, the bond-slip mechanism between fiber and matrix cannot be ne-glected. This is also confirmed by the pullout tests con-ducted on unbonded fibers (Kohno and Mihashi, 2005). Although their pullout work is nearly equal to zero when fibers are orthogonal to crack surfaces, significant values of the maximum load and of the pullout work can be equally obrved in the ca of inclined f
ibers. De-pending on the way this mechanism is taken into con-sideration, the models reported in the current literature can be classified into different groups (Shah and Ouy-ang, 1991).
Classical approaches assume that the condition of perfect bond persists till either stress or energy criterion is exceeded on the interface of fiber and matrix. In models founded on stress criterion, debonding begins and slip between fiber and matrix takes place after the maximum admissible value of bond stress is reached (Stang et al., 1990). Constant bond stress are assumed to be prent in the debonded zone (pure frictional model), while in the remaining part of the fiber the con-dition of perfect bond (zero slips) is assumed. Several pullout models have been founded on this criterion, like the analytical solution propod by Morton and Groves (1974), and the most recent numerical approach intro-
1Assistant Professor, Department of Structural and Geotechnical Engineering, Politecnico di Torino, Italy.
E-mail: alessandro.fantilli@polito.it
2Professor, Department of Structural and Geotechnical Engineering, Politecnico di Torino, Italy.
248 A. P. Fantilli and P. Vallini / Journal of Advanced Concrete Technology Vol. 5, No. 2, 247-258, 2007
duced by Katz and Li (1995). In addition to the debond-ing phenomenon, the last model is able to take into ac-count the stiffness contribution of the fiber portion which protrudes from the matrix (assumed to be in the linear elastic regime), while the embedded part of fiber interacts with the matrix similar to a beam on elastic foundation (Leung and Li, 1992).
Very few models have been bad on the energy crite-rion. In this ca, only when the energy relea rate of debonding reaches its critical value, does slip between materials begin (Shah and Ouyang, 1991). Becau of difficulties in measuring the critical interface debonding energy, the approaches cannot be applied to real cas, with the exception of the model propod by Brandt (1985) to define the optimal inclination angle. Besides stress and energy criteria, the pullout of in-clined fibers can be investigated by means of the so-called cohesive interface models, in which bond stress are only due to slip between steel and cement-bad matrix (Shah and Ouyang, 1991). Such models, as well as the tension stiffening investigation of reinforced con-crete structures, consist of the classical equilibrium and compatibility equations (Fantilli et al., 1998). Cohesive interface models are ldom ud in the ca of fiber pullout, becau bond properties of fiber and
matrix cannot be generalized, but have to be measured in each single ca (Shah and Ouyang, 1991). Moreover, the mathematical solution of the problem does not generally carry out analytical formulae, and therefore numerical iterative procedures are needed. This has been done by Fantilli and Vallini (2003a) to model the complete pull-out respon of aligned steel fibers in cementitious ma-trixes. Such a model, bad on the definition of a suit-able bond-slip relationship (Fantilli and Vallini, 2003b), is here extended to the analysis of fibers inclined with respect to crack surfaces.
2. Equilibrium and compatibility equations of a cohesive interface model
The bridging action of a fiber, initially inclined of αrespect to the crack surfaces of a cementitious matrix, is shown in Fig. 1. The crack width 2w is produced by a horizontal displacement (w = pullout displacement) im-pod to the whole composite (Fig. 1b). The final posi-tion of the fiber is illustrated in Fig. 1c, where, due to symmetry, only half fiber is shown.
When w > 0, the points named A, A’ and A”, which coincide for w = 0, will be parated as a conquence of a slip s between fiber and matrix, and of matrix spalling failure that affects the length d (measured on the original position of the fiber). According to Leung and Li (1992), the pullout load P, which has to be ap-plied in order to produce the displacement w, can be computed by splitting the
fiber into two parts (Fig. 1c): the block delimited by the points A” and B” (named block A”-B”), which protrudes from the matrix, and the block delimited by the points B” and C’(named block B”-C’), which is embedded in the matrix. C and C’ are the position of fiber’s end before and after the applica-tion of w, respectively.
Fig. 1 Pullout of an inclined fiber: a) undeformed state (w = 0); b) deformed state (w≠ 0); c) position of the fiber at a given displacement w (or load P ).
A. P. Fantilli and P. Vallini / Journal of Advanced Concrete Technology Vol. 5, No. 2, 247-258, 2007 249
2.1 The block A”-B”
The initial position of the fiber, defined in Fig. 1c by the points A and B and by the angle α, changes in con-quence of the displacement w . The new position, de-fined by the point A” and B” in Fig. 2, is univocally defined by the kinematical variables α, d , and w , and by the effects of matrix deformation (i.e. the rotation θB and the displacement ηB of the point B”). Therefore, the real length l 0 of the block A”-B” and the complete rotation δ of the fiber can be respectively computed with the following equations:
()()2
20cos sin sin cos w d d l B B +−++=
αηααηα (1)
⎟⎟⎠
⎞⎜⎜⎝⎛+−+−−=−w d d tg B B αηααηααπ
δcos sin sin cos 21 (2) where π = 3.1426. The angle δ can be considered as the
sum of three different rotations:
t i B θθθδ++= (3) where θt
= rotation produced by shear actions; θi = rotation produced by bending moment. Both elastic
and plastic components (named θe and θp ) have to be
included in the last rotation:
p e i θθθ+= (4)
When all the contributions are known, the apparent length l t of the block A”-B” (Fig. 2) and the slip s are obtained as follows:
z l l i t Δ+=θcos 0 (5)
d l s t −= (6)
The apparent length reduction Δz , which appears in the Eq. (5), is a function of the deflection ηi (z ) of the fiber. If a sinusoidal function is assumed for ηi (z ), the apparent shortening has the following f
orm:
z y l i l l dz z z z
2
22
1621πη=⎟⎠⎞
⎜
⎝⎛∂∂=Δ∫
if ⎟
⎟⎠⎞
⎜⎜⎝
⎛−=z y i l z l 2cos 1πη (7) where l z = l 0 cos θi = z component of l 0 (Fig. 2); l y = l 0 sin θi = max
imum deflection of the fiber (y component of l 0). The static conditions of the block A”-B” is depicted in Fig. 3a . Due to symmetry, the bending moment M A =0,
while both shear and normal forces, defined respectively
in the y and z directions, are constant in each cross-ction of the fiber (T A =T B and N A =N B ). Therefore, by
considering cond order effects, the bending moment at
the point B” results:
i A i A B l N l T M θθsin cos 00−= (8)
Under the hypothesis of linear elastic behavior, the
rotations θt and θe of the fiber can be written as func-tions of T B and M B
, respectively:
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f f t B
t A G f T =θ (9) f
f z B
e J E l M 24πθ= (10) where,
f t = dimensionless shear constant (for a circular
cross-ction f t = 32/27); G f = shear modulus of the fi-ber; E f = Young’s modulus of the fiber; A f = area of the fiber cross-ction; J f = moment of inertia of the fiber cross-ction.
In the block A”-B” of length l z = n Φ ( Φ = diameter of the fiber cross-ction; n = integer number), it can be of practical interest to evaluate the ratio q T between shear and flexural rotations:
n
q e t T Φ
谎话
==402.0θθ (11) From Eq.(11) it is possible to obrve the necessity of taking into account the shear contribution to rotation in ca of narrow crack widths (that is l z < 10 Φ).
2.2 The block B”-C’
For the embedded part of the fiber, modeled as a beam on elastic foundation (Leung and Li, 1992), it is possi-ble to introduce a relationship between the kinematical variables (θB and ηB ) and the static actions (T B and M B ):
⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡=⎭⎬⎫⎩⎨⎧B B B B M T d d d d 2221
1211θη
where the coefficients d ij of the deformability matrix (that is, the inver of the stiffness matrix) have to be evaluated as functions of the matrix foundation stiffness K . If the Young’s modulus E m and the Poisson’s ratio νm of the matrix are known, K is given by (e Fig. 3c and Appendix 2):
()
651
.212
m m E K ν−=According to the classical books on foundations (Hetenyi, 1946; Bowles, 1988), the differential equation for the deflection curve of a beam supported on an elas-tic foundation is bad on the factor β, who inver is usually called characteristic length. In the prent ca, it connects the flexural stiffness of the fiber ( E f J f ) and the matrix foundation stiffness previously computed: ()
44
2
4
104.64Φ−==πνβf m
m f f E E J E K If the length of the considered block l BC = l i - d - s ≥ π/β (Fig. 3b ), the embedded part of the fiber can be
considered as a long beam and therefore the coefficients d ij of Eq. (12) are tho of a mi-infinite beam with free end (Hetenyi, 1946; Bowles, 1988):
f f J E d 31121β= f f J E d d 2211221β== f
怎么给文件夹设密码f J E d β1
22= (15) On the contrary, if l BC ≤ 0.25 π/β , the supported part of the fiber can be considered absolutely rigid (like a short beam), thus d ij can be determined by simple static considerations:
BC l K d 411=
221126BC l K d d == 32212
BC
l K d = (16) When the block B”-C’ is of medium length, 0.25 π/β < l BC ≤ π/β , the coefficients d ij of Eq. (12) are computed by means of a linear interpolation of Eq. (15) and Eq. (16).
The axial load N B in the cross-ction B” of the fiber can be considered as a function of the bond load N bs and Fig. 3 Free body diagrams of the inclined fiber: a) ac-tions in the block A”-B”; b) actions in the block B”-C’; c) stress-diffusion in the ction F-F.
A. P. Fantilli and P. Vallini / Journal of Advanced Concrete Technology Vol. 5, No. 2, 247-258, 2007 251
B T B M bs B T K M K N N ++= (17) where the coefficients K T and K M are computed with the hypothes that the embedded part of the fiber lies on an elastic matrix foundation and by assuming a suitable friction coefficient γ between materials. If l B
C ≥ π/β (Fig. 3b ), the coefficients K T and K M are respectively (Hetenyi, 1946; Bowles, 1988):
()γγ4.1≅=∫BC
l B B T dz T y T K
K (18a)
()Φ≅=∫γγBC
l B B M dz M y M K K (18b)
In the ca of short beam ( l BC ≤ 0.1 π/β) the previous coefficients can be written as: ()γγ35≅=∫BC
l B B T dz T y T K
K (19a)
()BC l B B M l dz M y M K K BC γγ3≅=∫ (19b) For medium lengths, 0.25 π/β < l BC ≤ π/β , the coeffi-cients K T and K M are computed by means of a linear
interpolation of Eqs. (18) and Eqs. (19).
Since the bond force N bs is a function of the slip s fm (ζ) between fiber and matrix within the block B”-C’ (Fig. 3b ), it can be obtained by solving the classical tension-stiffening problem, which consists of the fol-lowing system of equilibrium and compatibility equa-tions (Fantilli and Vallini, 2003a):
()()⎪⎪⎪
⎪⎩
⎪
⎪⎪⎪⎨⎧Φ−==−=ζτζσσζεζ4d d A N d ds f做梦生男孩
f f f fm
(20) where εf = axial strain in the fiber; σf = axial stress in the fiber; and τ(ζ) = bond stress at fiber-matrix inter-face. To solve Eqs. (20), the boundary conditions [N (ζ=0) = N bs and N(ζ = l BC ) = 0] and a suitable bond-slip relationship τ(s) have to be introduced. In this way, all the possible bond mechanisms (e.g., slip softening or slip hardening) are taken into account in the computa-tion of normal force N bs . Thus, according to Shah and Ouyang (1991), the propod approach can be classified in the family of cohesive interface models. When N bs is known, Eq. (17) can be inrted into Eq. (8) (N A = N B and T A = T B ), in order to obtain a new equation for the shear force T B :
()()
i T i i
bs i M B B l K l l N l K M T θθθθsin cos sin sin 10000−++= (21)
3. A possible solution of the problem
From the equations written in the previous ction, the
complete pullout diagrams (load P vs. displacement w ) can be theoretically defined and compared with tho measured experimentally. This is possible after defining the constitutive relationships of materials, their interac-tion [that is, τ(s ) function], and by introducing a nu-merical procedure for the solution of the problem.
3.1 The bond-slip relationship and the friction coefficient between fiber and matrix
For smooth steel fibers in a cementitious matrix, the model propod by Fantilli and Vallini (2003b) can be adopted. It consists of an improvement and an extension
of the classical model propod by Model Code 90 (CEB, 1991) for smooth steel reinforcing bars. In par-ticular, both for bars and fibers, the post peak softening
is introduced in conjunction with the size effect pro-duced by fiber diameter on bond strength. The ascend-ing branch and the post-peak stage of the propod
bond-slip relationship (Fig. 4a ) are respectively defined by the following equations: 5
.01max ⎟⎟⎠
⎞
⎜⎜⎝⎛=s s ττ if 1s s ≤ (22a)
()()s s k fin fin e −−+=1max ττττ if 1s s > (22b)
where τmax = bond strength; s 1 = slip at bond strength; τfin = asymptotic value of bond stress; k = coefficient. The parameters are defined in Fig. 4b as a function of bond conditions, of the type of smooth bar (hot rolled or cold drawn), and of the compressive strength f c of the matrix.
The maximum bond stress is here considered as a function of the fiber diameter according to the Bazant’s
size effect law (Bazant et al ., 1995) for hot rolled bars
and for cold drawn wires, respectively:
c f Φ+=
5.12572.1max τ (23a)
c f ⎟⎟⎠
⎞
⎜⎜⎝⎛−Φ+=2.05.12572
.1max τ (23b)
where τmax and f c are measured in MPa, while Φ is
measured in mm.
The bond-slip relationship previously defined cannot be ud for all kind of fibers. Nevertheless, if a new τ-s has to be defined for other types of fiber (hooked, twisted, etc.), the procedure propod in Fantilli and Vallini (2003b) can be adopted. In particular, after defin-ing the shape of the relationship, the possible variation of its parameters should be measured, at different scales, by means of pullout tests on aligned fibers.
The cold or hot manufacturing of reinforcement steel produces not only different bond stress between bars
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(or fibers) and matrixes, but also different friction coef-ficients γ . Since no experimental campaign has been devoted to evaluate all the possible values of such coef-ficient, it must be considered as a free parameter, which can vary within the range 0.1÷0.3 in ca of steel fibers in cementitious matrixes.
3.2 The mechanical behavior of steel fibers
The pullout tests by Leung and Shapiro (1999) clearly show the importance of yield strength f y of in
clined steel fibers in a cementitious matrix. For this reason, it is necessary to take into account the nonlinear respon of fibers, which is here included in the bilinear stress-strain σf -εf relationship depicted in Fig. 5a . In this diagram, the linear elastic branch is univocally defined by the Young’s modulus E f , whereas the non linear stage, as-sumed to be perfectly plastic, is only defined by the yield strength f y .
Yielding of an inclined steel fiber is generally reached around point B” (Fig. 2), where fiber cross-ction is contemporarily subjected to normal, shear and bending actions (N B , T B and M B , respectively). At this stage, the plastic rotation θp [Eq. (4)], which is zero during the elastic stage, can be incread indefinitely. The defini-tions of the yield surface of a fiber, having a circular cross-ction and subjected to N B , T B and M B , is there-fore of primary importance. For instance, this is possible by means of Hodge’s approach (Chen and Han, 1988). However, in the ca of a circular cross-ction made of an elastic-plastic material (Fig. 5a ), it provides a yield-ing surface which can be adequately modelled by the following ellipsoidal equation:
2
02
02
⎟⎟
⎠⎞
⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛=T T N N M
M B B B λ (24) where M 0 , T 0 and N 0 are, respectively, the limiting yield
values of bending moment, shear and normal forces of a circular cross-ction (Chen and Han, 1988). If the val-
ues of N B , T B and M B give λ > 1, yield conditions are violated, thus there must be an increa of θp until Eq. (24) gives λ = 1.
3.3 The mechanical behavior of damaged ma-trix
欢度国庆节手抄报图片Undamaged matrix generally behaves linearly. When failure conditions are reached, damage occurs and pieces of matrix are progressively broken away from the fiber. As Fig. 1c shows, the damaged z
one progressively increas its length d with the increa of w . To be more preci, the position of point B” should be clor to the point C’, if the resultant of the applied loads exceed matrix strength. Referring to the beam on elastic foun-dation depicted in Fig. 3b , the normal force H y trans-ferred by the fiber to the matrix is (Hetenyi, 1946; Bowles, 1988):
()()B B
B
B
y T M
dz T y K dz M y K
H 2
4/40
/80
π
πβ
β
β
+
==+=∫∫ (25)
The resultant of applied loads can be obtained by combining H y and the friction forces H z of Eq. (17), which are suppod to be applied at point B” (Fig. 3b ):
B T B M z T K M K H += (26)
If the linear Mohr-Coulomb failure criterion is as-sumed for the cementitious matrix (Fig. 5b ), the frac-tured cone surface has its axis parallel to y under pure compression (Fig. 5c ); whereas, in pure tension, the axis is parallel to z (Fig. 5d ). In the cas, matrix resis-tances are given respectively by:
()c y f d R 2
新朝王莽sin 2απ= (27a) ()ct z f tg d R 2sin ψαπ=
(27b)
of cold drawn wires and hot rolled bars (Fantilli and Vallini, 2003b).
