Revista Ingeniería de Construcción Vol. 29 N^{o}1, Abril de 2014www.ricuc.cl PAG. 115-127

**Modeling a bending resistance modulus for a cementitious material compound based on properties in fresh state**

**Fernando Toro*, Marisol Gordillo****¹**, Silvio Delvasto***, Jr Holmer Savastano*****

* Universidad San Buenaventura, Cali. COLOMBIA.

** Universidad Autónoma de Occidente, Cali. COLOMBIA

*** Universidad del Valle, Cali. COLOMBIA

**** Universidade de Sao Paulo. BRASIL

**ABSTRACT**

In this research we studied the relationship between fresh state and hardened state of a fiber reinforced laminate compound (MCLRF), respectively, based on a matrix of Portland cement. The variables that have been worked in the fresh state were undulation ability, flow ability and drain ability and hardened in the form of resistance to bending, (bending resistance modulus). The samples were prepared with different raw materials (fique fiber, bentonite, pulp, silica fume, and Acronal). The ingredients were grinded by a mixer working at a constant speed, after that, the mixture was poured into a drainage chamber, and the water was extracted, forming a sheet with dimensions 130x50x6 mm, finally, it was immersed in a curing chamber for 28 days, for bending evaluation. The linear regression model proposed to predict the bending resistance modulus, considering the properties of in fresh state, was performed through the analysis of interactions between variables in formulation of the mixture used. From the regression model it was found that there is a clear correlation between bending resistance modulus and the predictive variables: undulation ability, flow ability and drain ability. Furthermore, the variables have a negative effect on the bending resistance modulus

**Keywords:** Corrugated tiles, regression models, laminated composite, module flexural strength

**1. Introduction**

Fiber cement materials are slender elements in a stretched shape of thin thickness, such as panels, tin ceilings, walls divisions, containers (bins and tanks), plain and corrugated tiles used for housing covers and ferrocement pieces (Kaufmann, J. et al., 2004). The processes developed by Hatschek, Mazza and Magnani are the most popular ones in tiles production. Over the late century, such processes have predominated in this industry mainly due to their advantages; such as: processing simplicity, economical raw materials and excellent final properties. These processes generally employ asbestos as fiber material, which is considered as a health-risk element. Consequently, technology researches on fiber cements have evolved to replace asbestos and to develop new formulas, so as to improve productivity and reduce energy consumption (Savastano Jr. et al., 2000; Negro, C. et al., 2005; Roma Jr. et al., 2004).

The production of corrugated tiles requires materials with appropriate usage properties, such as: adequate flow ability, which facilitates proper mixing and pouring processes; proper moldable capacity, so that tiles do not become cracked; dimensional stability; low density; high thermal isolation; reduced permeability and high flexural resistance (Dos Anjos M. A. et al., 2003; Onésippe C. et al., 2010; Delvasto S. et al., 2010). Summarizing, if there is a rheological properties imbalance during fresh state, provoked by variability of the mixture components and due to inappropriate parameters in the production process, then there will be problems to achieve hardened state properties.

The introduction of engineering statistical models is a tool progressively accepted, since it systematically enables the analysis of the effect of different variables over a given response variable. Modeling based on multiple regressions has been widely used to predict compression of fiber reinforced concretes and non-reinforced ones, which have achieved quite adequate settings ranging from 97 to 99% (P. Ramadoss, 2012; S. Eswari, P. et al., 2011; Mahmoud Sayed-Ahmed, 2012). It is worth mentioning that relationships between variables proposed by this article have not been found in literature available so far.

The purpose of the current research is to increase the knowledge on the relationship between rheological properties (such as undulation ability, flow ability and drain ability) and the bending resistance modulus of cementious mixtures reinforced with fique fibers, which are adequate for corrugated tiles production. Based on the fact that bending resistance modulus is affected by fresh state properties, a multiple regression model is adopted and a predictive equation is proposed.

**2. Experimental program**

**2.1. Materials**

The research employed Portland cement type I, produced by Argos Company. Physical-chemical properties were evaluated in accordance with Colombian regulations NTC 221, 110, 111, 33, 118 and 220, respectively, for a specific weight (3030 kg/m3); regular consistency (2.8%); flow ability (w/c) (0.63); Blaine fineness (336 m2/kg); initial setting time (143 min) and final setting time (195 min); cement setting time with 4% addition; initial time (197 min) and final time(340 min); compressive resistance at 5 days (18.87 MPa), 7 days (21.17 MPa) and, 28 days (27.23 MPa). The cement chemical characterization is shown on Table 1.

**Table 1.** Cement Chemical Characterization

Betonite´s chemical properties reported by the commercial technical datasheet are the following: Liquid limit wL (935), plastic limit wP (47%), material piece size < 75 µm (85), hygroscopic water content w (14%), solid density ρs (2.70 Mg/m3), montmorillonite content > 90%.

Table 2 shows the results from fique fiber characterization.

** Table 2.** Characteristics of the fique fiber

Table 3 shows the results from pulp characterization.

**Table 3.** Characterization of paper pulp

The sand employed is of rounded-edge grain and siliceous type, which has water absorption of 1.87%; fineness modulus of 2.95; and density of 2.66 g/ml.

Carbonate employed is Omyacarb 8, with the following chemical composition: Calcium Carbonate; 99.04 %; Magnesium oxide 0.385 %, Silicon oxide 0.315 %, HCl Insoluble components 0.26 %; with the following physical properties: Specific weight 2.7 g/cm³, pH 8.76 and humidity 0.04%. Laser grain size distribution was also performed delivering a specific surface area 1.77 m²/g, diameter (0.1): 1.226 µmm, diameter (0.5): 7.844 µmm, diameter (0.9): 26.305 µm; and uniformity of 1.02. Acronal® employed is an aqueous dispersion copolymer made of butyl-styrene acrylate. This material is used as base material to produce adhesive materials for a wide variety of applications, such as roofing tiles and wooden floors.

**2.2. Specimens preparation process**

The tiles production process consisted in adding and mixing the components, following a logical and chronological order. A water cement ratio 0.9w/c was employed in order to obtain an adequate flow ability and homogeneity. The mixture was poured into a mould covered with a piece of clothing on the base and then immersed in a drainage chamber. Drainage equipment consisted of a suction chamber, vacuum pump, moulds, filter, piping system and valves. Water was drained at a ratio of -2 to -4 bars during approximately one minute. Then a 130x50x6 mm plate was elaborated following its own weight waveform inside the mould. Afterwards the plate was immersed in a curing chamber for 28 days, at a relative humidity close to 100%. 96 samples were employed for 32 different mixtures, which dosages are shown on Table 4.

**Table 4.** Dosages employed for mixtures

**2.3. Tests developed with mortars**

Bending resistance test carried out on the fiber reinforced laminate element was determined by loading a transversal section laminate element with a span of three times its thickness. The configuration of bending resistance test considered three loading points and a speed of 1.5 mm/min (Figure 1). The values of the material bending resistance modulus (MOR) were calculated in accordance with Equation 1 (Odera, R.S. et al., 2011).

** (1)**

Where P is the maximum loading applied, L is the span between points, b: length, d: thickness

** Figure 1.** a) Configuration of three loading points for corrugated tiles. b) Representations of arrows measurements

Flow ability is the mixture ability to spread itself on a smooth surface. This ability is measured by the equipment called minislump, which is employed to evaluate flow ability on fluid mortars, as shown on Figure 2 (Kuder K. G. et al., 2007; Martinie L.,et al., 2010).

**Figure 2.** Equipment used to measure minislump

Drain ability is the ability the fiber-reinforce mixture has to eliminate water by means of vacuum drainage. Such ability is measured by exposing a high consistency mixture at a high vacuum pressure during a constant period of time. Then the water poured from drainage is determined by saturating the mixture with alcohol and employing thermal drying.

The mixture was prepared with a water/cement ratio of 0.9, so as to obtain the fiber homogeneous distribution to make up the plate; vacuum applied was (3 ± 0.5 bars) for a period of (1 min ± 1 s); then three 50g-samples were saturated in 99% pure alcohol and they were dried inside a ventilated furnace at 70°C for 24 hours. Samples were finally weighted.

Undulation ability: the undulation meter is a device, which design was developed during this research in order to measure molding ability or the ability to make up a fiber-cement plate in fresh state. The device is made up of three bands or conveyor belts. Each band or continuous belt is driven by two rollers or pulleys of a given diameter, which support the belts. One roller of each conveyor belt is manually moved by using a crank pulley at a constant speed. A fresh state cementious plate is placed on the band and conveyed towards one of the rollers (Figure 3). When cementious plates descend or fall down, they bend themselves in accordance to their weight and the corresponding rounded radius, which corresponds to the roller diameter of the belt employed. Consequently, the device allows the application of diverse loads on the material plate, which increase as long as the spinning radius decreases.

**Figure 3.** Undulation meter. a) Undulation device seen from the side. b)Contour of cylinders

**2.4 Description of the statistical methodology associated to the multiple regression models**

The interest variable (Y) or response variable is often influenced by more than one predictive variable (X). The Xs variables can be controlled by the researcher or not. The linear regression model presents the relationship between Y and Xs, (Equation 2). The general expression is as follows:

**(2) **

Where correspond to the model parameter, known as regression coefficients, ε is the random error which is not explained by predictive Xs. Assumptions for ε are the following ones: are constant and are independent.

This model is calculated by means of the expression: . The model can be expressed as a matrix (Equation 3) as:*y=Xß + ε*

**(3)**

Matrix X has n rows per k+1 column; therefore, columns are literally independent. Consequently, it is possible to employ the least squares method (Montgomery D. C., 2004). Since the X matrix contains k columns corresponding to the model´s k parameters and n rows corresponding to the number of records, there is another assumption to be met during the whole analysis, that is to say the number of records, n, shall be equal or higher than the number of parameters (k). It really makes sense because when estimating k parameters, it is necessary to count at least with k records (Uriel E. 2013).

**Estimation of Parameters:** The most common parameters estimation method is the least squares technique, where regular equations are expressed as , from which arises that the estimated regression equation is . The reasoning of the least squares technique considers the deviation of records Yi and their average value, thus determining the values βs, which minimize the addition of squares in such deviations (Montgomery D. C., 2004; Gelman, A. 2005).

**Hypotheses associated with the Regression Model:** The hypotheses are posed on the parameters of model *Ho*: *βi* = 0 and the statistical *t-student* test is employed. When trying to prove hypotesis , the variance analysis technique shall be employed.

The model was initially applied with all considered variables, (X1: Undulation ability or malleability, X2: Drain ability, X3: Flow ability), with a low importance level (< 0.05), that is to say that all applied variables in the regression equation are eliminated, if their importance level is higher than (> 0.05).

**Variance Analysis:** The variance analysis (Anova) is a technique that summarizes the model and it consists in dividing the total variance of records in their variance sources, in accordance with the proposed model (Gelman, A. 2005, Montgomery D. C. 2004). The total variance of Anova is contained in the addition of total squares (SCT), which is composed by the addition of total regression squares (SCR) and the addition of error squares (SCE), .

**Multiple Determination Coefficients:** The multiple determination coefficients correspond to a strength measurement of a linear association between random variables (Equation 4). This is a value oscillating between 0 and 1. Values close to one indicate that predictive variables explain the higher variance amount.

** (4)**

**Selection of the Best Set of Variables:** all variables the researcher considers might be associated with the response variable are initially considered. However, it is necessary to use a method to determine the values to be included, as they affect the average response. For such selection, the individual hypothesis tests for each parameter are analyzed; and the non-significant ones are excluded (values *p* > 0.05), as R2 considerably decreases when they are eliminated from this model (Montgomery D. C., 2004; Gelman, A. 2005).

All calculations involved in the multiple regression model were obtained by the statistical software Minitab 16 (Minitab® statistical software (version 16)).

**3. Results**

**3.1. Behavior of fiber in the matrix**

Longitudinal visualization of the fiber immersed in the cementious matrix is shown on Figure 4, which enables the determination of main characteristics of transition zones in the compound, such as: matrix partial bonding and micro-cracking. The developed treatment on the fiber, calcium hydroxide immersion, increased the contact area between the fiber and the cementious matrix perimeter. EDS analysis (Figure 4a, b, c) detected chemical elements normally present in the cement hydration products, with ratios of calcium-silica between 1.2 and 2.0 (Roma Jr. et al., 2008; De Andrade S. et al., 2009).

**Figure 4.** ESEM Microphotography at 193 days, 200 ím (punched the fiber center in 1), (punched outside the fiber in 2) and (punched outside the fiber in 3)

** Figure 4.a.** EDS Punched the fiber center in 1

**Figure 4.b.** Punched outside the fiber in 2

**Figure 4.c.** Punched outside the fiber in 3

**3.2. Relationship between rheology and bending resistance modulus**

Relationship between rheology and bending resistance modulus was determined by using the multiple regression model. Specifically for the behavior analysis of fiber-reinforced mortar, the bending resistance modulus in hardened state was used (y), as predictive variable. As interest variables, probably explaining the behavior, we considered Undulation ability (X1), Drain ability (X2) and Flow ability (X3) in fresh state.

Based on the statistical analysis using the linear model, the importance regression coefficient values are obtained for each aforementioned variable, as well as the importance levels (P) (Table 5) obtained to check the individual importance hypotheses of forecast variables (Xs) over linear regression. Similarly, it is observed that at importance levels lower than 0.011, the bending resistance modulus is explained by the variables: undulation ability, flow ability and drain ability. On the other hand the goodness of fit is excellent (R² = 98.3%), proving the experimental sample fits perfectly to the proposed model. Undulation ability as well as flow ability and drain ability have a negative effect on bending resistance modulus (coefficients -1.27, -0.65 and -0.66, respectively). However, the interactions undulation ability-flow ability and flowability-drainability have positive effects on the bending resistance modulus, while the bending resistance modulus rises with each increment in the involved interactions.

**Table 5.** Contrast of the regression coefficients of potential variables affecting the bending resistance modulus

Flow ability is posed by the Equation 5, where the bending resistance modulus is expressed in MPa, undulation ability is dimensionless, drain ability is expressed in percentage and flow ability in centimeters. Consequently, if such variables are available for the mortar in fresh state, then it is possible to determine the bending resistance modulus in hardened state.

** (5)**

Where: X1: Undulation ability or moldeability, X2: Drain ability, X3: Flow ability and Y: is the failure of bending resistance modulus.

**Validation of assumptions on the errors in the model:** The model proposed meets the error assumptions, that is to say, errors come from a regular distribution population with average 0 and variance σ². Besides, variance is constant and errors are independent, with importance values higher than 15%. Similarly, the required condition was also met: the greater amount of records, the greater amount of parameters to be estimated.

Table 6 shows the proposed regression variance analysis (Equation 5) where it is evident that it is significant (p-value of 0.000). Above indicates that, at least one coefficient is different from zero, thus indicating that regression is not only significant in the sample but also in the population. The random effects explanation is minimal (0.073) compared to the regression (4.269).

** Table 6.** Variance analysis of the proposed regression

The optimization to obtain values, from undulation ability, flow ability and drain ability (Figure 5) maximizing the bending resistance modulus takes place when lower values are achieved for undulation ability values lower than 3 approximately as well as flow ability values lower 16 cm approximately and drain ability values lower than 4.7%, thus predicting bending resistances higher than 6 MPa.

**Figure 5.** Contour Graph of bending resistance modulus

**Verification of predictions delivered by this model:** In order to verify the forecast ability of this model, 3 mixtures were elaborated (Table 7), which showed that values predicted by the regression equation are quite similar to real values obtained in the laboratory. Above, because fitting in the modeling was very adequate, so values are quite close to the results shown on Figure 5.

**Table 7.** Verification of the forecast ability of the regression model

**4. Conclusiones**

The results obtained lead us to the following conclusions:

• The analysis results were used to establish a predictive model of the bending resistance modulus based on the fresh state of a fiber-cement mortar to be used for corrugated tiles production. This model allowed us to predict the mechanical resistance in hardened state, at 28 days, by using the parameters of undulation ability, flow ability and drain ability during the first minutes of mixing process. The raw materials employed by this model were Cement, Fique Fiber, Bentonite, Pulp, Silica fume and Acronal).

• The applied multiple linear regression model properly represents the existing relation between undulation ability, flow ability, drain ability and the bending resistance modulus (R2 = 98.3%) of fiber reinforced mortars elaborated in this research, which are then employed in tiles production.

• Validation of multiple regression models intended to develop an advanced diagnosis of bending resistance modulus, by using properties such as: undulation ability, flow ability and drain ability. These models have an adequate approximation to real data.

• By means of Micro-photographic tests, it was observed that fibers are tightly embedded in the matrix, thus demonstrating the bonding strength between the fiber and the matrix (Pull out). The developed treatment (immersion in a hot calcium hydroxide suspension) increases the contact area between the fiber and the cementious matrix perimeter, leading us to suppose there will be a proper bending resistance.

**5. Acknowledgements**

Authors wish to thank the Universidad del Valle (Colombia), the Colombian Institute for the Development of Science and Technology "Francisco José de Caldas" (COLCIENCIAS), Raw Materials Excellence Center (CENM) that supported this research and, the Group of Construction and Environment of the Faculty of Animal Husbandry and Food Engineering (FZEA) at the University Of Sao Paulo (USP).

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E-mail:mgordillo@uao.edu.co

Fecha de Recepción: 02/08/2013 Fecha de Aceptación: 03/12/2013

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