A NATURAL FIBER-CONCRETE EXPERIMENTALCOMPUTATIONAL DESIGN THROUGH DYNAMICAL TIMOSHENKO THEORY

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

A NATURAL FIBER-CONCRETE EXPERIMENTALCOMPUTATIONAL DESIGN THROUGH DYNAMICAL TIMOSHENKO THEORY Flávio Pietrobon Costa* 1,, Ricardo de Carvalho Alvim*, †, Rosana de Albuquerque Arleo Alvim*2, Lucas Pereira da Silveira# *1

professor, State University of Santa Cruz, UESC DCET, Rodovia Ilhéus Itabuna, km 16, Ilhéus, Bahia, Brazil [email protected]

#

*1

professor, State University of Santa Cruz, UESC [email protected]

†

professor, State University of Santa Cruz, UESC in memoriam

bachelor student at production engineering, scientific beginner [email protected]

Keywords: Natural fibers–concrete composites, Experimental-computational modeling, Timoshenko’s Beam Theory. Summary: This work deals with a gap in the behavior of structural mechanics and design: determination of mechanical properties to allow the employment of natural fibers in concrete composition, resulting in a natural fiber composite. Bamboo and cocoa fibers was used as strength element to concrete beams. Tall structures in reinforced or prestressed concrete need more strength construction elements, as non-slim structures components. For beams, to deal with this problem, was employed the Timoshenko Beams Theory (TBT), at small configuration changes, taking in account the dynamics of concrete-natural fibers beams. In this first work the damping internal mechanisms were not considered, to acquire the behavior of the influence in incorporating the natural fibers to concrete. This research performs correlation analysis between experimental and computing solutions. Numerical formulation considers the shear coefficient introduced by TBT, correcting distortion in the total rotation of beam cross section, not considered in the Classical Euler – Bernoulli Theory. Laboratory essays consider the design and making of proof CPs Timoshenko’s beams. Computational solutions are obtained in a finite element transient semi-discrete model. Numerical and experimental examples were performed, and results present an intrinsic potential to next development. With this approach, the natural fiber content rate to concrete was characterized, acquiring an optimal performance in the structure response. Results show an increment of composite concrete tensile strength in relation to a pure concrete one. Cocoa fiber incorporated to the composite concrete answer to a best performance of elastic modulus with an increment of elastic limit to the material.

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

1 INTRODUCTION Natural fibers have been touted as an alternative material suitable for use in construction, with the potential for cost reduction, offering quality and durability, incorporation a strength compatible with conventional technologies. The behavior of structures have been extensively studied in the literature, but gaps remain requiring validation of specific theories, especially when considering the construction of tall structures, and in the design of more resistant, flexible, and elastic, constructive elements, as fencing blocks, masonry, covering surfaces, and also floor elements. The use of unconventional materials, such as composites of natural fibers, allows the incorporation of those desired characteristics. The design of composites was obtained with the incorporation of cocoa or bamboo to a concrete matrix. The interest was is project the employment of those composites in more strength structural elements, for use in tall buildings or slenderness structures. This work is a first approach in this target. With those employments in mind, to deal with the requirement for stronger structural, and at the same time more elastic, elements and materials, the Timoshenko’s Beam Theory, TBT, was used to design the proof specimen for laboratorial essays, and for computational simulations. TBT is suitable to project and design beams with those characteristics point below, and has the advantage of consider the occurrence of growth in transversal displacement in relation to Euler-Bernoulli slenderness beams. The focus of this work was to acquire knowledge about (1st) the behavior of composites Timoshenko’s beams elements under loads, at free dynamic behavior to evaluate the influence of natural fibers incorporation on amplitude, or transversal displacement, answer, and (2nd) also about theirs mechanical properties, in special on the longitudinal elastic modulus, on the strength to compressive and shear loads, and about the influence of intrinsic water incorporation to the material, along others mechanical characteristics. The development of new compounds arises from the need to combine the characteristics of two or more materials in order to improve product performance in certain specific applications. In construction, fragile cement based matrices are used, largely on their applications. These matrices mostly derived from mineral binders suffer plastic deformation rupture, and although resistant to compressive forces, do not support requests from large tensile, shear or dynamic loads. In tropical countries, the waste generated by agribusiness plant fiber can be a source of raw material for the production of building components. Depending on the quantities available and the geographical dispersion, as the costs of natural fibers are low, use this as reinforcement of brittle cement based matrices materials has aroused great interest in developing countries, because not only because of its low cost, but cause of it’s availability, energy savings and also as regards environmental issues [27]. However, the increase in tensile strength even greater evidence depends in particular on the fibers and ground natural state. Moreover, the determination of it’s benefic properties is already an research nest. Another aspect of this research is to analyze the resistance of lightweight concrete reinforced with fibers, since its main application is in the manufacture of masonry blocks, tiles and floors. Therefore, the need to seek an adequate and useful cement matrix, requires that the material be light and strength enough for shipping and handling during the construction process. Obtaining new materials for construction in environmentally sustainable conditions is a challenging way. When studying a new composite material, the main objective is to understand the

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

potential of it to use in practical applications or to substitute other products that are more expensive or that have environmental disadvantages. This research line so far is developing a standard operating procedure for the fabrication of lightweight composite cement matrix reinforced with fibers of bamboo and coconut, in order to obtain accurate results, optimized and low deviation between samples. For these reasons this design, studied composites are made from lightweight cement mortar, with modifications to the incorporation of cement partly replaced by additions of minerals, and use of bamboo fibers and coconut fibers all fresh, untreated surface. From historical works of [1, 2], studies of the deformation shear effects, had been addressed by several researchers. Sought this theory, TBT, corrects the classical theory of Euler-Bernoulli, which disregards the effect due to shear, getting underestimated values of displacement and deflection of the beam, under external loads. The introduction of a dimensionless correction factor, the shear coefficient, k, attempted to obtain values of displacement and strains compatible with the the real ones. The numeric formulation for the shear deformability evaluation was developed on a semi-discrete finite elements approach, adopting an energy method based on the minimization of a variational formulation. The object of this study correlates the response to dynamic loads of beams, developed with natural fibers, with the study of the efficiency of the Theory of Timoshenko beams for the study of structures of composite materials, simulating a static response to the dynamic response calibration by computational numerical analysis, and using this to develop a dynamic free vibratory model. The study then contributes in an area little explored in the technical literature. The response of Timoshenko beam vibration has already been studied with analysis results concerning to the oscillation of the beam excited by impact load. Composites TBT beams has being a developing research, approached on our computational model. This 1st line of this work finds application in vibration control industrial, construction of composite structures with natural fibers in composite materials, and the use of natural materials in structural systems construction. With relation to our 2nd guide line, that target to evaluation and determination of mechanical and physical properties of natural fiber –cement based composites; some laboratory experimental essays were performed, to acquire data that answer to what is the composite laboratory specimen, CP, responses to external loads. Those essays gives also elastic and strength properties. The experiments and laboratory analyzes have considered the preparation of fibers from bamboo and coconut, milled in a Wiley mill, achieving high surface of contact with the concrete, oven-dried to determine the moisture content. Analysis of resistance to compression and tension in flexion of the material was carried out with a content of 5% fiber, both parts incorporating bamboo fiber as for the coconut fiber. Body-ofproof, to 28 days of age, where tested: for direct compression and traction in flexion. Laboratory essays shows a little decrease in compressive resistance, detected in consequence of some compactation difficulties of CPs development, meanwhile, for the test results in the bending tensile essays, was detected an increase of 37.97% (coconut fiber) and of 35.95% (bamboo fiber) of the tensile strength when compared to pure cement matrix related CPs performance. The use of vegetable fibers to improve the behavior of building materials has been a recent development, due to its low cost, availability, and energy savings. There is interest in strengthening brittle matrix fiber cement, and seek better structural solutions. This target is linked to increased tensile, flexural, and shear resistance, but also answer to impact and dynamic excitation, preventing or delaying the appearance of cracks and higher material

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

toughness. Cement matrix composites reinforced with continuous fibers from bamboo pulp indicated that the content of refined fibers around 8%, showed remarkable improvements in mechanical properties of the composites compared to the homogeneous matrix [22]. Other studies showed that tiles made from fiber-reinforced composite residual sisal "baler wine" topper, eucalyptus pulp, mauve and banana, and combination of coconut fibers and eucalyptus pulp, acquires a higher resistance to bending loads. This solution is unique, in relation to the systems without the addition of fibers. [23] demonstrated the efficiency of mass determination in the lower order of higher frequency energy beams and concrete floors, of major concern in the analysis of dynamic structures. The mass reduction system, allowed by adding natural fibers to concrete, is of great interest for practical purposes, for project and design of composites. Those results indicated the potential of its use in the reinforcement of brittle matrices, based on Portland cement, with the increase of its ductility [24]. Researches developed in UESC, State univertity of Santa Cruz, show the better performance of cement – natural fibers composites, in relation to solely cement matrix structures. Similar research, with the use of composite materials have been developed in UEFS, using natural sisal fiber [26], and long steel fibers, with successful results, also with consideration of the theory of Timoshenko beams, and slender beam [25]. Section 2 deal with the computational model performed to analyze the structural dynamic behavior of those composites, and section 3 shows the laboratory essays methodology. Section 4 presents some experimental (at computational or laboratory) results with comments. 2 COMPUTATIONAL MODEL 2.1

Conceptual model

As being previously developed [10, 14] the computational application model is a prismatic beam of symmetrical section related to the plan defined by longitudinal axis, x, and transversal axis, z. The Cartesian axis are oriented as a direct tried, resulting in transversal axis, y, oriented in normal direction of symmetric plan determined by x and z-axis. Models to compute simulation are non-damped free vibratory beams with length of 0,24 m, with 0,04 m in height and 0,04 m in width. This gives a aspect radius of 0.167, inside the TBT requirement for a Timoshenko’s beam. Load pattern is computed as a double uniformly distributed load of 100 kN/m and a midspan point shock load of 1,0 kN, for instantaneous impact, inducing free vibrations of beam models. This computational analysis is focused on determining composite cement – to – natural fibers dynamical performance, in relation to pure cement CPs beam, at TBT assumptions. Taking in account symmetrical bending of beams with little configurations change, and also considering a corrected rotation of transversal section in coupled bending and shear effect analysis. Arnold, Madureira and Zhang [28] studied the range of applicability of ReissnerMindlin [3 to 9] and Kirchhoff-Love theories in 3D thin plate elastic models showing that Reissner-Mindlin has a convergent solution while Kirchhoff-Love fails for shear effect. [10] is one of the few works in the literature that effort to determine limit of validity to Timoshenko’s Beams models to deal with structural shear effects, at concrete material in an effort to support carbon nanotubes, CNT, design. In an first analogous way, Harik [17] and Pietrobon-Costa [14] proposed ranges of validity of the continuum beam model for CNT in a way to realize the extreme properties and benefits of CNT in stiffness and strength – to -

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

weight high ratios. One dimensional beam models were used and response was plotted in transversal displacement ratio versus aspect ratio, h/L. Transversal displacement ratio was posed between displacement, in relation to shear effect consideration: with, w(G0), a TBT approach, and without, w(G4), and Euler – Bernoulli Theory approach. The variational formulation is developed to support the semi-discrete finite element formulation. A finite differences approach was used to deal with acceleration terms. Results were obtained taking in account a dynamic equilibrium condition for the structural system. Small configuration changes induce linear geometric behavior, without the effect of damping in the dynamic answer. In this way some assumptions were posed to solve the problem: (1) loads acts transversally, time independent, in the plan of symmetry xz; (2) material is homogeneous, isotropic and linearly elastic; (3) normal tensions σy and σz are small when compared to the normal tension σx; (4) displacements w act transversally to longitudinal axis of the beam; (5) transversal sections are plane and normal to the beam longitudinal axis, in the not deformed configuration, but comes plane and non-warped, even not normal to the axis, after the bending. 2.2 Variational formulation A displacement field, u, is defined, based in assumption 5, considering that traverse sections suffer only an rotation angle ψ, different from the declivity ∂w/∂x of longitudinal axis. In the scope of the Linear Theory of the Elasticity, it’s posed a deformationdisplacement relationship between differential operator B and displacement field, ε = Bu. Hooke’s Law derives from this last relation, to the tension-deformation relationship, where D is the elasticity matrix, in the form σ = Dεε, or σ = DBu. Displacement field, differential operator, and elasticity matrix are given respectively by:  u  − zψ ( x , t )     u = v =  (1) 0  w   w ( x , t )     

∂  ∂x  0    0 B= ∂   ∂y ∂  ∂z ∂   ∂z

0 ∂ ∂y 0 ∂ ∂x

 0  0  ∂ ∂z     ∂ ∂x  ∂  ∂y 

(2)

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

0 0 0  E 0 0 0 E 0 0 0 0   0 0 E 0 0 0    kE 0 0  0 0 0 (3) D= 2(1 + ν )    kE 0 0  0 0 0 2(1 + ν )   kE  0 0 0 0 0  2(1 + ν )   where ψ is the beam longitudinal axis rotation angle, k is the shear coefficient to beams, in this work taken to rectangular section, that is the Cowper approach, ν is the Poisson’s coefficient, and E, G are the longitudinal and transversal elasticity modulus. Action functional, Φ(u,t), is developed from Hamilton’s Principle, considering kinetic, strain and potential (from external loads) energies, respectively T, U and P: t2

Φ(u, t ) = ∫ (T − U − P) dt

(4)

t1

among all the dynamic ways of acceptable configurations between arbitrary time instants, t1 and t2, which satisfies Newton’s Law. The minimal, that is the extreme, of the action functional, gives the condition of dynamic stability of the structural system, t2

δ Φ (u, t ) = δ ∫ (T − U − P) dt

(5)

t1

where: 1 1L 2 2 ∫ ρu& dΩ = ∫ ∫ ρu& dA dx 2Ω 2 −A 1 T 1L U = ∫ ε ⋅ σ dΩ = ∫ ∫ ε T ⋅ Dε dA dx ⇒ 2Ω 2 −A L 1 ⇒ U = ∫ ∫ u T ⋅ BDBu dA dx 2 −A T=

(6)

L

P = − ∫ p T ⋅ u dx 0

where ρ denotes mass density, and p = [0 0 pz] is the applied loads vector. This development gives the functional, representative of total internal system energy: t2 L 1L 1L (7) Φ (u, t ) = ∫ ( ∫ ∫ ρ u& 2 dA dx − ∫ ∫ u T ⋅ BDBu dA dx + ∫ p T ⋅ u dx ) dt 2 −A t1 2 − A 0 which extreme form is given by: t2 L

L

L

t1 − A

−A

0

δ Φ (u, t ) = ∫ ( ∫ ∫ ρ u& ⋅ δu& dA dx − ∫ ∫ u T ⋅ BDBδu dA dx + ∫ p T ⋅ δu dx ) dt

(8)

Taking in account the equilibrium condition, that is given by δΦ(x,t) = 0, then after integrating by parts the terms of velocity, for arbitrary 1st variation of u& , in t1 and t2, that are arbitrary, it’s obtained: L

L

L

−

−

0

&& ⋅ δu dx − ∫ BDBAuδu dx + ∫ p T ⋅ δu dx = 0 ∫ ρAu

(8a)

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

or in matrix form, && + K u = F (8b) Mu Grouping terms, in eq. (8), and integrating by parts again the terms of first space derivatives, results in obtain for the one dimensional, 1D, weak form of the displacement equation: L

&& δw dx − ρ Iψ&&δψ dx − EIψ , x δψ , x −kGA( w , x −ψ ) δ ( w , x −ψ ) + p zδw ) dx = 0 ∫ (− ρAw

(9)

−

where I is the inertia momentum. The strong, or differential form, considering first variations of the unknowns as arbitrary at first and final instants, reading, && − −kGA( w , xx −ψ , x ) − p z = 0 ρAw (10) ρ Iψ&& − EIψ , xx −kGA( w , x −ψ ) = 0 The respective boundaries conditions in edge supports are showed in fig. 1 for beam models. 2.3 Finite Differences (FDM): time marching approach, and reduced integration A central finite difference scheme is adopted for first derivatives, in reduced integration form, where f is a general designation for the unknowns, and sub-index refers to generalized ith discrete node segment: f i + 12 − fi - 12 (11) f,x =

λ

The linear behavior of the problem induces a linear variation for acceleration between two successive time instants, so for second order time derivatives it provides (Newmark, 1959): &f&t 2 = 4 f t 2 − 4 (f t1 + f& t1 ∗ ∆t ) − &f&t1 (12) i i i i i ∆t 2 ∆t 2 for ∆t meaning time interval between t2 and t1 time steps. By substitution in the weak variational form of displacement equation model, reads the M coefficients matrix, related to time evolution, related to the time intervals. 2.4 Finite Elements (FEM) Numeric Formulation: space discretization approach In a Galerkin finite element semi-discrete approach, defining spaces for trial, u, and weighting, w, functions: u ∈ U = {u ∈ [H1 ]3 | u = u in ΓD } w ∈ W = {w ∈ [H10 ]3 | w = 0 in ΓD } u = v + u∴v ∈ W reading, in this FEM formulation, for finite dimensional functions: nno

u h = ∑ N j u ij j =1

nno

v h = ∑ N j v ij j=1

nno

w h = ∑ N j w ij j=1

so, equation (8a) holds:

(13)

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

n

nno −1 nno L

n

nno −1 nno L

T T ∑ ∑ ∑ ∫ ( ρANi N j&u&ij + B DBANi N ju ij ) w i dx = ∑ ∑ ∑ ∫ (p N i w i ) dx

t =1

i=2

j =1 0

t =1

i=2

(14)

j =1 0

for each element in the discrete mesh, representative of the beam. As posed by Hughes (1995), and again for each element results in obtain the equivalent to equation (8b) in Galerkin FEM formulation: && + Ku = F (15) Mu Where: nno nno L

M = ∑ ∑ ∫ ρANi N jdx i = 2 j=1 0

nno nno L

K = ∑ ∑ ∫ B T DB ANi N jdx

(16)

i = 2 j =1 0

nno nno L

F = ∑ ∑ ∫ pT N i dx i = 2 j =1 0

Considering the 1D case, local coefficients of those matrixes are assembled in global ones in association with their degrees of freedom, by a proper correspondence relation law, between locals and global degrees of freedom. 3 LABORATORIAL ESSAYS PREPARATION 3.1

Specimen manufacturing

The CPs where industrialized, as 5 m radius cylindrical CPs of 10 cm height, fig. 1, for compressive essays, and 4 cm x 4 cm x 24 cm long CPs, fig.2, for traction flexural essays. Those last dimensions are related to obtain a 0.167 aspect radius, a Timoshenko’s aspect radius characteristic. For preparation of the fibers used in the experiment, initially they were milled in a Wiley Mill of knifes, to increase the contact surface and finally dried in an oven to determine the moisture content of the same, fig. 3 and 4.

Figure 1 – Shear in flexural (beams) ,and Figure 2 – Molds for specimen manufacture: compressive (cylindrical), 5 % natural fibers shear in flexural (beams) ,and compressive to cement composites, CPs specimen (cylindrical) Materials employed to make the CPs were: a) Natural fibers: bamboo or coconut one, b) Locally regional sand, c) Cement Portland II, with a characteristic resistance (28 days) of 30 MPa,

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

d) metakaolin, used for partial replacement of cement at 30%, e) fly ash, for a partial replacement of cement mass at 20%, f) fresh pure water.

Figure 3 - Wiley Mill of knifes

Figure 4 – Milled cocoa fibers

Figure 5 – Milled bamboo fibers

For the analysis of tensile and compressive strengths in bending the material, concrete – to – natural fibers composite structures where performed with 5% of fibers incorporation. The laboratory specimen were tested for direct compression and traction in flexion, after completing 28 days of age; the ideal age to target the maximum resistance performance. 3.2

Materials granulometry

Workability of concrete is strongly influenced by the dimensions of the aggregated materials, as sand, and, in our research, of the natural milled cocoa or bamboo fibers. The finest the sand and shortest the fibers, more homogeneous will be the obtained composite concrete, more workable and free of void points, resulting in a strongest one. After the milling of coconut or bamboo, different particle sizes were obtained from them. Screening those milling products it was specified the percent of fiber (or sand) particles dimensions with relation to the openings of a collection of strainers. For this, a sieve shaker was used, fig. 5, where the strainer’s screens have openings 150µm to 2.36 mm, divided into 8 screens.

Figure 5 – Sieve shaker with sieve screens

Figure 6 – Percent of retained sand particles at each sieve characteristic mesh

Sand, cocoa, and bamboo fiber granulometry curves are viewed in figs. 6, 7 and 8, as

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

relation curve of percent of retained particles to sieve openings in the screens set.

Figure 7 – Bamboo fibers granulometry

Figure 8 – Cocoa fibers granulometry

As showed the sand granulometry answer to a medium sand, with a decay between 0.3 and 0.6 mm opening associated to a retained proportion near 75 % of particles with a diameter equal to mesh 50, i.e, 0.3 mm opening. Bamboo fibers that pass that same mesh is little than 12 %, and for cocoa fibers it was obtained a retention greater than 53 % to mesh 20, i.e, 0.85 mm, with a retention near 13 % more at mesh 30, or 0.6 mm opening. Both bamboo and cocoa fibers may be classified as medium particles, as be the sand. 3.3

Humidity measurement

Humidity determination of the concrete compounds blend is a significant property to adjust the water – to –cement percent ratio, in the sense to allow the acquisition of a perfect curing time of the concrete. Humidity was evaluated with the employment of NBR – 14.929:2003, the specific Brazilian norm, where the procedure is: 1) measure the initial mass; 2) put the sample in a stove for 1 hour at 105 º C; 3) remove and re-measure the mass; calculating the difference; 4) submit the sample to a new 1 hour drying, and then weight it again; 5) this cycle must be repeated until the occurrence of variation between a weighing and another be less than 0.5% of the final heavy mass; 6) the remaining mass is considered to be dry. In this approach the humidity, H, was evaluated by: Mi − Ms H= ⋅ 100% (17) Ms Coconut fiber and bamboo fiber, humidity’s are given in figs. 10 and 11, respectively. The bamboo fiber and coconut ones showed levels of humidity are respectively at 13% and 19.33%.

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

Figure 10 – Cocoa draying performance 4

RESULTS

4.1

Laboratory experimental essays

Figure 11 – Bamboo draying performance

Compression tests were performed on cylindrical specimens of 5 cm x 10 cm, with the use of a machine EMIC PCM 100C, fig. 12, with a maximum capacity load of 100 tons, at a growing in steps load of 0.25 MPa/min. Essays specimen, CPs,of this test were weighed and measured for a comparison between the load strength to densities ratios, of the specimens groups, fig. 13.

Figure 12 – Compressive press

Figure 13 – Compressive maximum stress to density ratios: (1st bar) pure cement matrix, (2nd bar) coconut – cement composite, (3rd bar) bamboo – cement composite

For the determination of compressive strength, three essays were performed for each kind of composite, each one employing tree body of proofs: pure cement, and with cocoa and bamboo fibers. Based on the results of the compression test, it was observed that the incorporation of coconut fiber and bamboo in the cement matrix promotes a resistance to compression 15.63% (coconut fiber) and 42 1% (bamboo fiber) below of compressive resistance of pure cement matrix.

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

For the determination of the tractive strength, a tensile bending essay was used, performed in a machine Gotech AI-7000 servo controlled, with a nominal capacity 20 KN, fig. 14. That bending essay considers the Timoshenko’s beams models in a bi-simple supported sample, fig. 15.

F

Figure 14 – Bending tractive strength determination essay

Figure 15 – Bending tree points essay scheme

For the tensile strength determination, curves of load, F (kN), to transversal displacement, d(mm), were plotted in relation to the beam central section. Those curves connect the maxima supported forces, before the plastic rupture, to the correspondent maxima displacement arrows, indicated as the A1, E6 and F2 curves, respectively for the pure cement matrix, for cement – cocoa composite matrix, and for bamboo – cement composite, fig. 16.

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

1.5 ×10

3

A1- Pure cement E6- Cocoa composite F2- Bamboo composite

1 ×10

3

500

0

0

0.1

0.2

0.3

0.4

Figure 16 – Tensile flexural strength experimental curves For the tensile strength determination, were plotted curves of load, F (kN), to transversal displacement, d(mm), at the beam central section. Those curves connect the maxima supported forces, before the plastic rupture, to the correspondent maxima displacement arrows, indicated as the A1, E6 and F2 curves, respectively for the pure cement matrix, for cement – cocoa composite matrix, and for bamboo – cement composite, fig. 16. The A1 series withstands a force 748.72 N, the series E6 answer to 1373 N, and was acquire a maximum force of 1286 N for F2 series It is observed that the pure cement shows a fragile behavior in relation to other series. In the force - displacement relationship it’s observe a change in failure mode of the composite coconut fibers and bamboo fibers, since they have higher strengths than the pure cement, but does not provide a smooth curve as the pure cement. In fig. 17 it’s showed the tensile strength for each of those materials, obtained of the tractive bending essay, as a ratio to respective densities. If for compressive strength was detected a less performance with natural fiber incorporation into the cement matrix, meanwhile for those results, in the bending tensile strength an increase of 37.97% (coconut fiber) and of 35.95% (bamboo fiber) were detected, when compared to pure cement matrix performance. Results plotted in fig. 18 shows an increase in longitudinal elastic modulus, E (MPa), for the cocoa – cement and for bamboo – cement composites, in relation to pure cement matrix, answering for 3.55, 6.24 and 5.16 GPa, respectively.

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

Figure 17 – Elastic modulus, longitudinal (GPa): pure cement, in 1st bar, cocoa composite (2nd bar),and bamboo composite, in last bar 4.2

Figure 18 – Ratio of tensile strength to density of laboratory CPs

Computational numeric essays

To construct knowledge about the dynamic answer of composite Timoshenko’s beams, in this first approach, some examples were processed, coneidering the free vibration of simple supported beams with the same pattern of the sample laboratory ones, as said being 4 cm x 4 cm x 24cm beams models. The numerical solutions were acquired by the computation of those just cement matrix, and composites with coca and bamboo fiber incorporated to the cement at a 5 % ratio, employing a self-developed computational code already tested and validated [10, 14]. The solution was constructed with employment of a semi-discrete finite element formulation, with employment of the constant acceleration method to deal with the time marching process of the problem, for a time step of 0.5 sec with process of 1000 time steps. Considering the beam with rectangular section discretized into 98 sections (number of nodes equal to 99), this is an optimal discretization for model analysis [10, 14, 16]. An impact load was applied to each beam model at 106 N at the central beam point, as excitation load. Responses where analyzed in terms of maximum transverse displacement of the beam, that is in the region located at node 50 for simple supported ones. Elastic and permanent loads properties were inserted in those computational models taking in account the results previously obtained in this work, as said those of the laboratory essays. With those assumptions fig. 19 shows the beams behavior for dynamic response, of pure cement matrix and cement – cocoa fiber composite materials. Fig. 20 gives the same answer behavior as a comparison of pure cement and bamboo fiber – cement composite ones.

IV ECCOMAS Thematic Conference on the Mechanical Response of Composites COMPOSITES 2013 F Pietrobon-Costa, R Carvalho Alvim, R A Arleo Alvim, L Pereira da Silva © IDMEC 2013

Figure 19 – Pure cement and cocoa – cement composite Timoshenko’s beams: free vibration dynamic behavior

Figure 20 – Pure cement and bamboo – cement composite Timoshenko’s beams: free vibration dynamic behavior

Cocoa – cement and bamboo - cement Timoshenko’s composites beams shows maxima transversal displacements of 0,00162 m and 0,00196 m, where the pure cement beams answer with a displacement of 0,00287 m. 4.3

Conclusions and connections

The incorporation of natural fibers to cement matrix materials, as intrinsic component, is already a challenge to researches. The presence of natural fibers in the composites returns a better performance in magnifying the dynamic response with a reduction of oscillating amplitude, a greater tractive strength when we compare the composites to pure cement matrix specimen, and the acquisition of a more tensile resistant, also more workable, and flexible composite, with the natural fibers incorporation, in special with cocoa one. The achieved granulometry for milled, both bamboo and cocoa fibers, to be medium particles as also the local sand. Those results are relate to some difficulty to a better homogenization of those aggregates of the cement composites. As result the laboratory essays gives compressive strengths of 15.63% (coconut fiber) and 42 1% (bamboo fiber) below of compressive resistance of pure cement matrix. This trouble must be verified with more essays, for a more energetic mixing of components of those composites, as also with more intense compaction procedures, in the sense that the compactation method employed in this work was shaking in a shake table. Meanwhile future essays will perform in a trace band of 2.5 to 50 % of natural fiber in to cement matrix ratio, that we evaluate will gives a greater compressive strength that had been acquired in composite blocks essays already in course in UESC LAMER and CPqCTR laboratories. For tensile strength essays the achieved resistance to tension forces, in bending tensile strength essays, had given an increase of 37.97% (coconut fiber) and of 35.95% (bamboo fiber) when compared to pure cement matrix performance. Those results are the answer to the natural fibers more cohesive structure as consequence of the fibers web arrangement incorporated to the cement matrix. Another positive gain of the natural fibers presented into those composite materials the gains in the elastic behavior, resulting in a better workable material, in special in relation

Pietrobon-Costa, F.; Carvalho Alvim, R.; ArléoAlvim, R. A.; Siveira, L.P.

to the cocoa fiber composite, when the longitudinal elastic modulus perform to 6.24 GPa. Bamboo one return 5.16 GPa for this same composite property, when pure cement matrix gives 3.55 GPa. For those results cocoa – cement matrix composite answer to a 75.78 % greater elastic modulus then the pure cement matrix Timoshenko’s beams. Cracking essays will be performed in next future to evaluate the gains in cracking prevention to beams response. Finally, the dynamic answer to impact excitation gives a better performance to the cocoa – cement composite then the pure cement CPs. Cocoa – cement and bamboo - cement Timoshenko’s composites beams shows maxima transversal displacements of 0,00162 m and 0,00196 m, where the pure cement beams answer with a displacement of 0,00287 m. It was detected a reduction in the dynamic displacement answer to vibratory excitation of Timoshenko’s beams, when adding natural fibers to the cement matrix. The performance in this response gives an improvement 43,55 % for cocoa – cement composite, and of 31,71 % for the bamboo – composite one. BIBLIOGRAPHY [1] S. Timoshenko, On the vibration of bars of uniform cross section, Philosophycal Magazine 43, series 6 (1922) 125-131; [2] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics 6 (1945); [3] R. D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics 3 (1951); [4] G. R. Cowper, The shear coefficient in Timoshenko's beam theory, Journal of Applied Mechanics 33-2, trans. ASME 88 E (1966); [5] J. G. Oliveira, N. Jones, The influence of rotatory inertia and transverse shear on the dynamic plastic behavior of beams, Journal of Applied Mechanics 46 no. 2 (1979); [6] M. Levinson, An accurate simple theory of the static and dynamics of elastic plates, Mechanics Research Communications 7 no. 6 (1980); [7] M. Levinson, D. W. Cook, Thick rectangular plates - I: The generalized Navier solution, International Journal of Mechanics Science 25 no. 3 (1983); [8] M. Levinson, D. W. Cook, Thick rectangular plates - II: The generalized Lévy solution, International Journal of Mechanics Science 25 no. 3 (1983); [9] M. S. S. Borges, Análise do efeito do esforço cortante na deformação de vigas formulação geral do problema e determinação do coeficiente de cisalhamento; DSc Seminário de Qualificação, COPPE/PEC - UFRJ (1996); [10] F. Pietrobon, Análise numérica da flexão dinâmica de vigas com a consideração da deformabilidade por cortante e da inércia de rotação, MSc Thesis COPPE/PEC-UFRJ Rio de Janeiro (1998); [11] N. Perrone, R. Kao, A general finite difference for arbitrary meshes, Computer & Structures 5 (1975) 45-58; [12] L. Lapidus, G.F. Pinder, Numerical solutions of partial differential equations in science and engineering, John Wiley & Sons Inc, New York (1999); [13] J. C. Tannehil; D. A. Anderson, R. H. Pletcher, Computacional fluid mechanics and heat transfer; Taylor & Francis Publishers, Washington (1997); [14] Pietrobon, Flávio C . On Timoshenko's beams coefficient of sensibility to shear effect. TEMA. Tendências em Matemática Aplicada e Computacional, v. 9(3), p. 447-457, 2008. [15] D. H. Buragohain, S. C. Patodi, Large deflections analysis of plates and shells by

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