ANALYSIS OF BAMBOO AS A FUNCTIONALLY GRADED MATERIAL

ANALYSIS OF BAMBOO AS A

FUNCTIONALLY GRADED MATERIAL

G. LOKESHA* *

Associate professor, Department of Mechanical Engineering, Vemana Institute of Technology, Bangalore, India Email ID: [email protected], +91-9448694378

M. VENKATARAMA REDDY

Principal, Sri Krishna College of Engineering, Bangalore, India

T. YELLA REDDY3

Formerly of Department of Mechanical Engineering, UMIST, UK.

Abstract

Bamboo is an optimized natural composite that exploits the concept of Functionally Graded Material (FGM). Biological structures such as bamboo have complicated micro-structural shapes and material distribution, and thus the use of numerical methods such as finite element method can be a useful tool for understanding the mechanical behavior of these materials. This paper explores techniques such as finite element method to investigate the structural behavior of bamboo. Two-dimensional models of bamboo cells were built and simulated under tensile load, compression load and bending load cases, using ANSYS 12.1 version with two material options, one with isotropic material properties (averaged Young’s modulus) and the second with FGM properties (spatially varying Young’s modulus). In this study the stress obtained from FGM model are much higher than those obtained from Isotropic material model and the maximum stresses are noted at the outer diameter. This is due to the fact that the higher stiffness of that fiber-dense region and also the stress redistribution through the bamboo wall.

Keywords: Bamboo, functionally graded material / structure, finite element analysis

1. INTRODUCTION

Bamboo (Latin: Bambusa) is a group of perennial evergreens that belong to the true grass family Poaceae (subfamily Bambusoideae, tribe Bambuseae). There exist 75 genera and about 1250 species. These tall grasses only produce a primary shoot without secondary growth, contrarily to most woods. Among plants, bamboo has a unique structure which resembles that of a unidirectional fiber reinforced composite with many nodes along its length. Furthermore, bamboo’s growth is very fast producing an adult tree in one year.

From the analysis of the microstructure of bamboo [Xu (2001)], it was found that bamboo is a biological natural composite that can be regarded as Functionally Graded Material. Bamboo mainly consists of two materials, a fiber material surrounded by a matrix material. On average the distribution of the volume concentration of fibers is found to be 60% at the outer side of the stem and 20% at the most inner side of the wall. Most work in the literature that characterizes bamboo is experimental, dedicated to estimating strength and stiffness properties. Few works treating the modeling of natural fibers have been found in the literature and these deals primarily with simplified analytical model. Most significantly, the mesostructure was highly graded, with the volume fraction of the fibers in the bamboo culm increasing from ground to the top [Fumio (1995)]. The volume fraction of fibers also increases across the thickness of the bamboo culm, from outside to inside surface.

Considering that biological structures, such as bamboo, have complicated shapes and material distribution inside their domain. The use of numerical methods such as finite element method (FEM) can be useful tools for understanding the mechanical behavior of these materials. The objective of this work is to explore computational technique, including FEM to investigate the structural behavior of bamboo

matrix of Em=15 GPa. Here the Poisson’s ratio of 0.35 is used. The estimated variation of Young’s modulus through the bamboo thickness is given by expression [Emilio (2006)]

       E(r) =3.75℮(2.2r/t) GPa (1)

where ‘r’ denotes the position through thickness of the cell wall starting at the inner surface and ‘t’ denotes the thickness of the cell wall. The modulus at the inner surface is 3.75 GPa. The variation expressed by this equation corresponds to a common bamboo species known as Moso bamboo. For the 13 mm cell wall thickness of the model used in this study, the equation gives a maximum modulus of 33.84 GPa at the outer edge of the wall.

2. MATERIAL MODELS 2.1 Isotropic material

The first model considers a homogeneous isotropic material model with average Young’s modulus determined from the following expression.

(2)

The average modulus is obtained by integrating Eq.2 between r = 0 to r = 13. Which yields E= 13.67

GPa. This is close to the value of E=15 GPa for bulk material reported by Nogata and Takahashi [7], which was obtained through the rule of mixtures.

2.2. Functionally graded material

The second material model considers the continuous gradation of the young’s modulus through thickness of the cell wall as described by eq.(1). The objective of comparing these material models is to find differences in displacements and stresses computed by numerical method.

3. BAMBOO GEOMETRY FOR ANALYSIS

The bamboo geometry is a hallow cylinder with periodic stiffeners called diaphragms, located at positions called nodes. A bamboo cell is the section of culm between two diaphragms. The diameter of the culm is lightly tapered, being largest near the ground. Figure 1 shows the geometry of the modelled cell. A fillet was created to transition the diaphragm into the cell wall. The maximum specimen height equal to the outer diameter [Methods of tests for bamboo (2008)].

4. LOADS AND BOUNDARY CONDITIONS

For compression load case, loading and boundary conditions were same as that of tensile case. Compression load was applied to the faces of all elements on the opposite end. Figure 2b shows a schematic of the bamboo cell model compression boundary conditions

For bending load case the cell was assumed to be a cantilever with one end fixed and a concentrated load applied at the free end in transverse direction. Figure 2c shows the details of constraints and loads.

Figure 2. (a)Bamboo discretization boundary conditions and applied load for tension load  (b) For compression load & (c) For bending load case

5. RESULTS AND DISCUSSION

This section presents the results of finite element analysis and their comparison with available analytical solutions. Fig. 3a & Fig.3b shows the stresses for tension case for isotropic and functionally graded material models respectively. In isotropic case same stresses are obtained throughout the thickness, as it is true because of constant E value throughout. In functionally graded material model maximum stress is obtained at the outer layer, because of high E value,

Fig. 3a Tension case: Axial stress plot  (Isotropic material model)       Fig.3b Tension case: Axial stress plot (FGM)

Figure 4a & Fig.4b shows the stresses for compression case for isotropic and functionally graded material models respectively. In isotropic case same stresses are obtained throughout the thickness, as it is true because of constant E value throughout. In functionally graded material model maximum stress is obtained at the outer layer, because of high E value,

Fig. 4a Compression case: Axial stress plot (Isotropic model) Fig.4b Compression case: Axial stress plot (FGM)

Therefore the stress distribution in the functionally graded material differs greatly with isotropic material properties and leads to remarkable stress redistribution in the bamboo cell. This is due to the fact that the fibre density from inner diameter to outer diameter increases exponentially and hence the stiffness of the bamboo cell. The stress plots also demonstrate that the material gradient through cell wall has greater influence on the local

cell wall stresses.

Fig. 5a Stress plot: Isotropic material model Fig. 5b Stress plot: Functionally graded material

Table 1, shows the comparison of Finite Element Analysis results with analytical calculations.. Table 2, shows the displacements and stresses normalized with analytical solutions for better understanding of the results.

Table 1 Comparison of FEM results with analytical solutions

Table 2 Normalized displacements and stresses

6. CONCLUSIONS

Finite element simulation of bamboo structure using two material models and three loading conditions are preformed. Functionally graded material model is used to approximate the continuously varying material properties of the bamboo across the cell wall thickness and its influence in the mechanical behaviour of the bamboo was studied. Average material properties are derived to simulate the Isotropic material model. The finite element results are compared with the actual closed form solutions.

As can be seen from the displacement results for isotropic and functionally graded material models are closely matching. It should be sufficient to model the bamboo with simple elastic modulus obtained by rule of mixtures or an average modulus obtained from FGM variation. This simple approximation will provide suitable numerical accuracy for capturing the “global” deflection response of a bamboo structure.

To estimate the local features, such as stresses near supports, pin connections or holes etc., it is necessary to employ a numerical procedure that accurately models material gradients through the cell wall. In this study the stress obtained from FGM model are much higher than those obtained from Isotropic material model and the maximum stresses are noted at the outer diameter. This is due to the fact that the higher stiffness of that fiber-dense region and also the stress redistribution through the bamboo wall.

The analytical calculations and FEM (isotropic material model) computations agree well for all the loading cases but are between 10 % to 20 % less than those computed using FEM for a FGM model. Keeping this in view analytical calculations can be performed with an appropriate margin for each of the loading case.

Stress (MPa)

Case Name Analytical Isotropic FGM

Tension 2.95 2.96 5.84

Compression 2.95 2.97 6.09

Bending 54.57 55.3 101.95

Deflections (mm)

Case Name Analytical Isotropic FGM

Tension 0.0598 0.0598 0.071

Compression 0.0169 0.0171 0.0193

Bending 2.923 2.903 3.114

Normalized stress (MPa)

Case Name Isotropic FGM

Tension 1.00 1.98

Compression 1.01 2.06

Bending 1.01 1.87 Normalized displacements (mm)

Case Name Isotropic FGM

Tension 1.00 1.19

Compression 1.01 1.14

Acknowledgement: The author is thankful to the Principal and Head of the Mechanical Engineering of Vemana Institute of Technology for their kind encouragement. The laboratory facilities provided by the parent body are gratefully acknowledged.

References

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[2] Emilio Carlos Nelli Silva, Matthew C, Walters and Glaucio H. Paulino, Modeling bamboo as a functionally graded material, J. Mater. Sci. 754-759, (2006)

[3] Fumio Naogata and Hideaki Takahashi, Intelligent Functionally Graded Material, Composite Engineering. 5(7), (1995), 743 -751. [4] Liese, W. “Anatomy and properties of bamboo”, Recent research on bamboos. Proceedings of the International Bamboo Workshop.

Hangzhou, China October 6-14, 1985.

[5] ‘Methods of tests for bamboo’. Bureau of Indian Standards, Manak Bhavan, 9 Bahadur Shah Zafar Marg, New Delhi 110002 (2008). [6] Nogata F, Takahashi H, Composite Engg, 5:743 (1995).

[7] Ulrike G.K. Wegst, The mechanical performance of natural materials, PhD thesis, University of Cambridge, UK. (1996),