Available online at www.rilem.net Materials and Structures Vol (Month Year) pp-pp
A physical model for the prediction of lateral stress exerted by self-compacting concrete on formwork G. Ovarlez1, N. Roussel2* (1) Laboratoire des Matériaux et Structures du Génie Civil, Champs sur Marne (France). (2) Laboratoire Central des Ponts et Chaussées, Paris, (France). * to whom correspondence should be addressed. Received: 20 03 2005; accepted: 03 08 2005
ABSTRACT In this paper, a model is proposed to describe the evolutions of the lateral stress exerted by self compacting concrete (SCC) on a formwork during and after casting. The predictions of the model are compared to pressure drop measurements after the end of casting carried out on real formwork simultaneously with measurements of the evolution of the apparent yield stress of the cast concrete. Then, the predictions of the model during the casting phase are compared to results from the literature and show that the proposed model is able to explain and predict the experimental observations and the quantitative evolutions. 1359-5997 © 2005 RILEM. All rights reserved.
RÉSUMÉ Nous présentons dans cet article un modèle permettant de décrire l’évolution de la contrainte latérale exercée par un Béton AutoPlaçant (BAP) sur la paroi du coffrage pendant et après le coulage. Nous comparons les prédictions du modèle à des mesures de chute de pression latérale à la fin du coulage associées à des mesures de l’évolution au repos du seuil d’écoulement du matériau. Puis, nous comparons les prédictions du modèle pendant la phase de coulage à des mesures expérimentales tirées de la littérature. Le modèle proposé s’avère capable d’expliquer l’ensemble des observations expérimentales ainsi que de prédire les évolutions quantitatives de la pression latérale.
1. INTRODUCTION Self Consolidating Concrete (SCC) flows readily under its own weight and achieves good consolidation without any mechanical vibration. During casting, given the high fluidity of this type of concrete, it can be expected that a hydrostatic pressure will be reached in the formwork and formworks were prudently designed by taking into account this high pressure. Such an approach, however, increases the cost of the formwork and limits the maximum allowable placement height, which was advertised as an advantage of SCC. In cases around the world, the pressure was, when monitored, reported as being very high but, in other cases, opposite results were reported. It was concluded that the thixotropic behavior of the SCC had to play a role [1, 2]. During placing, the material behaves indeed as a fluid but, if cast slowly enough or if at rest, it builds up an internal structure and has the ability to withstand the load from 1359-5997 © 2005 RILEM. All rights reserved. doi:10.1617/*****
concrete cast above it without increasing the lateral stress against the formwork. This conclusion was drawn by observing the decrease in pressure in the first hours after casting. Even though the hydration process has not yet started [3-4], the lateral stress on the formwork wall steadily decreases. The only phenomenon that can occur at this time in the fresh concrete is a flocculation [5, 6] if we assume that the material is stable (no segregation). This flocculation is linked to the strong thixotropic behavior of these modern concretes. Rheological behavior of self compacting concretes, as many concentrated suspensions, may be approximated by a yield stress model [7, 8]. This type of model allows the description of the steady state shear flow of the material: τ = τ0 + f(γ) where τ is the shear stress and γ is the shear rate. τ0 is the yield stress and f is a positive increasing function of the shear rate. f(γ) and τ0 may be measured using a concrete rheometer (BTRheom [9], BML [10], two-point test [11] and others [12, 13]. Ref. MS1878
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However, self compacting concretes are also known for their evolving rheological behavior. Even if their steady flow may be described by the above models, the characteristic time to reach it may be rather long [14] and, after a long time of rest, the stress to be applied for flow to occur may be one or two orders higher than the yield stress measured when the material is flowing [15]. These two aspects answer to the two parts of the definition of a thixotropic material: After a time of rest, when a thixotropic material is submitted to a stress or a shear rate, its apparent viscosity is a decreasing function of the duration of the flow. The material comes back to its initial state after a time of rest. These evolutions may be explained by a reversible decrease of the flocculation state under shear or an increase if the material is left at rest. This flocculation and deflocculation phenomena have recently been studied by Jarny and co-workers [16] in the case of a white cement paste. This flocculation process is also called sometimes “abnormal setting” [17] in the literature. It has however nothing to do with setting as materials that do not display any setting property such as bentonite clay behaves just the same once they have stopped flowing. When the material is left at rest, its apparent yield stress increases: τ0 = g(t) with g(t) an increasing function of time. The most complete experimental work linking thixotropy and formwork pressure found in the literature has been carried out by J. Assaad and K.H. Khayat and their co-workers [2, 4-6]. It deals with the two aspects of the problem quoted above: quantifying the thixotropic behavior [5] and measuring the formwork pressure [6]. Instead of using the H-shaped impeller of a Tattersall concrete rheometer, a four-bladed vane impeller was used. This resulted in less slip flow of fresh concrete and an increase in the sheared surface during rotation. This allowed measurement of the apparent viscosity and the yield stress after a 4 minutes resting time. Test results on various SCCs show that thixotropy is not an inherent property of a given slump flow consistency. In [2,6], the lateral stress and its evolution in time was measured in columns filled with the same SCC at various casting rates. The experimental results are given in [2]. An adequate model should be able to explain the following observations at least qualitatively. _ When the SCC is injected from the bottom of the formwork, the lateral stress on the wall is equal to the hydrostatic pressure whereas it is not always the case when the concrete is poured from the top [1]. _ The casting section plays a role. It is considered that the maximum pressure should be smaller in formworks of limited cross sections [6]. _ There is a scale effect on the pressure drop rate once casting is over [6]. The drop in the ratio lateral stress/hydrostatic pressure increases with the height of concrete cast. _ There is a contradiction in the literature between two statements: the maximum lateral pressure is slightly
affected by casting rate [6] and the casting rate plays a major role [1]. In this paper, a model is presented and its physical meaning is justified on a theoretical point of view. Second its predictions are compared to pressure drop measurements after the end of casting carried out on real formwork simultaneously with measurements of the evolution at rest of the apparent yield stress of the cast concrete. Finally, the predictions of the model during the casting phase are compared to results from the literature and show that the proposed model is able to explain and predict the above experimental observations.
2. THEORETICAL MODEL In the following, calculations of the stress field performed in a rectangular formwork with an elastic modeling and a Tresca plasticity criterion are presented. It is shown that they give the same mechanical relation between vertical and horizontal stresses as the Janssen model used to describe granular materials; the lower normal stresses value at the walls in a yield stress fluid is then computed without any mechanical assumption; finally, the Janssen parameter value in the case of SCC is discussed.
2.1. Stress field in a formwork filled with a yield stress fluid We consider here that SCC is characterized by a yield stress τ0, which is a increasing function of the resting time. For the sake of simplicity, we consider that the yield criterion is a Tresca criterion i.e. τ0 is the maximum shear stress sustainable by an internal plane. Moreover, we assume in this section that, at stresses below the yield stress, the SCC behaves as an elastic material [18]. We now recall the general framework of homogeneous isotropic linear elasticity, and then predict the behavior of an elastic material confined in a rectangular formwork. The elastic theory gives, for small deformations, a linear relation between the stress tensor components σij and the strain tensor components εij. For an isotropic elastic material, we get Eεij = (1+νp)σij - νpδijσkk (1) where E is the Young modulus, and νp the Poisson ratio. In the following, we use the coordinates x in the width direction, y in the thickness direction; the vertical direction z is oriented downwards. The top surface is the plane z=0; the walls are the planes x=±L/2 and y=±e/2. We now confine an elastic medium of density ρ in this rigid rectangular formwork of length L and width e. We can compute the elastic solution of the stress profile in the formwork. The boundary conditions we impose is the Tresca condition everywhere at the walls (2) σ xz ( L / 2, y , z ) = σ yz ( x, e / 2, z ) = τ 0 and infinitely rigid walls i.e. displacements (3) ux(±L/2,y,z)= uy(x,±e/2,z)=0 Using the stress-strain relation (1) and internal equilibrium relation ∂iσij=-ρ gj we find
G. Ovarlez , N. Roussel / Materials and Structures Vol (year) pp-pp
σ zz ( z ) = ( − ρ g + 2τ 0 (1/ L + 1/ e ) ) z
(4)
σ xx ( z ) = σ yy ( z ) = K ( − ρ g + 2τ 0 (1/ L + 1/ e ) ) z
(5)
σ xz ( x, y, z ) = −2τ 0 x / L and σ yz ( x, y, z ) = −2τ 0 y / e
(6)
In the above approach, the Janssen parameter K, defined as the ratio of horizontal to vertical stresses, is related to the Poisson ratio νp: K=νp/(1-νp) (7) For a free elastic medium, the Poisson ratio effect is a transverse dilatation; for a confined elastic material, the Poisson ratio effect is a transverse redirection of stresses. Therefore, the stresses at the walls may be less than the hydrostatic pressure for two reasons: if the stress redirection parameter K is less than 1, or if some shear stress is supported by the walls. As a priori the shear stress at the walls can take any value between 0 and τ0, the normal stresses at the walls are always a value between this profile and the hydrostatic value. Note a limitation of the model: for low values of K, the yield criterion may be attained in the bulk but we will show that, in the case of SCC with standard air contents, this is not the case. However, note that that there are several problems with this solution: _ First, the Tresca condition at z=0 is not compatible with a free top surface for which τ0(±L/2,y,z)=τ0(x,±e/2,z)=0. _ Second, the vertical displacement is parabolic, which is not compatible with a flat displacement imposed by a rigid base: the solution may have to be slightly modified near the bottom of the column. _ Finally, and of greater importance, the Tresca condition may be satisfied somewhere in the bulk if the material is compressible (i.e. K≠1): e.g., at the center of the formwork x=y=0, the maximum shear stress on a internal plane is
τm =
1 1 − 2ν p 2 2 1 −ν p
1 1 ρ g − 2τ 0 + z L e
(8)
However, we will show that, in the case of SCC with standard air contents, K value is near 1, so that this concern may not be a problem. Moreover, if K≠1 the Tresca condition at the walls is not satisfied along the walls but in some other plane: in that case, the derived calculation is a good approximation as long as the yield stress τ0 is much higher than the maximum shear stress τm. It is worth noting that we obtain here a Janssen like equation: σxx=σyy=Kσzz. The Janssen model, which captures the physics of pressure saturation in silos [19], is based on the equilibrium of material slices taken at the onset of sliding everywhere at the walls, with the assumptions (i) of perfect frictional contacts at the walls, and (ii) that vertical and radial stresses are proportional: σrr(z)= Kσzz(z) (9)
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It has been shown [20] that this model agrees very well with experiments performed in granular columns. However, we see that there is no need for force chains, as in granular materials, for writing such a proportionality between vertical and radial stresses. Actually, this relation holds (at least asymptotically) for an elastic material confined in a column with frictional contacts [20,21] or yield stress at the walls (in the present work), and gives the same pressure profile as the Janssen model [20,21]; moreover, it has been shown that an elastic modeling is more appropriate regarding changes in boundary conditions even for granular materials [20,21]. Note however that our model, which considers a constant yield stress at the walls, is different from the granular material case where a frictional shear stress, which is proportional to normal wall stress, is considered. It has to be noted that boundaries on stresses at the walls may be computed out of any mechanical model (the assumption of an elastic behavior below the yield stress is not compulsory to obtain a solution), just by considering it as a Tresca material. The Tresca criterion limits the stress that the material imposes on the walls: (10) σ xz ( L / 2, y, z ) ≥ −τ 0 and σ yz ( x, e / 2, z ) ≥ −τ 0 Moreover, the material should not yield in the bulk, i.e.
σ xx ( x, y, z ) − σ zz ( x, y, z ) ≤ 2τ 0 and σ yy ( x, y, z ) − σ zz ( x, y, z ) ≤ 2τ 0
(11)
(here we suppose that σ xx , σ yy ≤ σ zz ). The equilibrium equation of horizontal slices can be written: L 2 e2
∂σ zz (x , y, z ) dxdy = ∂z
∫ ∫
−L 2 e 2
e2
− ρgLe − 2
∫σ
−e 2
xz
(L
2 , y, z )dy − 2
(12)
L 2
∫σ
xz
(x , e
2 , z )dx
−L 2
If we assume that σxx(x,y,z) and σyy(x,y,z) do not vary much with x and y so that: L/2 e/2
∫ ∫ ( −σ
yy
( x, y, z ) ) dxdy = − σ yy ( z ) Le and
xx
( x, y, z ) ) dxdy = −σ xx ( z ) Le ,
-L/2 − e / 2
L/2 e/2
∫ ∫ ( −σ
-L/2 − e / 2
combining Eq. (12) with Eqs. (10) and (11), we obtain a lower boundary on normal stresses at the walls similar to that obtained with elasticity, without any assumption on the mechanical modeling of the material.
−σ yy ( z ) ≥ ρ gz − 2τ 0 z (1/ L + 1/ e) − 2τ 0 and −σ xx ( z ) ≥ ρ gz − 2τ 0 z (1/ L + 1/ e) − 2τ 0
(13)
The main difference with the precedent prediction (Eq. (5)) is for depths of order or less than the width: the boundary can be lower than in the elastic model. Finally, it has to be noted that, in such an approach, the shear stress at the walls is between 0 and τ0, depending on
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the local deformation. For a purely non compressible fluid in transition towards a solid behavior, this means that, even if the yield stress of the material is evolving, there should not be any change at the walls as there is no deformation. However, in the case of concrete, this deformation can occur as the material slightly consolidates under its own weight [6] and settlement of the surface can be measured. A vertical deformation of order 0.003 was measured by [6]. It induces a shear deformation at the walls of order 0.003*H*2/e which is of order 0.05 (H=4m, e=20cm). This is sufficient to enable full shear stress mobilization as the critical deformation of cement paste is of the order of 0.0001 [18]. This means that the assumption that the shear yield stress is fully mobilized at the wall is valid and stays valid as the yields tress is increasing with the resting time.
2.2. Discussion on the k parameter value in the case of concrete As above, as a first approximation, we can consider that the material is incompressible, i.e. νp≈ 0.5 and K = 1. However, in concrete, there is some air trapped in the material which renders it compressible. It is then of importance to evaluate the effect on the Poisson ratio. We use the Mori-Tanaka homogenization scheme [22]; in this framework, the Poisson ratio of an incompressible isotropic elastic material including a fraction φ of spherical air inclusions can be computed
νp=
6+ϕ 6+ϕ i.e. K = 12+11ϕ 6+10ϕ
(14)
This yields K=0.97 for φ=2 %, K=0.945 for φ=4 %. This means that, just because SCC is not vibrated and has a rather high amount of trapped air, the lateral stress on the wall may be up to 10% lower than the hydrostatic pressure.
2.3. Comparison with existing models Empirical correlations can be found in the literature [2] but the only attempt to predict the lateral stress on formworks and its evolution based on a physical approach of the phenomenon is the one proposed by Vanhove and coworkers [23]. It is based on the assumption that the shear stress at the walls is due to a dynamic or static friction that is ruled by a Coulomb law. This approach, as the one proposed here, is inspired by the physics of granular media. The main difference with the model proposed in this work is that the shear stress at the walls is limited by dynamic or static friction depending on the normal stress instead of simply considering that it is comprised between 0 and the yield stress. Moreover, we now relate the Janssen parameter K to the elastic properties of the material. However, in the model proposed in [23], several points seem to have been dealt with rather quickly. _ First, the use of the Coulomb law is justified in their papers by the fact that they measure in a dynamic experiment a proportionality between shear stress and normal stress when shearing the material. This is not sufficient as it can be demonstrated that, for many suspensions (even when they are not subject to frictional
flows), this proportion exists without any friction at the interface [24-25], but with purely viscous dissipation. It is the natural consequence of the fact that both shear and normal stresses are proportional to the shear rate. Therefore, it does not imply that the yield stress is proportional to normal stress. _ Second, the value of the friction coefficient is surprisingly low compared to what is measured in the case of real granular materials. _ However, the measured value of this friction coefficient is still too high to get a quantitative agreement with the experimental results and the model finally predicts a pressure on the formwork far lower than what is measured. A fitting coefficient is then introduced by the authors to correct the quantitative prediction; e.g., this fitting coefficient implies that only 15% of friction has to be mobilized when SCC is poured from the top; this is in contradiction with most experiments performed on granular materials, where static friction is almost fully mobilized at the walls after the filling [26-27]. _ Finally, if the shear stress at the walls is due to dynamic friction, its direction should be opposed to the direction of the concrete flow. When concrete is poured from the top, the lateral stress should then be lower than the hydrostatic pressure as the friction is directed upwards. This is the case in their paper; the lateral stress at the bottom of the formwork is 26% lower than the hydrostatic pressure. But, when the concrete is injected from the bottom, the friction is directed downwards and the lateral stress should then be higher than the hydrostatic pressure. It is not the case in their experimental results as the lateral stress is also 25% lower than the hydrostatic pressure. In the opinion of the present authors, the correct conclusion of this strong quantitative discrepancy that should have been reached in [23] was that the physical phenomena they were considering were not the ones that were really occurring. As it will be demonstrated in this paper, there is no need to introduce a dynamic friction to explain the influence of the casting rate on the lateral stress. It is not a matter of how fast the concrete is flowing in the formwork but it is about how much time the concrete can rest before the next “layer” is cast.
3. COMPARISON WITH EXPERIMENTAL MEASUREMENTS 3.1. Measurement of the evolution of the yield stress The concrete rheometer used here is a BTRHEOM [9]. The measurement procedure used in this work is described in [28] and its adaptation to SCC in [29]. It consists in applying an increasing torque until flow starts. The shear stress is then computed from the imposed torque. It has been adapted to our purpose here by reducing the maximum allowed rotation speed before stoppage in order to reduce the de-flocculation of the sample that occurs when the yield stress is measured. This de-flocculation can not of course
G. Ovarlez , N. Roussel / Materials and Structures Vol (year) pp-pp
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be completely eliminated from the measurement procedure and, as the measures were all successively carried out on the same sample on the building site, it has to be kept in mind that the yield stress values obtained may be lower than the ones that could have been reached if the sample had really been left at rest. A typical measurement is shown in Fig. 1.
Figure 1. Yield stress measurement for a SCC after 40 minutes rest. The flow starts when the shear stress reaches 270 Pa. The initial yield stress of this SCC (coming out of the truck) was approximately equal to 30Pa and its spread to 730mm.
3.2. Formwork lateral stress measurement The procedure and sensors used to measure the lateral stress on formwork are described in [30]. A rather complete description of an experimental device specially developed for this type of measurements can also be found in [3]. The formwork studied here is 10.00 m high, 5.44 m long and 0.20 m wide. The concrete was injected from the bottom of the formwork and the results presented were obtained on the 3 sensors that were 0.55 m, 1.95 m and 3.36 m respectively above the bottom of the formwork. As it can be seen on Fig. 2, the initial lateral stress at the end of the casting was equal to the hydrostatic pressure. This could have been awaited as the concrete was injected from the bottom at a very high casting rate (21.4 m/h) and thus stayed fluid during the entire casting phase. However, once the casting was over, the lateral stress quickly decreased because of the flocculation of the concrete, which was now at rest.
Figure 2. Percentage of the initial pressure in terms of time for the three pressure sensors. Formwork height = 10.00m.
3.3. Method employed for the analysis of data Our objective in this section is to show that the evolution of the shear stress needed to obtain the measured lateral stress evolution using the model proposed here is equal to the evolution at rest of the SCC yield stress measured using the BTRHEOM. An experiment performed on a concrete formwork provides a σxx(±L/2,y0,z) (or σyy(x0,±e/2,z)) profile, i.e. σxx(±L/2,y0,zi) for a series of depths zi . In order to test the model, we can compute the mean shear stress at the walls between depths zi and zi+1, within the model framework, and compare it to the yield stress measured by the BTRHEOM. From Eq. (5), we compute the shear stress τi between depths zi and zi+1:
τi =
1 σ xx (zi+1 )-σ xx (zi ) 1 +ρ g 2 zi+1 -z i (1/L+1/e)
(15) where ρ = 2265 kg/m3. The shear stress needed at the wall and the measured yield stress evolutions are plotted on Fig. 3. The quantitative agreement is very good, knowing the uncertainty on the yield stress measurements.
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where the concrete is not at rest varied from 0.2 m to 0.4 m as shown on Fig. 4. It was for most of the calculations of the same order as the thickness of the formwork.
Figure 3. Comparison between the shear stress at the wall needed to obtain the measured pressure and the measured yield stress in terms of resting time.
4. PRACTICAL APPLICATION OF THE PROPOSED MODEL The fact that the lateral stress, initially equal to the hydrostatic pressure, decreases once the casting is over and that we are able to predict the rate of this decrease in terms of the thixotropic behavior of the concrete has a very limited interest on the building site. To reduce the cost of formwork and increase the maximum allowable placement height, we need to understand what happens during casting and to be able to predict the maximum lateral stress that is reached during casting. From now on, we will focus on concrete poured from the top of the formwork as injection from the bottom prevents the concrete from flocculating. The yield stress of an injected SCC stays very low and, as no vertical stress can be supported by the walls, the lateral stress always stays more or less equal to the hydrostatic pressure.
4.1. Pressure evolution One could object that concrete, even when poured from the top, is not at rest in the formwork. We carried out numerical simulations of a formwork filled with a Bingham fluid for different yield stresses (from 20 Pa to 100 Pa), casting rates (from 2 m/hour to 36 m/h) and formwork widths (from 0.2 m to 0.4 m). The numerical techniques used to model flow of concrete are those used in [31]. In the simulations, the concrete was poured 1 meter above the surface of the already cast material. We considered that the flow was sticking at the walls. The criterion used to determine the extent of the sheared zone was based on the shear rate value (lower or higher than 0.1 s-1). In the range of parameters tested, the computed thickness of the zone
Figure 4. Sheared and resting zone when SCC is cast from the top of the formwork. In the black zone, the shear rate is greater than 0.1s-1.
4.2. General evolution of the apparent yield stress at rest We carried out many measurements on various SCCs. We obtained for the vast majority of our results typical evolution of the apparent yield stress at rest similar to the one shown on Fig. 5. We systematically tried to fit a linear relation to the obtained results: τ 0 ( t ) = τ i0 + A thix t (16) where τ i0 is the initial yield stress of the SCC, t is the resting time and Athix is a flocculation coefficient that is fitted from the experimental results. The initial yield stress (of the order of magnitude of a few tens of Pa) can most of the time be neglected compared to the increase due to resting and flocculation. In the range of resting times studied and for all the SCCs we tested, a linear relation could be fitted to the yield stress vs. resting time curve with Athix varying between 0.1 and 0.2 Pa/s. Of course, it can be imagined that the flocculation phenomenon reaches a limit and slows down but, when we tried to carry out measurements over more than two hours, we measured irreversible changes in the behavior that seemed linked to the beginning of the hydration process [29,32-33]. For now, we will assume that the evolution of the yield stress at rest is linear with rest time and may be described using Eq. (17).
G. Ovarlez , N. Roussel / Materials and Structures Vol (year) pp-pp
τ 0 ( t ) = A thix t
(17)
As our own experiments were limited in numbers and, as this type of measurements is not available in the literature for other SCC(s), it was difficult to know whether Athix = 0.1-0.2 Pa/s is a low, normal or high value. It will be shown in the next section that it may be considered as rather low.
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integrated to compute the lateral stress at the bottom of the formwork: H −e K (18) σ xx = σ yy = ρgHLe − 2(L − e ) τ 0 (z )dz Le 0
∫
Figure 6. Sheared and resting zones and their respective dimensions when concrete is cast from the top of a rectangular formwork (width e) or a column (radius r). Figure 5. Example of a yield stress evolution measured with the BTRHEOM.
4.3. Computation of the maximum pressure during casting We assume here that the casting rate R is constant (at a time t after the beginning of casting, H = Rt). We will deal simultaneously with the case of a rectangular formwork of width e shown on Fig. 6 but also give the final relation for a column of radius r. We will consider here only the lateral stress at the bottom of the formwork as it is the most important for a practical point of view. As demonstrated in section §2.1, at a depth H, it is equal to the hydrostatic pressure ρgH reduced by the amount of vertical stress supported by the walls. This vertical stress, also demonstrated in section §2.1, takes a value between 0 and the yield stress τ0 of the concrete. We will assume here that the vertical deformation of the concrete under its own weight is always sufficient for the shear stress to reach its maximum value τ0. Because of the thixotropic behavior of concrete, this yield stress increases when the material is at rest, which is, as shown in §4.1, the case everywhere in the formwork except in the upper layer (thickness e). At the bottom of the zone where the concrete is at rest, the resting time is maximum and is equal to (He)/R. At the top of this zone, it is equal to zero. The yield stress of the concrete thus varies with depth and has to be
Using Eq. (17) and H=Rt, Eq.(18) becomes in the case of a rectangular formwork, with L>>e: (H − e )2 A thix σ xx = σ yy = K ρgH − (19) eR or, for a column of radius r, (H − r )2 A thix σ xx = σ yy = K ρgH − (20) rR Generally, H is much larger than e (or r) and e(r) can be neglected in Eqs (19) and (20) to give: HA thix σ xx = σ yy = KH ρg − in the case of a rectangular eR formwork HA thix σ xx = σ yy = KH ρg − in the case of a column rR The relative lateral stress σ’ is the ratio between the lateral stress and the associated hydrostatic pressure. The relative lateral stress during casting at a depth H is then: HA thix σ (21) σ′ = xx = K1 − ρgH ρgeR This relation is able to explain the experimental observations listed in the introduction. It indeed predicts lower lateral stress in formworks of limited cross section (low values of e). Moreover, it explains the scale effect as H still appears in the expression of the relative lateral stress.
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Finally, it gives an explanation to the contradiction found in the literature. According to the values of Athix, e and H, there exists two regimes as shown on Fig. 7: in the first regime, the influence of the casting rate on the lateral stress is very strong, in the second regime, there is almost no influence of the casting rate on the resulting lateral stress.
This critical rate is of course equal to zero at the beginning of casting (H = 0). This means, that, when casting starts, it is not possible to prevent the lateral stress from increasing. However, as H is increasing, it becomes possible to greatly reduce the lateral stress by reducing the casting rate to the critical value. In the two cases from the literature the following values of the critical casting rate were computed when the height of the concrete is equal to half the height of formwork (middle of the casting phase): it was equal to 0.6 m/s for [6] and 0.9 m/s for [1]. To conclude, on a practical point of view, the SCC tested by Billberg [1] (Athix = 0.6 Pa/s) seem the most suitable to high formwork casting. These are the ones that flocculate the fastest and therefore the ones that allow the fastest casting rate for a given formwork pressure or the ones that generate the lowest formwork pressure for a given casting rate. The main difference with the other SCC studied here seems to be the amount of limestone filler used as a powder additive between 90 and 140 kg/m3.
5. CONCLUSION
Figure 7. Comparison between literature experimental results and predictions of the proposed model in the case of a rectangular formwork [1] and a column [6]. The values of the parameters necessary to compute Eq. (19) or (20) are given in [1,6]. The only parameter that is fitted in the predictions of the model on Fig. 7 is Athix. It is almost equal to 0.6 Pa/s for the SCC tested by Billberg [1]. This value is an average value as the additives were different from one concrete to another. Athix is equal to 0.2 Pa/s for the SCC tested by Khayat and co-workers [6]. These values are both in the same order of magnitude as the values obtained for the SCC we tested (0.1-0.2 Pa/s), which confirms the quantitative validity of the approach proposed here. If the casting rate is low enough, it is possible to imagine that the pressure increase due to the increasing height of concrete is compensated by the increasing shearing stress supported by the walls. This critical casting rate fulfills the following condition: 2HA thix ∂σ xx = K ρg − (22) =0 ∂H eR and thus 2HA thix R crit = (23) eρg
We have proposed in this paper a model that physically links the consequences of thixotropy (flocculation at rest and increase of the apparent yield stress with resting time) and the evolution of the lateral stress during and after casting. The following points have to be emphasized: The lateral stress is equal to the hydrostatic pressure when the casting rate is high or when the concrete is injected from the bottom of the formwork because the material is not able to flocculate and thus keeps on behaving as a fluid. The lateral stress does not reach the hydrostatic pressure when the material is cast from the top of the formwork slowly enough to flocculate and withstand the load of concrete cast above it. The lateral stress decreases quickly after the end of the casting as the concrete, now at rest, is able to develop a higher yield stress and starts behaving as a solid. The simple physical model (Eq. (5)) presented here seems to be able to describe quantitatively the evolution of the lateral stress on the formwork in term of the thixotropic evolution of the yield stress at rest measured using a concrete rheometer. A practical application of the model has been proposed. It allows the prediction of the maximum lateral stress reached during casting in terms of the height of concrete, the geometry of the formwork, the casting rate and the ability of the concrete to flocculate at rest (Eq.(19) and (20)). According to the casting rate, the existence of two regimes have been demonstrated: in the first one, the casting rate plays a major role on the maximum lateral stress, in the second one, there is almost no influence of the casting rate on the maximum lateral stress, which is, in this case, close to the hydrostatic pressure. When the flocculation rate is high enough (fast increase of the yield stress) and the casting rate low enough, the maximum value of the lateral stress stays lower than the
G. Ovarlez , N. Roussel / Materials and Structures Vol (year) pp-pp
hydrostatic pressure during casting. In that case, the predictions of the model were compared with results from literature and proved to be able to explain most of the experimental observations.
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