Abstract: Bamboo stem’s cross sectional area is never a perfect circle, but almost ellipse. Each ellipse shape has a unique value of eccentricity (e) as parameter to denote its circularity. A perfect circle has a zero value of eccentricity. Conventional calculation for bamboo flexural properties as designated by ISO 22157-1:2004 resulted an overestimated or underestimated value compared to the actual value because of the perfect circle cross sectional assumption. Inappropriate geometrical assumption of cross sectional area derived inaccurate value of moment of inertia hereafter affected to the measured flexural properties. Thirty six bamboo stems from 4 species namely Ampel (Bambusa vulgaris Schrad.), Tali (Gigantochloa apus (Bl.Ex Schult.f) Kurz), Gombong (Gigantochloa verticillata (Willd.) Munro) and Mayan (Gigantochloa robusta Kurz.) were harvested and it was found that the eccentricity (e) value of bamboo stem could vary from 0.000 to 0.508. This paper studied the effect of eccentricity to the flexural properties of bamboo and aimed to create the strength ratio (Ce) between actual elliptical shape and assumed perfect circle shape. It was reported that the conventional calculation arise an under estimate result if the major axis arranged horizontally, while overestimate result will be get if the major axis arranged vertically. So the modulus of rupture (SR) which is calculated by conventional calculation should be adjusted by the strength ratio of eccentricity (Ce) in order to define more precise value. This study result the exact relationship between Ce value and eccentricity for both conditions. For simplicity, the graphical sketches were made too.
INTRODUCTION
Bamboo is natural product which traditionally has become the rural community’s main choice for many purposes in South East Asia villages because it is cheap and easy to find in their neighborhood; some bamboo species are used for building material (pillars, walls, roof and floor), e.g., Bambusa bambos (L.) Voss, B. blumeana (J.A. and J.H. Schulthes), B. tulda Roxb, B. vulgaris, Dendrocalamus asper, Gigantochloa apus (J.A. and J.H. Schultes) Kurz, G. atter (Hassk.) Kurz, G. levis (Blanco) Merrill, G. pseudoarundinaa (Steudel) Widjaja, G. robusta Kurz and G. scortechinii Gamble (Dransfield and Widjaja, 1995). People commonly build their bamboo houses based on the traditional experiences without any engineering calculations. Since the demand for green and sustainable construction arises and spreads globally (Tam et al., 2004; Lam et al., 2011; Kamar et al., 2010), recently bamboo construction attracts the engineer’s attention because of its artistic, high performance, natural resources sustainability and environmentally friendly (Chele et al., 2012; Yu et al., 2003; Chung and Yu, 2002; De Flander and Rovers, 2009). Many researcher reported the advantages of bamboo for environment (Bahtiar et al., 2012; Van der Lugt et al., 2006, 2012), its properties compared to another materials (Hamid et al., 2012; Verma et al., 2012; Sakaray et al., 2012; Jiang et al., 2012; Yu et al., 2008; Huang et al., 2012; Li and Shen, 2011) and its sustainability (Vogtlander et al., 2010; Nath et al., 2012).
As natural product, bamboo stem properties are influenced by many factors during its growth period, e.g., genetic and habitat condition (Kleinhenz and Midmore, 2001). These factors create the variability in size and physical shape, so every stem could have varied diameter size, taper and eccentricity (Nugroho and Bahtiar, 2012). Nugroho and Bahtiar (2012) conducted some researches of bamboo taper effect on its flexural properties. It was reported that the taper value didn’t affect to flexural properties on center point bending test, but the previous study on third point loading bending test showed that taper played significantly to its flexural properties. So the bamboo modulus of rupture (SR) should be adjusted by its taper strength ratio (Ct) when it was defined by third point loading bending test. Conventional method to measure the SR of bamboo stem as designated in ISO 22157-1:2004 based on third point loading bending test resulted under estimate values than the actual ones because of no-taper assumption. Adjusting the resulted testing value with the corresponding strength ratio will result more precise value. Beside taper effect, the eccentricity on bamboo stem will affected to its flexural properties which will be studied in this paper.
Bamboo stem commonly assumed as hollow cylinder shape (Sharmaa et al., 2013; Wegst, 2011; Inoue et al., 2011; Schulgasser and Witztum, 1992). In fact, a perfect circle of natural product (including bamboo stems) may never be found. The cross sectional area of bamboo stems are naturally more similar to ellipse than circle. There are always maximum and minimum diameters on every pieces of cross sectional area. Some standards (e.g., ISO 22157-1:2004) designated the average value of diameter as standard value to calculate the bamboo mechanical properties. This unapropriate geometrical assumption created an over or under estimate value compared to the actual properties because the inaccurate value of moment of inertia of plane area. Moment of inertia is directly related to the beam stress and strain (Nash, 1998) which is became the basic equation to calculate the flexural properties of beam. An overestimate mechanical properties of material could become dangerous in structural planning because the building could collapse since the overload condition, while the under estimate value created non-efficient building. A precise value of each material mechanical properties play important role in building construction planning. So it is important to study the effect of eccentricity on bamboo mechanical properties in order to plan the bamboo construction more reliable.
Eccentricity term is commonly used in physical and planetary science (Olson and Deguen, 2012; Correia et al., 2011). Eccentricity is the parameter to measure the circularity of ellipse shape. The eccentricity value for a perfect circle is 0 (zero), while the value becomes higher for the thinner ellipse shape. This study aimed to derived the exact mathematical relationship between eccentricity value and its effect on the bamboo stems flexural properties which determined by its strength ratio (Ce). Then this mathematical relationship was applied for eccentricity range value which was obtained from survey and harvested bamboo stems. Finally, this study resulted strength ratio formulae which could be applied as adjustment factor to gain more precise value of bamboo flexural properties.
MATERIALS AND METHODS
Survey on bamboo eccentricity: First, a survey was conducted on 5 bamboo shops in Bogor, West Java-Indonesia to measure the dimensional properties of available bamboos. We choose 20-40 bamboo stems on every shop randomly. At the same time we harvested 36 bamboo stems from 4 species in Arboretum Bamboo-Bogor Agricultural University: 9 stems from each species, then measuring its dimensional properties.
Strength ratio of eccentricity (Ce) derivation: Eccentricity effect on bamboo’s flexural properties defined by deriving it theoretically based on beam’s maximum stress concept. The ratio of maximum stress on ellipse (actual) and circle (assumed) cross sectional shape is denoted as strength ratio of eccentricity (Ce). The exact relationship between eccentricity (e) and its strength ratio (Ce) derived mathematically.
Ce value range for bamboo: The Ce value for overal range of bamboo stems eccentricity could be justified by substituting the range of eccentricity value which resulted from survey and harvested stems into the obtained mathematical equation.
RESULTS AND DISCUSSION
Survey on bamboo eccentricity: A survey was conducted in 5 bamboo shops in Bogor. The basal and top diameters of 162 bamboo Tali (Gigantochloa apus (Bl.Ex Schult.f) Kurz) stems which have 50-110 cm length were measured. The maximum diameter was defined as major axis and minimum diameter was the minor axis. The result was shown in Table 1. The basal eccentricity varied from 0.00 to 0.47 and the top varied from 0.00 to 0.51. Then 36 bamboo stems from 4 species, namely: Ampel (Bambusa vulgaris Schrad.), Tali (Gigantochloa apus (Bl.Ex Schult.f) Kurz), Gombong (Gigantochloa verticillata (Willd.) Munro) and Mayan (Gigantochloa robusta Kurz.), were harvested: 9 stems from each species. The measurement found that the bamboo cross sectional shape could vary from perfect circle into ellipse.
| Table 1: | Dimensional properties of tali stems |
| d: Average diameter, a: Major axis (maximum diameter), b: Minor axis (minimum diameter), e: Eccentricity, N = 162 | |
| Table 2: | Eccentricity of bamboo stems |
| N = 4, a: Major axis, b: Minor axis, e: Eccentricity | |
Zero eccentricity which means a perfect circle shape found in Tali and Ampel, but it was not found in Gombong and Mayan. As seen on Table 2, the overall eccentricity for 36 measured bamboo stems was 0.000-0.508. It was similar with the survey result on the shops. This condition proved that most of bamboo cross sectional plane was more similar to ellipse than circle shape. Meanwhile some researchers assumed the circle cross sectional area of bamboo stem in order to make more simple calculation for their study (Sharmaa et al., 2013; Wegst, 2011; Inoue et al., 2011). In their study, the diameter was defined as average of maximum diameter (major axis) and minimum diameter (minor axis). Since there is exact relationship between geometrical shape and beam’s stress and strain (Nash, 1998), bending test with perfect circle cross sectional area assumption may result unprecise value of bamboo’s flexural properties. In order to minimize the difference of assumed and actual value, a strength ratio should be applied (Nugroho and Bahtiar, 2012). Kretschmann (2010) defined: “the strength ratio is the hypothetical ratio of the strength of a piece of lumber with visible strength-reducing growth characteristics to its strength if those characteristics were absent”. On this study, strength ratio was defined as the hypothetical ratio of strength of a piece of bamboo stem with ellipse cross sectional shape compared to its strength if ideal circle shape applied.
Strength ratio of eccentricity (Ce) derivation: Bamboo stem’s cross sectional area is commonly assumed as a perfect circle, while its actual shape is almost ellipse (Fig. 1). Ellipse shape has major (a) and minor (b) axis which are the longest and shortest diameters, respectively (Bressoud, 1991). In order to calculate more simply, in some studies the circle diameter (d) which calculated as average of maximum and minimum diameter of ellipse shape is commonly chosen as the standard value (Sharmaa et al., 2013; Wegst, 2011; Inoue et al., 2011). So the mathematical relationship between a, b and d usually be defined as Eq. 1:
| (1) |
The strength ratio of eccentricity (Ce) denoted as the ratio of maximum stress in actual ellipse shape (σe) and the assumed cylindrical shape (σc) (Eq. 2):
| (2) |
| Fig. 1(a-b): | Sketch of assumed cylindrical shape compared to the actual ellipse shape which the major axis coincides with (a) Absis and (b) Ordinate |
Since the bending stress is known as Eq. 3 (Nash, 1998), so the eccentricity strength ratio could be define as Equation 4 because the maximum length from centroid (c) for circle is a half diameter (d/2) while for the ellipse is a half minor axis (b/2):
| (3) |
| (4) |
Substituting Eq. 1 into 4 it becomes:
| (5) |
Since the moment of inertia for circle (Ic) and ellipse (Ie) shape are denoted by Eq. 6 (Nash, 1998) and Eq. 7 (Symonds et al., 1996), respectively, Eq. 5 could be solved become Eq. 8:
| (6) |
| (7) |
| (8) |
Eccentricity is the ratio of the distance of any point on a conic section (ellipse, parabola, hyperbola or circle) from a focus to its distance from the corresponding direction.
| Fig. 2(a-b): | Strength ratio of ellipse bamboo when major axis arranged (a) Horizontally and (b) Vertically during bending test |
This ratio is describing the shape of a conic section and the value is constant for any particular conic section (Jennings, 1994). By this definition, eccentricity (e) is defined as Eq. 9, so ratio of minor axis (b) to major axis (a) of ellipse could be defined as Eq. 10:
| (9) |
| (10) |
Substituting Eq. 10 into 8, we get the exact relationship between eccentricity with its strength ratio as seen in Eq. 11 and the graphical sketch is shown in Fig. 2a:
| (11) |
As seen on Fig. 2a, strength ratio value for a perfect circle shape is 1 (one), while for ellipse shape is always higher than 1 (one). It is proved that the perfect circle assumption on conventional bending test resulted an under estimate flexural properties value when the major axis (a) configured horizontally during testing. Equation 11 and Fig. 2a are suitable for major axis (a) arranged coincided with horizontal axis (absis) (Fig. 1a). Different result will arise when the testing conducted with major axis (a) configured vertically as shown in Fig. 1b. If the major axis (a) arranged coincided with vertical axis (ordinate), the Ce value could be derived by similar way become Eq. 12 and the graphical sketch is shown in Fig. 2b:
| (12) |
Figure 2b showed that the strength ratio commonly lower than 1 (one). This condition proved that the conventional flexural properties are over estimate compared to the actual value if the major axis (a) configured horizontally during the bending test.
Strength ratio of eccentricity (Ce) value range for bamboo: As mentioned before, during the survey it was found that most of bamboo stems cross sectional plane varied from perfect circle into ellipse. Most of them were ellipse. Applying Eq. 11 for bamboo in bending test which its major axis arranged horizontally, the strength ratio value was 1.000-1.087. The detail strength ratio for all 4 species was shown in Table 3. So the conventional bamboo’s flexural properties value which calculated within circle shape of bamboo stem assumption could make 0-8.7% under estimate value. The under estimate flexural properties value will made the oversize structural component. The building will be stronger but more expensive. Meanwhile, Table 3 also showed the strength ratio for bamboo in bending test when its major axis arranged vertically. The values were gained by applying Eq. 12. For overall eccentricity range the strength ratio value was 1.000-0.935. It means perfect circle shape assumption on bamboo bending test caused 0-6.5% over estimated value compared to the actual modulus of rupture (SR) which tested by vertically arranged major axis ellipse shape configuration. This condition could be dangerous because it leads the engineer to design smaller size structural component than the demand. In extreme condition, the building could be collapse before estimated maximum load applied.
| Table 3: | Strength ratio of eccentricity for bamboo species |
| N = 4 | |
CONCLUSION
Cross sectional shape of bamboo stems could vary from perfect circle into ellipse. The eccentricity which denoted the circularity of the shape affected to the measurement of bamboo stem’s flexural properties. The relationship between eccentricity and its strength ratio was determined by mathematical equation and it was proved that circle assumption on bending test lead under estimate value if the major axis arranged horizontally on test configuration and lead over estimate value if the major axis arranged vertically. The measured modulus of rupture (SR) could be 0-8.7% lower or 0-6.5% higher than the actual value.
ACKNOWLEDGMENT
The authors thank “Direktorat Jendral Pendidikan Tinggi (DIKTI)”-Indonesian Ministry of Education for the support and research funding.