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Conferência Brasileira sobre Materiais e Tecnologias não-convencionais na Construção Ecológica e Sustentável. BRASIL NOCMAT 2006 - Salvador,BA 29 de outubro a 01 de novembro de 2006

A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS Cruz, L.A.Ta ; Ghavami, K.b; Álvarez, J.J.Aa,* a

Universidad del Valle – COLOMBIA; [email protected], [email protected] b Pontifícia Universidade Católica do Rio de Janeiro – PUC-Rio [email protected] *Corresponding Author

ABSTRACT Bamboo is a natural, anisotropic and heterogeneous material widely used in structural applications. There is almost a complete lack of documentation about the anisotropic elastic properties of bamboo which avoids the generation of reliable models to simulate the mechanical response of the culm. A transversely isotropic model was adopted here for bamboo, characterized by five independent elastic constants. Experimental test were undertaken to determine the circumferential Yong’s modulus and the out-of-plane shear modulus. These elastic constants were calculated first using simple formulas of elasticity and were validated next using finite element anisotropic models considering true geometry as well as simplified geometries. Results showed that the use of idealized geometries and simple formulas provides a good estimation of elastic constants. Load-deformation curves displayed an elastic behavior with a high percentage of recovery. The out-of-plane shear moduli were 930 MPa and 710 MPa for Metake and Moso bamboos, respectively. In addition, the cirumferential moduli were 393 MPa and 1450 MPa for Guadua Angustifolia and Bamboo Mosso, respectively. These results confirm the significant anisotropic behavior of bamboo, which must be a factor to be included in a reliable mechanical model of the culm. The transversly isotropic model appears to be a simple alternative to consider main anisotropic effects of bamboo, however, more experiments have to be undertaken to explore advantages and limitations of this model. KEY WORDS:Bamboo, anisotropy, elastic constants, finite elements, characterization procedures

Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A INTRODUCTION Bamboo is a natural, anisotropic and functionally graded composite material [1, 2, 3] with physical and mechanical properties well suited for structural applications. The economical and environmental advantages of bamboo have been revealed in several studies [3, 4] which have further stimulated its uses and waken the interest of the scientific community in this material. Bamboo’s structural applications include: steel substitution in structural concrete elements, concrete slabs with bamboo permanent shutter forms [3], temporary scaffolds [5, 7], structural columns [5], and reinforcement elements in bahareque walls [8]. The bamboo culm is a complex structure formed by a hollow cylinder reinforced with transversal diaphragms appearing at variable distances along its length (figure 1). The regions near the transversal diaphragms are called nodes, which have an external diameter slightly larger than that of the inter-nodal region. The thickness of bamboo wall remains almost constant in the inter-nodal region and increases near the nodes. In addition to these geometric variations, bamboo’s complexity is further increased by its heterogeneity and anisotropy. These characteristics should be considered in a mechanical model of the culm. The finite element method (FEM) has shown to be an excellent tool to analyze a variety of problems in engineering, being able to take into account heterogeneous and anisotropic materials with irregular geometries, like the bamboo culm. The only documented study of a FEM model of a bamboo culm is that of Silva et al. [1] who conduct a 3-D FEM study to analyze tension, torsion and bending response of an idealized geometry using isotropic, orthotropic and functionally graded models, where the orthotropic elastic properties were obtained from homogenization theory and not from experimental characterization.. Results of the orthotropic model under tension were different than those of the isotropic model [1] suggesting the necessity to incorporate anisotropy for a proper simulation of the mechanical response of the material. Experimental studies have focused mainly on the determination of bamboo’s longitudinal elasticity modulus, which is currently the only elastic constant with a bamboo-specific standardized protocol [9]. One of these studies was accomplished by Ghavami and Marinho [6] who conducted compression and tensile tests instrumented with bi-directional strain gages to determine the longitudinal elasticity modulus and longitudinal-circumferential Poisson’s ratio of specimens from the bottom, middle and top part of bamboo stems, obtaining average results of 13.85 GPa and 0.30 for the elasticity modulus and the Poisson’s ratio respectively. Nogata and Takahashi [10] studied the variation of the longitudinal elasticity modulus through the bamboo wall due to the graded distribution of the longitudinal fibers obtaining values of 33.84 GPa and 3.75 GPa for the external and internal parts of the wall, respectively. Méndez and Vallecilla [11] conducted wave propagation tests on specimens of Guadua angustifolia finding a value of 13.69 ± 1.76 GPa. for the Young´s modulus. Sánchez and Prieto [12] adapted the ASTM D 198 timber standard to obtain from bending tests a shear modulus of 644 ± 280 MPa for Guadua angustifolia. As far as we know, experimental methods for the determination of other elastic constants that consider bamboo anisotropy have not yet been reported. A mechanical model of bamboo able to simulate the response under different types of load conditions would be an important tool to understand the role of the geometrical and material parameters in the mechanical response of the culm. In order to consider the anisotropy of bamboo, a transversely isotropic constitutive law was adopted here, described by five Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS independent elastic constants. To better characterize the model, experimental procedures were carried out for the determination of bamboo’s shear modulus and circumferential Young’s modulus from pure torsion and diametric compression tests, respectively. The proposed tests

Transversal Diaphragm

Node

Bamboo Wall Internodal Region

Transversal Diaphragm

Node

Figure 1. Geometric characteristics of the bamboo culm.

were designed to suit bamboo’s geometry; the torsion test uses equations that are considerably simpler than those used in the ASTM D 198 standard for the determination of the shear modulus, and the diametric compression test is the first reported procedure for the determination of the circumferential elasticity modulus. Transversely isotropic FE models where used to simulate the tests and to evaluate how the specimens geometric irregularities and anisotropy could affect the results. METHODS AND MATERIALS A transversely isotropic model was adopted here, characterized by five independent elastic constants which are the Young’s moduli in the axial (Ez) and circumferential directions (Eφ), the out-of-plane ( ν zφ) and in-plane (ν rφ) Poisson’s ratios, and the shear modulus (Gzφ) (figure 2).

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Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A



Figure 2. Definition of the cylindrical coordinate system used with bamboo.

Determination of the torsion shear modulus (Gzφ) The principle used for the determination of Gzφwas to measure the twist angle φof a bamboo culm subjected to twisting couples (torques) T. As a first approximation we used the mechanics of materials formula [14] that relates φand T in a tube with a circular constant cross section, defined as T L  J G

(1)

where L is the length of the tube, J is the polar moment of inertia and G is the shear modulus in the longitudinal – circumferential directions, i.e. Gzφ. Hence, the slope S of an experimental torque vs. twist curve allowed the calculation of Gzφas

L G z   S . J

(2)

Two specimens of approximately 1.3 m long and 10 cm diameter were tested under torsion: one of the species Bambusa metake (Metake bamboo) and the other of the species Phyllostachys pubescens (Moso bamboo). Both specimens were cut in the state of São Paulo, Brazil at an approximate age of three years. The Metake bamboo (specimen 1) had 3 internodal sections and the Moso bamboo (specimen 2) had 4 internodal sections. Nodal diaphragms were removed in specimen 1 in order to study their influence on torsion behavior. Diaphragms were left intact in specimen 2. Each specimen was loaded under torsion (figure 3) by hanging two boxes from two horizontal steel bars firmly attached to the free end of the specimen. One box was directly attached to the free end of one bar in order to apply a vertical downward force, while the second box was attached to the other free end by a cable passing trough a pulley, in order to exert a vertical upward force. Torque increments were accomplished by simultaneously depositing weights of 9.8 N (1 kgf) in both recipients. The unloading procedure was accomplished in a similar way, using the same load steps. The clamped end of the cantilever was fixed by 4 wooden pieces (figure 3), that were cut to fit bamboo’s curvature. To improve the contact at the fixed end, 2 mm thick foam paper was placed between the wooden pieces and the specimen. For the Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS measurement of the twist angle, a vertical wooden ruler was fixed to the specimen’s free end. A dial indicator (D.I.1) was used to measure the lateral displacement of the ruler at 50 cm from the specimens’ longitudinal axis. The twist angle was then calculated based on this lateral deflection using an arctangent relationship. A second dial indicator (D.I.2) was used to determine the vertical displacement of the specimen’s free end. D.I. 1

PULLEY

D.I. 2

Figure 3. Experimental setup for the torsion test and detail of the clamped end.

Since equation 1 is valid for an isotropic homogenous material with a perfect circular geometry, the proposed methodology for the estimation of the torsion shear modulus was validated with two finite element models of specimen 2 that considered anisotropy and the variations of the geometry. To develop the geometry of model 1, external diameters and thicknesses were measured (cutting the specimen after the test) every 2 cm along the length of the specimen in the internodal regions and every 0.5 cm near the nodes. Hence, the mesh was finer near the nodes, where the specimens geometries varied more rapidly (figure 4). In the radial direction, the mesh was divided into three equal parts. A second idealized mesh of specimen 2 was used to determine the role of the nodal diaphragms and the variation of the dimensions of the specimen in the torsional response (model 2). The idealized mesh maintained the same number of divisions than the firs model, but had constant dimensions, made equal to the averaged dimensions of the original mesh, namely, external diameter of 9.2 cm, wall thickness of 0.89 cm and length (constant on the first mesh as well) of 1.29 m. Nodal diaphragms were also neglected in model 2. Both models were analyzed with the program Algor (Algor Inc., Pittsburg, USA) using eightnode transversely isotropic brick elements with their elastic constants defined as Ez = 13.7 GPa [6], E φ= 1.45 GPa (obtained from diametric compression tests, described below), and ν zφ = 0.31 [6]. Since no experimental procedures have been reported for the determination of ν rφ in bamboo, this elastic constant was taken equal to ν . The shear modulus G was varied to zφ zφ fit the experimental slope.

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Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A

Nodal Diaphragm

Figure 4. Mesh of nodal region in specimen 2 (left) compared with picture of the same region.

To constrain the model, all 390 nodes in contact with the wooden pieces of the experimental setup were completely fixed. A 100 N-m torque was prescribed by applying 6 twisting couples at the free end. The twist angle φwas calculated from the lateral displacement of the node shown in figure 5.

Figure 5. Modeling of the torque and control node.

Determination of the circumferential Young’s modulus (Eφ) To determine the circumferential Young’s modulus (Eφ), bamboo rings were subjected to diametric compressive tests (figure 6), where the deflection ν 0 was calculated as follows [15]

 PR PR 3   2   PR v0      4 EA EI 4  4Gr As

(3)

where P is the diametric load, Eφis the circumferential modulus, R is the average radius, A is the area of the cross section under bending (given by L × h from figure 6), Gφr is the in-plane shear modulus, As is the shear area of area “A”, and I is the moment of inertia of area “A” about axis 1 (figure 6). In equation 3, the first, second and third terms represent deformation due to axial force, bending moment and shear force, respectively. A preliminary parametrical analysis showed that for radius/thickness ratios (R/h in figure 6) higher than 4.3 the shear contribution was less than 5% of the total deflection, which is consistent with the geometry of Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS bamboo specimens under investigation. Therefore, the third term in equation 3 was neglected so that the calculation of the slope S of a force-deflection experimental curve allowed the approximate determination of Eφas

  R R 3  2   E      4 A S I 4     

(4)

Figure 6. Ring subjected to diametric compressive load.

The diametric compression test was made with two groups of bamboo rings. The first group was intended to study the variation of E φalong the height of the culm. This group consisted of 54 rings chosen from the upper (18 rings), middle (18 rings) and bottom (18 rings) culms (i.e. nine rings per culm) of Colombian Guadua angustifolia. The rings had an external diameter of 110 ± 10 mm, a thickness of 12 ± 3 mm and were cut to have their length equal to one fourth of their external diameters. The second group was intended to study the influence of the length of the rings in the value of Eφand consisted of 12 rings cut from the bottom part of a Moso bamboo. Three rings were cut with each of the following lengths: One fourth, one half, three fourths and once their external diameter (figure 7). These rings had an external diameter of 87 ± 1 mm and a thickness of 8.0 ± 0.2 mm. Slightly different experimental setups were used for the two groups of rings. The first group was tested in a soil unconfined compression tester (EI25-3602 Soiltest) in which the speed of the deflection was controlled manually at an approximate rate of 1 mm/min. Load lectures were taken when deflection reached multiples of 0.127 mm (0.005 in). The rings’ deflection and the applied load were measured with a precision of 0.025 mm and 1.5 N. The second group was tested in an automatically operated universal testing machine (Instron 5500 R) (figure 8). The crosshead speed was set up at 1 mm/min and load acquisition was made every two seconds. The rings’ deflection and the applied load were measured with a precision of 0.01 mm and 0.1 N. Both groups of rings were loaded until rupture (figure 8).

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Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A Four FE models of rings from the second group were modeled to study the influence of geometric irregularities, anisotropy and length of the rings on the validity of equation 5. To take into consideration irregularities of the geometry, meshes were created over digitalized

Figure 7. Moso bamboo rings (second group) after tests.

Figure 8. Experimental setup for the second group of rings and details of the fracture patterns.

images of the traced cross sections of four rings. One ring was modeled for each of the characteristic lengths used, i.e. one fourth, one half, three fourths and once their external diameter (table 1). Table 1. Main characteristics of the meshes used to model the diametric compression tests.

Mesh 1 2 3 4

Average diameter [cm] 8.61 8.67 8.79 8.56

Length [cm] 8.96 (D) 6.73 (3D/4) 4.63 (D/2) 2.38 (D/4)

Average thickness [mm] 7.79 8.37 8.14 7.81

Number of nodes 2736 2016 1440 864

Number of elements 1944 1404 972 540

All four models used eight-node transversally isotropic brick elements with elastic constants equal to those of the torsion model. To simulate the boundary conditions (figure 8), all displacement degrees-of-freedom of the nodes located in the external face of the inferior part of the ring were constrained. Given that it is not possible to constrain rotational degrees-offreedom of brick elements, the lateral stability of the models was achieved by constraining lateral displacements of the nodes located in the interior face of the superior part of the ring. A prescribed displacement of 2 mm was assigned to all of the nodes located in the exterior Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS face of the superior part of the ring (figure 9). The application of this prescribed displacement generated vertical reactions in the constrained nodes. The Young´s modulus Eφwas obtained adjusting the experimental and the FE force-deflection curves. The t-student test for paired data was used to determine if there was a statistical significant difference. Prescribed displacement (2 mm)

Lateral constraint

Full constraint

Figure 9. Boundary conditions for the diametric compression models .

RESULTS AND DISCUSSION Torsion test (Gzφ) A linear elastic behavior (R2 > 0.99) was observed in the torque-twist angle curve for both specimens (figure 10). During the unloading process, specimens 1 and 2 recovered 96% and 99% of their angular deformation, respectively. The vertical deflections measured by the indicator D.I.2 varied randomly between -0.07 mm to 0.02 mm for specimen 1 and from 0.11 to 0.04 mm for specimen 2, implying that no significant accidental bending force was applied to the specimens. 80 70 Specimen 1

To rq ue [N-m]

60 50 40 30 Specimen 2

20 10 0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Twist angle [rad]

Figure 10. “Torque vs. twist angle” curve for specimens 1 and 2. Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A The shear modulus of Bambusa Metake resulted to be 30% higher than that of Moso Bamboo (Table 2). Given the limitations of the present study, i.e. only one specimen tested, no definitive conclusion can be reached about the validity of this result for the entire population of both species. However, results for both species are consistent respect to the high degree of elastic recovery, which is a valuable property for construction materials. For the Moso Bamboo the torsion shear modulus (Gzφ) obtained with eq. (1) was 8 % higher than those obtained with FE models 1 and 2, respectively (Table 2). The small difference between FE models indicates that both, variations of geometry respect to the circular form as well as node diaphragms, do not contribute significantly to the torsion response of the element. In addition, the relatively low difference between results of the FE models and that of eq. (2) validates the use of this equation to determine the shear modulus of bamboo under torsion. Table 2. Experimental results for specimens 1 and 2.

Measured Parameter Slope [N-m/rad] Length [m] External diameter [cm] Internal diameter [cm] Polar Moment of Inertia [m4] Shear Modulus (Gzφ) [MPa]

Specimen 1 Metake bamboo Experimental 4468 1.00 10.10 8.62 4.80E-06 930

Specimen 2 Moso bamboo Experimental FEM 1 2888 1.00 9.20 7.42 4.06E-06 710 659

FEM2

660

Shear moduli obtained in this study were near the 644 MPa, reported by Sánchez and Prieto [12] for Guadua angustifolia, which further confirms the value of this experimental setuconfiguration for the determination of shear modulus of bamboo. The relation between the shear modulus obtained in this study and the Young´s modulus in the longitudinal direction reported by Ghavami et al [6] for Guadua Angustifolia is 0.048, which is near 0.053 and 0.037 documented for Douglas-fir and Spruce timber [16], typical of highly anisotropic materials like wood. Diametric Compression Test (Eφ) Brittle failures were observed in all rings, which is consistent with the lack of circumferential fiber reinforcement. A slight softening behavior was observed in all experimental curves (figure 11) which may be due to crack initiation at some level of stress. Guadua Angustifolia rings It was not possible to test 5 of the 54 specimens due to small fissures in the internal side of the rings’ wall. Three of these 5 specimens were from the bottom part of bamboo stem number 5, which impeded the estimation of a circumferential elasticity modulus representative for that position of stem 5. The other 2 fissured rings belonged to the middle part of stems 2 and 6. Young’s modulus from each location resulted to be statistically different than those of the others positions (Table 3), being higher in the upper portion and lower in the bottom of the culm. The average value of the Young´s modulus (392.5 MPa) is only about 3% of

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS theYoung’s modulus in the longitudinal direction (~ 13.6 GPa) which is consistent with the anisotropy of the material, which has fibers aligned in the longitudinal direction. Table 3*. Average values and standard deviations of the circumferential elasticity moduli of the first group of rings of Guadua Angustifolia (MPa) Stem 1 Stem 2 Stem 3 Stem 4 Stem 5 Stem 6 Location Eφ S.D. Eφ S.D. Eφ S.D. Eφ S.D. E φ S.D. Eφ S.D. Top 608 58 675 75 352 45 332 70 721 46 455 37 Middle 466 23 573 12 222 21 218 30 482 40 337 41 Bottom 306 23 361 46 170 6 136 36 --381 74

Moso rings The twelve tested rings exhibited an approximated linear behavior (R 2 0.97) until attaining failure. No statistical difference was found for the circumferential modulus from rings with different lengths. The average modulus (1.45 GPa) was around 3.5 fold that obtained for Guadua Angustifolia in the present study. Since we have not found documented values of the circumferential modulus, even for other types of wood, no further comparison can be undertaken. The Young’s modulus obtained with the FE model (table 5) resulted to be 7% different than that obtained with eq. 4, suggesting that the irregularities of the geometry respect to the circular form can be neglected. 500 y = 119.87x + 103.27 450

2

R = 0.85

400 y = 73.90x + 23.10 2 R = 0.98

350

Load (N)

300 250 200 150 100 50 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Deflection (mm) Bottom part specimen

Middle part specimen

Figure 11. Typical load-deflection curves. Specimens from the middle and bottom parts of stem 4, exhibiting the 2 highest (0.98) and lowest (0.85) R coefficient. Table 4. Average values and standard deviations of the circumferential elasticity moduli of the second group of rings of Moso bamboo

Ring's Length L=D L=3D/4 L=D/2 L=D/4

Eφ- Ring 1 (MPa) 1678 1351 1188 1577

Eφ- Ring 2 (MPa) 1385 1492 1585 1348

Eφ- Ring 3 (MPa) 1644 1283 1554 1278

Eφ– Average (MPa) 1569 1375 1442 1401

S. D. (MPa) 160 106 221 156

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Cruz, L.A.T; Ghavami, K.; Álvarez, J.J.A Table 5. Eφvalues from the finite element models and relative difference with respect to the value obtained with equation 4.

Model L=D L = 3D/4 L = D/2 L = D/4

Relative difference (%) 6.6 0.7 5.4 1.3

Eφ(GPa) from FE models 1.55 1.46 1.37 1.47

2000 Ring 1/3 - L=D

y = 650.19x - 11.47

1800

R

2

= 0.99

1600 CompressiveForce(N)

Ring 1/3 - L=3D/4

y = 486.80x - 32.25

1400

R

2

= 1.00

1200 1000

Ring 2/3 - L=D/2

y = 340.95x - 35.50

800

R

2

= 1.00

600 Ring 1/3 - L=D/4

400

y = 167.76x - 3.07 R 2 = 0.99

200 0 0.0

0.5

1.0

1.5

2.0 Deflection (mm)

2.5

3.0

3.5

4.0

Figure 11. Typical load vs. deflection curves. Specimens from the middle and bottom parts of stem 4, exhibited 2 the highest (0.98) and lowest (0.85) R coefficient .

4. CONCLUDING REMARKS A transversely isotropic model has been proposed in this study for the mechanical characterization of bamboo. In addition, experimental procedures were developed for the determination of the out-of-plane shear modulus and the circumferential Young’s modulus, using a pure torsion test and a diametric compression test, respectively. The tests were executed in specimens of Guadua Angustifolia, Metake and Moso bamboo and validated with transversely isotropic finite element models that incorporated bamboo geometric features. Both experimental procedures were suited to fit bamboo characteristic geometry and were found to be appropriate for its elastic characterization. The transversely isotropic model showed encouraging results to represent the main anisotropic attributes of bamboo. However, more refinements have to be introduced in the model to represent other important features, like the non-homogeneous distribution of fibers in the wall. These topics should be the focus of future studies as well as a better understanding of the advantages and limitations of the transversely isotropic model. Other experiments should also be conducted to fully determine all five elastic constant of this model, i.e. the inof-plane Poisson’s ratio. Bamboo structural designers are encouraged to use the proposed methodologies to make characterizations of bamboo species that can lead to design values and appropriate safety factors. The results of these characterizations can also be used to build finite element models of the bamboo culm that better represent its anisotropy and therefore allow a more detailed study of its mechanical behavior.

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A TRANSVERSELY ISOTROPIC MECHANICAL MODEL FOR THE BAMBOO CULM: EXPERIMENTS AND FINITE ELEMENT MODELS REFERENCES 1. NELLI, E.; WALTERS, M.; PAULINO, G. Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials. Submitted to Journal of Materials Science. Dec. 12, 2004. 2. A. K. RAY; S. MONDAL; S. K. DAS; P. RAMACHANDRARAO. Bamboo – A functionally graded composite – correlation between microstructure and mechanical strength. Journal of Materials Science 40. 2005. 3. GHAVAMI, K. Bamboo as reinforcement in structural concrete elements. Cement and Concrete Composites 27. 2005. 4. CLEUREN H. M.; HENKEMANS A. B. Development of the bamboo sector in Ecuador: Harnessing the potential of Guadua angustifolia. J. Bamboo and Rattan, Vol. 2, No. 2, p. 179-188, 2003. 5. CHUNG, K. F.; YU, W. K. Axial buckling of bamboo columns in bamboo scaffolds. Engineering Structures, 27. Elsevier 2002. 6. GHAVAMI, K.; MARINHO, A. Propriedades físicas e mecânicas do colmo inteiro do bambu da espécie Guadua angustifólia. Revista Brasileira de Engenharia Agrícola e Ambiental, v.9, p. 107-114, 2005. 7. CHUNG, K. F.; YU, W. K. Mechanical properties of structural bamboo for bamboo scaffoldings. Engineering Structures, 24. Elsevier 2002. 8. GONZALEZ, G.; GUTIERREZ, J. Structural performance of bamboo ‘bahareque’ walls under cyclic load. Journal of Bamboo and Rattan, v. 4, no. 4, 353-368, 2006. 9. ISO/DIS 22157 International Standard. Determination of physical and mechanical properties of bamboo. 2001. 10. NOGATA, F.; TAKAHASHI, I. Intelligent functionally graded material: Bamboo. Composites Part B: Engineering. Vol. 5, no. 7, pp. 743-751. 1995. 11. MÉNDEZ, L.; VALLECILLA, C. Sistema Constructivo de Casas en Tierra Timagua: Análisis Experimental de Conexiones. Undergraduate Thesis. Universidad del Valle. Cali - Colombia. 2003. 12. SÁNCHEZ, J; PRIETO, E. Comportamiento de la Guadua angustifolia sometida a flexión. Undergraduate Thesis. Universidad Nacional de Colombia, 2002. 13. ASTM D 198. Standard methods of static tests of timber in structural sizes. 14. GERE, J; TIMOSHENKO, S. Mecánica de Materiales. International Thomson Editors, Cuarta Edición, 1997. (Spanish) 15. ROARK, R. J.; YOUNG, W. C. Formulas for Stress and Strain. McGraw-Hill, 5th ed. 1975. 16. BEER; JOHNSTON. Mechanics of Materials. McGraw-Hill, New York, 1992.

Conferência Brasileira de Materiais e Tecnologias Não-Convencionais: Materiais e Tecnologias para Construções Sustentáveis. BRASIL NOCMAT 2006, Salvador, Bahia, Brasil. 29 de outubro a 01 de novembro de 2006. ISNB: 85-9873-07-5

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