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Hybrid approach to determine the mechanical parameters of fibers and matrixes of bamboo
Construction and Building Materials 35 (2012) 191–196
Contents lists available at SciVerse ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Hybrid approach to determine the mechanical parameters of fibers and matrixes of bamboo Zhou Aiping, Huang Dongsheng ⇑, Li Haitao, Su Yi School of Civil Engineering, Nanjing Forestry University, PO Box 210037, Nanjing, China
a r t i c l e
i n f o
Article history: Received 2 August 2011 Received in revised form 22 March 2012 Accepted 23 March 2012
Keywords: Bamboo Fiber Matrix Natural bamboo composite Micromechanics
a b s t r a c t In the view of micromechanics, bamboo can be idealized as a 2-phase composite consisting of vascular bundles (fibers which serves as reinforcement) and ground tissues (serves as matrixes). To determine mechanical parameters of fibers and matrixes is essential to quantitatively evaluate the mechanical properties of bamboo. This is significant to control the mechanical properties of engineered bamboo materials as expected in manufacture. However, it is impossible to measure the mechanical parameters of the fibers and the matrixes of bamboo at first hand because they cannot be separated from bamboo without damage. This paper presents a hybrid approach to determine the mechanical parameters of fibers and matrixes of natural bamboo. By micromechanical analysis, macro tensile experiments, and microscopic image analysis for bamboo samples, the mechanical properties of natural bamboo associated with the properties of fibers and matrixes and with the volume friction of fibers were quantitatively established. The results indicated that the Young’s moduli and the tensile strength of bamboo in longitudinal direction are determined by those of fibers and matrix, and are linearly related to the volume fraction of fibers. Using this character, the mechanical parameters of fibers and matrixes were determined. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Bamboo, known as a second forestry in China, is a renewable material with extensive application prospects. It has been expected to be a sustainable alternative for more traditional construction materials, such as concrete, steel and timber [1]. Earlier researches show that the energy consumption to produce 1 m3 per unit stress projected in practice for bamboo is great less than that for materials commonly used in civil construction, such as steel or concrete [2–5]. The ratio of strength to mass per volume of bamboo is much higher than that of concrete, steel and wood [1]. This makes bamboo an attractive alternative to traditional building materials. However, raw bamboo cannot fulfill the requirements of modern buildings because of their great variety in mechanical properties and geometry shapes. Therefore, if bamboo was used in modern construction, it is essential to process raw bamboo into engineered materials. Parallel Strand Bamboo (PSB), a new type engineered bamboo material made from bamboo strips (Fig. 1), which have outstanding properties and regular shapes, can meet the needs of modern buildings. Fig. 2 shows a 2-story building constructed by using PSB material, which was designed by the authors, serves as demo house in Qingchuan area where was severely ruined by the strong quake on May 12, 2008. But it is difficult to
⇑ Corresponding author. Tel.: +86 2585428890. E-mail address: [email protected] (D. Huang). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.03.011
commonly use PSB as a structural material, because no method has been developed to control its mechanical properties as expected in manufacturing till now. The parameters used in the demo house design were obtained by testing for the structural members. As a matter of fact, the mechanical properties of PSB are based on the properties of natural bamboo strips and their constitutional manner inside. One effective way to quantitatively control the mechanical properties of PSB may to select bamboo strips of mechanical properties in a desire range and to lay them out as designed. For this reason, to learn how to select bamboo of certain mechanical properties is an elementary step to control PSB properties. In the view of micromechanics, natural bamboo itself can be idealized as a 2-phase composite consisting of long and parallel cellulose fibers (vascular bundles) embedded in a ligneous matrix (ground tissues) [7]. The mechanical properties of bamboo are rested on the properties of fibers and matrixes, and on the distribution of fibers [6,7]. If the mechanical parameters of the phases and the volume friction of fibers are known, the mechanical properties of bamboo can be evaluated. However, these parameters of fibers and matrixes cannot be measured at first hand because they cannot be separated from bamboo without damage. Many researches have been carried out to investigate the mechanical behaviors of bamboo associated with the distribution of fibers along the longitudinal and radial directions of bamboo shell [7,8]. Young’s moduli and tensile strength of bamboo related to volume variation of fibers were suggested by statistic analysis in many references
192
A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
Fig. 1. Reformed bamboo materials.
fibers, the elements must have different fiber density. It is feasible to consider that the fibers are uniformly distributed over the cross-section of each element as long as it is narrow enough. Ignoring the microstructure of fibers and matrixes, they can be considered as an isotropic body. The natural bamboo strip can be represented by an ideal piece of transverse isotropic composite of the 2-phases, referring to Fig. 3b. Assuming that the longitudinal fibers in bamboo culm are continuously parallel to each other and the cross-section areas of them remain constant between two nodes, the volume fraction of the fibers can be calculated by their area percentage in any section cut. Referring to the Descartes coordinate system shown in Fig. 3b, let x1 denote the axis along the longitudinal direction of the culm, and x2 and x3 denote the other two axes in the plane perpendicular to x1. The mechanical properties of bamboo can be analyzed as follows. 2.2. Elastic parameters Let V denote the volume of representative element of bamboo strips, and Vf and Vm represent the volumes of fibers and matrixes, respectively. rf = Vf/V, rm = Vm/V are the volume fraction of fibers and matrixes, respectively. According to the isotropic and continuously assumption and referring to the coordinate system shown in Fig. 3b , the stress–strain relationship of fibers and matrixes can be expressed as
Fig. 2. A reformed bamboo building.
[9–14]. But these cannot reveal the mechanism of which the mechanical properties of fibers and matrixes affect the properties of bamboo. The mechanical parameters of fibers and matrixes remain unknown. This paper presents a hybrid approach to determine the mechanical parameters of fibers and matrixes of natural bamboo. Using this method, the associations of bamboo mechanical properties with the properties of fibers and matrixes, as well as the volume friction of fibers were quantitatively established. Furthermore, the Young’s moduli and the longitudinal tensile strength of fiber and matrix of bamboo were investigated by macro tensile experiments, mechanical analysis, and microscopic image analysis. 2. Micromechanical analysis 2.1. Modeling philosophy Fibers are graded distributed in the cross-section of bamboo culm along the radial direction. The density of them near the outside is greater than that near the inside of the culm, as shown in Fig. 3a. In order to mathematically establish the micromechanical model of bamboo, strips paralleling to the culm’s longitudinal direction were selected from every location along its radius as representative elements as shown in Fig. 3b. Due to the graded distribution of
rf11 ¼ Ef11 ef11 rm11 ¼ Em11 em11
ð1Þ ð2Þ
where rf11 and rm 11 are the longitudinal stress of fibers and matrixes, respectively, and similarly, ef11 and em 11 represent the corresponding strains for each. Ef11 and Em are longitudinal Young’s moduli of the 11 fibers and the matrixes, respectively. In the sense of macro scale, the average performances of bamboo are usually concerned and the stress–strain relationship of bamboo can be written as follows:
r11 ¼ E11 e11
ð3Þ
or
r11 ¼ rf rf11 þ rm rm11
ð4Þ
where r11 and e11 are macro average longitudinal stress and strain of bamboo strip, respectively, and E11 is its average longitudinal Young’s moduli. The prior studies have proved that the 2-phases of bamboo composite are deformed accordantly along the longitudinal direction in elastic state [15]. This suggests that ef11 , em 11 and e11 are equal to each other. Substituting Eqs. (1) and (2) in Eq. (4) gives
r11 ¼ ðrf Ef11 þ rm Em11 Þe11 m
Comparing Eq. (3) with Eq. (5) and considering r + r = 1, the average Young’s moduli along the longitudinal direction of bamboo composite associated with that of its 2-phases and with the volume fraction of fibers is obtained as follows:
Outside Matrix
x3
Inner side
ð5Þ f
x2
Vascular bundle
Fig. 3. Graded distribution of fibers and micromechanical model of bamboo culm.
193
A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196 Table 1 The relationships of mechanical parameters of fibers, matrixes, and bamboo. Knowing parameters Ef11 ,Em 11 ,
m,r
Lame parameter Ef11 f f Þð12 f
f
kf ¼ ð1þm
m
m Þ, k
Young’s moduli m
m Em 11 m Þð12 m Þ
¼ ð1þm
m
m
Shear moduli
m f E11 ¼ ðEf11 Em 11 Þr þ E11 , E12 ¼ E13 ¼
Ef11 Em 11 r m Ef11 þrf Em 11
m f Gm 12 , G13 ,r
ð6Þ
As the assumption above, the fibers and matrixes are regarded as isotropic elastic bodies, therefore, the relationships among elastic parameters k, G, E, and m can be given as follows [16]:
kf ¼
km ¼
Ef11 mf ð1 þ mf Þð1 2mf Þ m Em 11 m Þð1
m
ð1 þ m
2mm Þ
ð7aÞ
ð7bÞ
Gf12 ¼ Gf13 ¼
Ef11 2ð1 þ mf Þ
ð8aÞ
m Gm 12 ¼ G13 ¼
Em 11 2ð1 þ mm Þ
ð8bÞ
m where Gf12 , Gf13 , Gm 12 , and G13 are shear moduli of the fibers and the f matrixes, respectively; m and mm are Poisson’s ratios of them; kf and km are lame parameters for each. The transverse moduli (E12 = E13) and the shear moduli (G12 = G13) of the bamboo composite can be given by [15]
E12 ¼ E13 ¼
G12 ¼ G13 ¼
Ef11 Em 11 r m Ef11 þ r f Em 11 Gf12 Gm 12 r m Gf12 þ r f Gm 12
ð9aÞ
ð9bÞ
The Poisson’s ratio of bamboo composite can be given by [16]
m ¼ rf mf þ rm mm
ð10Þ
As analyzed above, if the longitudinal Young’s moduli, Poisson’s ratios of fibers and matrixes, and the volume fraction of fibers are known, the other elastic parameters of bamboo, fibers and matrixes can be evaluated. The relationships among these parameters are summarized in Table 1.
mf ¼ mm ¼ m
Gf12 Gm 12 rm Gf12 þr f Gm 12
E12 G12 ¼ G13 ¼ 2ð1þ mÞ
f m m f E11 ¼ r f Ef11 þ rm Em 11 ¼ ðE11 E11 Þr þ E11
Em 11
m Gf12 ¼ Gf13 ¼ 2ð1þmf Þ, Gm 12 ¼ G13 ¼ 2ð1þmm Þ
G12 ¼ G13 ¼
E12 ,m
Poisson’s ratios Ef11
mf ¼ mm ¼ m
microscopic image of a broken section of tensile bamboo specimen. It can be observed that the fibers had been drawn out of the matrix in one surface and left holes in the opposite one. This makes it difficult to accurately estimate the strength of fibers and matrixes by mechanical analysis. However, large amount of tests show that the tensile loading-stretch curves remain in straight until broken, therefore the point of ultimate strength of bamboo may be approximately taken the same point of elastic limit in the loading-stretch curve, i.e. the elongation of the 2-phases can be approximately regarded harmoniously before broken. From the view of this point, the tensile strength of bamboo can be expressed as [17]
ft ¼ rf ftf þ r m ftm ¼ ðftf ftm Þr f þ ftm ftf
ð11Þ
ftm
where ft , , and are the tensile strength of bamboo, fibers, and matrixes respectively. 3. Testing methodology Eqs. (6) and (11) show that Young’s moduli and strength of bamboo along the longitudinal direction are linearly related to the volume fraction of the fibers. This can be numerically determined by testing for bamboo specimens with various volume fractions of fibers. Consequently, the mechanical parameters in longitudinal direction of fibers and matrixes can be evaluated, and those in transverse direction can be calculated according the formulas presented in Table 1. Five-year old bamboo culms of about 10 cm diameters were selected to make experimental specimens. Making the moisture content of the culms less than 10% by drying them in a hot box on constant temperature of 60 °C for several days, and then wiping off the skin of culms, the specimens in the form of thin strip were cut out from every location of the culms along longitudinal and radial directions, as shown in Fig. 5. 3.1. Tensile test The configuration of standard tensile specimen is shown in Fig. 5c. Tensile loadings along longitudinal direction were applied to the each end of the specimens. The strains in the middle of the specimens in axial and in transverse directions were recorded simultaneously. Let the loading increase to 640 N from 0 and then unload it to 160 N at the rate of 7 N/s, and then reload it up to 640 N at the same rate. Repeating the loading and unloading cycle in the range of 160 N–640 N for five times hereafter, the mean values of the data recorded from the last four cycles were adopted to calculate the longitudinal Young’s moduli and the Poisson’s ratio for each specimen by
E11 ¼
Dr11 DF t ¼ De11 btDe11
ð12Þ
2.3. Tensile strength
m¼
e12 e11
ð13Þ
In most cases, fibers and matrixes in bamboo elongate coincidently when the composite subjected to tensile loading longitudinally. But in the ultimate of broken, slide between the 2-phases occurs and they are no longer elongating equally. Fig. 4 shows a
where E11 is the Young’s modulus of bamboo in longitudinal direction; Dr11 is the increment of stress; De11 and De12 are increments of strain at middle section along the axial and transverse directions respectively; DFt is the increment of loading; b and t are the width and thickness of the middle section of specimen respectively; and m is Poisson’s ratio of bamboo.
Broken fibers pulled out of the matrix
Holes left in the matrix due to fibers pulled out
Fig. 4. Broken section of a tensile specimen.
Node Culm
Sliced specimen
(a) Culm
Inner surfce
40mm
(b) section cut
4mm
Outer surfce
R=1
15mm
15mm
Section
7.5m m
A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
4mm
194
27.5mm 30mm
Thickness 4 mm
27.5mm
40mm
(c) Configuration of standard specimen
Fig. 5. Configuration of sliced tensile bamboo specimen.
(a) Counting fibers Fig. 6. Calculating the volume fraction of fibers.
Specimens testing for tensile strength are as the same as those of testing for Young’s moduli. Longitudinal loading was applied at the rate of 7 N/s monotonically until broken occurs. The strain in the middle section, the loading force, and the stretch value at the end of specimens were recorded simultaneously. The strength of the specimen can be calculated by
Fu ft ¼ bt
are linearly related to the variation of volume fraction of fibers. This is well agreement with relationship presented by Eq. (6), and it can be numerically expressed as Eq. (16) by data fitting.
E11 ¼ 29658r f þ 126
ð14Þ
Comparing Eqs. (6) to (16), it can be found that
where Fu is the ultimate loading applied to the specimens. b and t are width and thickness of the broken section in the middle of the specimens, respectively.
Ef11
3.2. Measuring the volume fraction of fibers
Em 11 ¼ 126
The volume fraction of fibers can be calculated by counting their numbers and measuring each area of them through microscopic image analysis of a section cut from the objective body, as shown in Fig. 6a and b shows how to measure the area of each fiber. Neglecting the hole in each fiber, the district surrounded by the dotline represents the area of a fiber. According to the continuous assumption, the volume fraction of fibers can be calculated by
rf ¼
Pn
s¼1 As
Ab
ð16Þ
Em 11 ¼ 29658
ð17aÞ ð17bÞ
Consequently, the Young’s moduli of the fibers and the matrixes along the longitudinal direction can be obtained respectively, i.e. Ef11 ¼ 29784 MPa and Em 11 ¼ 126 MPa. In order to validate the results obtained above, 12 specimens were tested and the results predicted by Eq. (6) and by test were compared in Table 2. It can be observed that the calculating values are well coincide with the test values.
ð15Þ
where As is the area of each fiber, and Ab is the area of broken section of specimen; n is the number of fibers.
4. Testing and analysis 4.1. Elastic constants One hundred and fifteen standard tensile specimens were tested according to the methodology described above. The longitudinal Young’s moduli were calculated by Eq. (12), and the volume fractions of fibers in the middle section of specimens were surveyed. The rule of Young’s moduli in longitudinal direction of bamboo variated to the volume friction of fibers is shown in Fig. 7. It can be observed that the Young’s moduli in longitudinal direction
Fig. 7. Longitudinal Young’s modulus connected to the volume fraction variation of fibers.
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A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
4.2. Poisson’s ratio The values of Poisson’s ratio calculated by Eq. (13) are given in Table 3. It suggests that the Poisson’s parameters of bamboo approximately remain in constant in despite of the volume fraction of the fibers keeping variation. As shown in Table 3, the mean value of 0.28 may be taken as the Poisson’s parameters of bamboo. Considering Eq. (10), the Poisson’s ratios of fibers and matrixes should be taken mf = mm = 0.28. Substituting Eq. (8a) and (8b) into Eq. (9b) and considering mf = mm = m and Eq. (9a), the shear moduli in transverse direction can be calculated by
G12 ¼ G13 ¼
E12 2ð1 þ mÞ
ð18Þ
4.3. Tensile strength of the phases
Table 4 Comparing of calculating and testing results of tension strength in longitudinal direction. Specimen number
1 2 3 4 5 6 7 8 9 10 11 12
Volume fraction of fibers
0.1612 0.2033 0.2334 0.3418 0.3353 0.3522 0.3716 0.3887 0.3911 0.4050 0.4255 0.4625
Tension strength in longitudinal direction (MPa) Calculating value
Test value
85.0 104.8 118.6 156.0 165.4 173.2 182.1 190.0 191.1 197.5 206.9 223.9
92.5 106.6 134.2 154.3 153.2 176.2 190.4 191.3 195.0 193.6 206.0 231.1
Tensile strength tests for the bamboo were implemented by the approach described above. The volume fractions of fibers in broken section were surveyed. The longitudinal strength related to the variation of volume fraction of fibers is shown in Fig. 8 according to the data analysis of 77 specimens. It can be concluded that the tension strength in longitudinal direction is linearly related to the volume fraction of fibers, which can be numerically expressed as
ft ¼ 448r f þ 11:4
ð19Þ
Therefore, the tensile strength of fibers and matrixes can be determined by comparing Eq. (19) with Eq. (11), i.e. ftf ¼ 459:4 MPa and ftm ¼ 11:4 MPa. Validation experiments of 12 specimens were carried out and the results were compared to that
Table 2 Comparing of calculating and testing results of Young’s moduli in longitudinal direction. Specimen number
Volume fraction of fibers
1 2 3 4 5 6 7 8 9 10 11 12
Young’s moduli in longitudinal direction (MPa)
0.1685 0.1923 0.2206 0.2991 0.3126 0.3254 0.3551 0.4015 0.4153 0.4231 0.4321 0.4924
Calculating value
Test value
5144.6 5853.5 6696.4 9034.4 9436.5 9817.7 10702.3 12084.3 12495.3 12727.6 12995.7 14791.6
5012.1 5793.6 9055.6 8824.1 9541.6 10539.0 10999.1 11864.4 12411.2 12344.4 12867.8 14994.9
Table 3 Poisson’s ratio of bamboo. Specimen number
Poisson’s ratio
Specimen number
Poisson’s ratio
Specimen number
Poisson’s ratio
1 2 3 4 5 6 7 8 9 10 11
0.28 0.30 0.29 0.29 0.29 0.31 0.29 0.27 0.30 0.30 0.27
12 13 14 15 16 17 18 19 20 21 22
0.29 0.29 0.29 0.27 0.27 0.29 0.24 0.24 0.27 0.29 0.31
23 24 25 26 27 28 29 30 31 32 33
0.29 0.30 0.28 0.29 0.29 0.30 0.29 0.27 0.24 0.27 0.27
Fig. 8. Longitudinal strength of bamboo related to the volume fraction variation of fibers.
predicted by Eq. (11) in Table 4. Well agreements of the two results can be observed. This indicates that the hybrid method to determinate the mechanical parameters of fibers and matrixes is feasible. 5. Conclusions The Young’s moduli and the tensile strength of natural bamboo in longitudinal direction are quantitatively determined by that of fibers and matrixes as well as the volume fraction of fibers, and are linearly related to volume fraction of fibers. The Young’s moduli of fibers and matrixes and of bamboo in transverse direction are rested with the longitudinal moduli of fibers and matrixes, the volume fraction of fibers, and the Poisson’s ratio of them. The Young’s moduli of fibers and matrixes of the samples along the longitudinal direction studied in this paper are approximately reached 30,000 MPa and 120 MPa, and the tensile strengths of them are about 450 MPa and 11 MPa respectively. The Poisson’s ratio of bamboo and its fiber and matrix may be taken as 0.28. Acknowledgment The research is supported by the Provincial Fund of Science, Jiangsu province (No. BK2009394). References [1] van der Lugt P, van den Dobbelsteen AAJF, Jassen JJA. An environmental, economic and practical assessment of bamboo as a building material for supporting structures. Constr Build Mater 2006;20:648–56. [2] Ghavami Khosrow. Bamboo as reinforcement in structural concrete elements. Cem Concr Compos 2005;27:637–49.
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[3] Ghavami Khosrow. Ultimate load behavior of bamboo reinforced light weight concrete beams. Cem Concr Compos 1995;17(4):281–8. [4] Janssen JA. Bamboo in building structures. PhD thesis, Holland: Eindhoven University of Technology; 1981. [5] Janssen JA. The importance of bamboo as a building material. Bamboos current research. In: Proc of the Int Bamboo Workshop, Kerala Forest Research Institute—India & IDRC—Canada, 1988. p. 235–41. [6] Nogata Fumio. Intelligent functionally graded material: bamboo. Compos Eng 1995;5(7):743–51. [7] Shigeyasu Amada, Yoshinobu Lchikawa, Tamotsu Munekata, et al. Fiber texture and mechanical graded structure of bamboo. Composites Part B 1997:13–20. [8] Tan T et al. Mechanical properties of functionally graded hierarchal bamboo structures. Acta Biomater 2011. http://dx.doi.org/10.1016/j.actbio.2011.06. 008. [9] Chung KF, Yu WK. Mechanical properties of structural bamboo for bamboo scaffoldings. Eng Struct 2002;24:429–42. [10] Amada S, Ichikawa Y, Munekata T, Nagase Y, Shimizu H. Fiber texture and mechanical graded structure of bamboo. Composites Part B 1997;28(13–20): 445.
[11] Gibson LJ, Ashby MF, Karam GN, Wegst U, Shercliff HR. The mechanical properties of natural materials. II. Microstructures for mechanical efficiency. Proc R Soc Lond A 1995; 450: p.141–62. [12] Ray AK, Mondal S, Das SK, Ramachandrarao P. Microstructural characterization of bamboo. J. Mater Sci 2004;39:1055–60. [13] Li XB. Physical, chemical, and mechanical properties of bamboo and its utilization potential for fiberboard manufacturing. Masters thesis, Louisiana State University; 2004. [14] Ghavami K, Rodrigues CS, Pacionik S. Bamboo: functionally graded composite material. Asian J Civil Eng 2003;4:1–10. [15] Jain S. Mechanical behavior of bamboo and bamboo composite. J Mater Sci 1994;29:5990–6. [16] Chen WF et al. Elasticity and plasticity. Beijing: China Architecture and Building Press; 2005. p. 140–6. [17] Aide-Memoire. Mechanics of material strength. Beijing: Science Press; 2009. p. 1247–65 [in Chinese].
Contents lists available at SciVerse ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Hybrid approach to determine the mechanical parameters of fibers and matrixes of bamboo Zhou Aiping, Huang Dongsheng ⇑, Li Haitao, Su Yi School of Civil Engineering, Nanjing Forestry University, PO Box 210037, Nanjing, China
a r t i c l e
i n f o
Article history: Received 2 August 2011 Received in revised form 22 March 2012 Accepted 23 March 2012
Keywords: Bamboo Fiber Matrix Natural bamboo composite Micromechanics
a b s t r a c t In the view of micromechanics, bamboo can be idealized as a 2-phase composite consisting of vascular bundles (fibers which serves as reinforcement) and ground tissues (serves as matrixes). To determine mechanical parameters of fibers and matrixes is essential to quantitatively evaluate the mechanical properties of bamboo. This is significant to control the mechanical properties of engineered bamboo materials as expected in manufacture. However, it is impossible to measure the mechanical parameters of the fibers and the matrixes of bamboo at first hand because they cannot be separated from bamboo without damage. This paper presents a hybrid approach to determine the mechanical parameters of fibers and matrixes of natural bamboo. By micromechanical analysis, macro tensile experiments, and microscopic image analysis for bamboo samples, the mechanical properties of natural bamboo associated with the properties of fibers and matrixes and with the volume friction of fibers were quantitatively established. The results indicated that the Young’s moduli and the tensile strength of bamboo in longitudinal direction are determined by those of fibers and matrix, and are linearly related to the volume fraction of fibers. Using this character, the mechanical parameters of fibers and matrixes were determined. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Bamboo, known as a second forestry in China, is a renewable material with extensive application prospects. It has been expected to be a sustainable alternative for more traditional construction materials, such as concrete, steel and timber [1]. Earlier researches show that the energy consumption to produce 1 m3 per unit stress projected in practice for bamboo is great less than that for materials commonly used in civil construction, such as steel or concrete [2–5]. The ratio of strength to mass per volume of bamboo is much higher than that of concrete, steel and wood [1]. This makes bamboo an attractive alternative to traditional building materials. However, raw bamboo cannot fulfill the requirements of modern buildings because of their great variety in mechanical properties and geometry shapes. Therefore, if bamboo was used in modern construction, it is essential to process raw bamboo into engineered materials. Parallel Strand Bamboo (PSB), a new type engineered bamboo material made from bamboo strips (Fig. 1), which have outstanding properties and regular shapes, can meet the needs of modern buildings. Fig. 2 shows a 2-story building constructed by using PSB material, which was designed by the authors, serves as demo house in Qingchuan area where was severely ruined by the strong quake on May 12, 2008. But it is difficult to
⇑ Corresponding author. Tel.: +86 2585428890. E-mail address: [email protected] (D. Huang). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.03.011
commonly use PSB as a structural material, because no method has been developed to control its mechanical properties as expected in manufacturing till now. The parameters used in the demo house design were obtained by testing for the structural members. As a matter of fact, the mechanical properties of PSB are based on the properties of natural bamboo strips and their constitutional manner inside. One effective way to quantitatively control the mechanical properties of PSB may to select bamboo strips of mechanical properties in a desire range and to lay them out as designed. For this reason, to learn how to select bamboo of certain mechanical properties is an elementary step to control PSB properties. In the view of micromechanics, natural bamboo itself can be idealized as a 2-phase composite consisting of long and parallel cellulose fibers (vascular bundles) embedded in a ligneous matrix (ground tissues) [7]. The mechanical properties of bamboo are rested on the properties of fibers and matrixes, and on the distribution of fibers [6,7]. If the mechanical parameters of the phases and the volume friction of fibers are known, the mechanical properties of bamboo can be evaluated. However, these parameters of fibers and matrixes cannot be measured at first hand because they cannot be separated from bamboo without damage. Many researches have been carried out to investigate the mechanical behaviors of bamboo associated with the distribution of fibers along the longitudinal and radial directions of bamboo shell [7,8]. Young’s moduli and tensile strength of bamboo related to volume variation of fibers were suggested by statistic analysis in many references
192
A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
Fig. 1. Reformed bamboo materials.
fibers, the elements must have different fiber density. It is feasible to consider that the fibers are uniformly distributed over the cross-section of each element as long as it is narrow enough. Ignoring the microstructure of fibers and matrixes, they can be considered as an isotropic body. The natural bamboo strip can be represented by an ideal piece of transverse isotropic composite of the 2-phases, referring to Fig. 3b. Assuming that the longitudinal fibers in bamboo culm are continuously parallel to each other and the cross-section areas of them remain constant between two nodes, the volume fraction of the fibers can be calculated by their area percentage in any section cut. Referring to the Descartes coordinate system shown in Fig. 3b, let x1 denote the axis along the longitudinal direction of the culm, and x2 and x3 denote the other two axes in the plane perpendicular to x1. The mechanical properties of bamboo can be analyzed as follows. 2.2. Elastic parameters Let V denote the volume of representative element of bamboo strips, and Vf and Vm represent the volumes of fibers and matrixes, respectively. rf = Vf/V, rm = Vm/V are the volume fraction of fibers and matrixes, respectively. According to the isotropic and continuously assumption and referring to the coordinate system shown in Fig. 3b , the stress–strain relationship of fibers and matrixes can be expressed as
Fig. 2. A reformed bamboo building.
[9–14]. But these cannot reveal the mechanism of which the mechanical properties of fibers and matrixes affect the properties of bamboo. The mechanical parameters of fibers and matrixes remain unknown. This paper presents a hybrid approach to determine the mechanical parameters of fibers and matrixes of natural bamboo. Using this method, the associations of bamboo mechanical properties with the properties of fibers and matrixes, as well as the volume friction of fibers were quantitatively established. Furthermore, the Young’s moduli and the longitudinal tensile strength of fiber and matrix of bamboo were investigated by macro tensile experiments, mechanical analysis, and microscopic image analysis. 2. Micromechanical analysis 2.1. Modeling philosophy Fibers are graded distributed in the cross-section of bamboo culm along the radial direction. The density of them near the outside is greater than that near the inside of the culm, as shown in Fig. 3a. In order to mathematically establish the micromechanical model of bamboo, strips paralleling to the culm’s longitudinal direction were selected from every location along its radius as representative elements as shown in Fig. 3b. Due to the graded distribution of
rf11 ¼ Ef11 ef11 rm11 ¼ Em11 em11
ð1Þ ð2Þ
where rf11 and rm 11 are the longitudinal stress of fibers and matrixes, respectively, and similarly, ef11 and em 11 represent the corresponding strains for each. Ef11 and Em are longitudinal Young’s moduli of the 11 fibers and the matrixes, respectively. In the sense of macro scale, the average performances of bamboo are usually concerned and the stress–strain relationship of bamboo can be written as follows:
r11 ¼ E11 e11
ð3Þ
or
r11 ¼ rf rf11 þ rm rm11
ð4Þ
where r11 and e11 are macro average longitudinal stress and strain of bamboo strip, respectively, and E11 is its average longitudinal Young’s moduli. The prior studies have proved that the 2-phases of bamboo composite are deformed accordantly along the longitudinal direction in elastic state [15]. This suggests that ef11 , em 11 and e11 are equal to each other. Substituting Eqs. (1) and (2) in Eq. (4) gives
r11 ¼ ðrf Ef11 þ rm Em11 Þe11 m
Comparing Eq. (3) with Eq. (5) and considering r + r = 1, the average Young’s moduli along the longitudinal direction of bamboo composite associated with that of its 2-phases and with the volume fraction of fibers is obtained as follows:
Outside Matrix
x3
Inner side
ð5Þ f
x2
Vascular bundle
Fig. 3. Graded distribution of fibers and micromechanical model of bamboo culm.
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A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196 Table 1 The relationships of mechanical parameters of fibers, matrixes, and bamboo. Knowing parameters Ef11 ,Em 11 ,
m,r
Lame parameter Ef11 f f Þð12 f
f
kf ¼ ð1þm
m
m Þ, k
Young’s moduli m
m Em 11 m Þð12 m Þ
¼ ð1þm
m
m
Shear moduli
m f E11 ¼ ðEf11 Em 11 Þr þ E11 , E12 ¼ E13 ¼
Ef11 Em 11 r m Ef11 þrf Em 11
m f Gm 12 , G13 ,r
ð6Þ
As the assumption above, the fibers and matrixes are regarded as isotropic elastic bodies, therefore, the relationships among elastic parameters k, G, E, and m can be given as follows [16]:
kf ¼
km ¼
Ef11 mf ð1 þ mf Þð1 2mf Þ m Em 11 m Þð1
m
ð1 þ m
2mm Þ
ð7aÞ
ð7bÞ
Gf12 ¼ Gf13 ¼
Ef11 2ð1 þ mf Þ
ð8aÞ
m Gm 12 ¼ G13 ¼
Em 11 2ð1 þ mm Þ
ð8bÞ
m where Gf12 , Gf13 , Gm 12 , and G13 are shear moduli of the fibers and the f matrixes, respectively; m and mm are Poisson’s ratios of them; kf and km are lame parameters for each. The transverse moduli (E12 = E13) and the shear moduli (G12 = G13) of the bamboo composite can be given by [15]
E12 ¼ E13 ¼
G12 ¼ G13 ¼
Ef11 Em 11 r m Ef11 þ r f Em 11 Gf12 Gm 12 r m Gf12 þ r f Gm 12
ð9aÞ
ð9bÞ
The Poisson’s ratio of bamboo composite can be given by [16]
m ¼ rf mf þ rm mm
ð10Þ
As analyzed above, if the longitudinal Young’s moduli, Poisson’s ratios of fibers and matrixes, and the volume fraction of fibers are known, the other elastic parameters of bamboo, fibers and matrixes can be evaluated. The relationships among these parameters are summarized in Table 1.
mf ¼ mm ¼ m
Gf12 Gm 12 rm Gf12 þr f Gm 12
E12 G12 ¼ G13 ¼ 2ð1þ mÞ
f m m f E11 ¼ r f Ef11 þ rm Em 11 ¼ ðE11 E11 Þr þ E11
Em 11
m Gf12 ¼ Gf13 ¼ 2ð1þmf Þ, Gm 12 ¼ G13 ¼ 2ð1þmm Þ
G12 ¼ G13 ¼
E12 ,m
Poisson’s ratios Ef11
mf ¼ mm ¼ m
microscopic image of a broken section of tensile bamboo specimen. It can be observed that the fibers had been drawn out of the matrix in one surface and left holes in the opposite one. This makes it difficult to accurately estimate the strength of fibers and matrixes by mechanical analysis. However, large amount of tests show that the tensile loading-stretch curves remain in straight until broken, therefore the point of ultimate strength of bamboo may be approximately taken the same point of elastic limit in the loading-stretch curve, i.e. the elongation of the 2-phases can be approximately regarded harmoniously before broken. From the view of this point, the tensile strength of bamboo can be expressed as [17]
ft ¼ rf ftf þ r m ftm ¼ ðftf ftm Þr f þ ftm ftf
ð11Þ
ftm
where ft , , and are the tensile strength of bamboo, fibers, and matrixes respectively. 3. Testing methodology Eqs. (6) and (11) show that Young’s moduli and strength of bamboo along the longitudinal direction are linearly related to the volume fraction of the fibers. This can be numerically determined by testing for bamboo specimens with various volume fractions of fibers. Consequently, the mechanical parameters in longitudinal direction of fibers and matrixes can be evaluated, and those in transverse direction can be calculated according the formulas presented in Table 1. Five-year old bamboo culms of about 10 cm diameters were selected to make experimental specimens. Making the moisture content of the culms less than 10% by drying them in a hot box on constant temperature of 60 °C for several days, and then wiping off the skin of culms, the specimens in the form of thin strip were cut out from every location of the culms along longitudinal and radial directions, as shown in Fig. 5. 3.1. Tensile test The configuration of standard tensile specimen is shown in Fig. 5c. Tensile loadings along longitudinal direction were applied to the each end of the specimens. The strains in the middle of the specimens in axial and in transverse directions were recorded simultaneously. Let the loading increase to 640 N from 0 and then unload it to 160 N at the rate of 7 N/s, and then reload it up to 640 N at the same rate. Repeating the loading and unloading cycle in the range of 160 N–640 N for five times hereafter, the mean values of the data recorded from the last four cycles were adopted to calculate the longitudinal Young’s moduli and the Poisson’s ratio for each specimen by
E11 ¼
Dr11 DF t ¼ De11 btDe11
ð12Þ
2.3. Tensile strength
m¼
e12 e11
ð13Þ
In most cases, fibers and matrixes in bamboo elongate coincidently when the composite subjected to tensile loading longitudinally. But in the ultimate of broken, slide between the 2-phases occurs and they are no longer elongating equally. Fig. 4 shows a
where E11 is the Young’s modulus of bamboo in longitudinal direction; Dr11 is the increment of stress; De11 and De12 are increments of strain at middle section along the axial and transverse directions respectively; DFt is the increment of loading; b and t are the width and thickness of the middle section of specimen respectively; and m is Poisson’s ratio of bamboo.
Broken fibers pulled out of the matrix
Holes left in the matrix due to fibers pulled out
Fig. 4. Broken section of a tensile specimen.
Node Culm
Sliced specimen
(a) Culm
Inner surfce
40mm
(b) section cut
4mm
Outer surfce
R=1
15mm
15mm
Section
7.5m m
A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
4mm
194
27.5mm 30mm
Thickness 4 mm
27.5mm
40mm
(c) Configuration of standard specimen
Fig. 5. Configuration of sliced tensile bamboo specimen.
(a) Counting fibers Fig. 6. Calculating the volume fraction of fibers.
Specimens testing for tensile strength are as the same as those of testing for Young’s moduli. Longitudinal loading was applied at the rate of 7 N/s monotonically until broken occurs. The strain in the middle section, the loading force, and the stretch value at the end of specimens were recorded simultaneously. The strength of the specimen can be calculated by
Fu ft ¼ bt
are linearly related to the variation of volume fraction of fibers. This is well agreement with relationship presented by Eq. (6), and it can be numerically expressed as Eq. (16) by data fitting.
E11 ¼ 29658r f þ 126
ð14Þ
Comparing Eqs. (6) to (16), it can be found that
where Fu is the ultimate loading applied to the specimens. b and t are width and thickness of the broken section in the middle of the specimens, respectively.
Ef11
3.2. Measuring the volume fraction of fibers
Em 11 ¼ 126
The volume fraction of fibers can be calculated by counting their numbers and measuring each area of them through microscopic image analysis of a section cut from the objective body, as shown in Fig. 6a and b shows how to measure the area of each fiber. Neglecting the hole in each fiber, the district surrounded by the dotline represents the area of a fiber. According to the continuous assumption, the volume fraction of fibers can be calculated by
rf ¼
Pn
s¼1 As
Ab
ð16Þ
Em 11 ¼ 29658
ð17aÞ ð17bÞ
Consequently, the Young’s moduli of the fibers and the matrixes along the longitudinal direction can be obtained respectively, i.e. Ef11 ¼ 29784 MPa and Em 11 ¼ 126 MPa. In order to validate the results obtained above, 12 specimens were tested and the results predicted by Eq. (6) and by test were compared in Table 2. It can be observed that the calculating values are well coincide with the test values.
ð15Þ
where As is the area of each fiber, and Ab is the area of broken section of specimen; n is the number of fibers.
4. Testing and analysis 4.1. Elastic constants One hundred and fifteen standard tensile specimens were tested according to the methodology described above. The longitudinal Young’s moduli were calculated by Eq. (12), and the volume fractions of fibers in the middle section of specimens were surveyed. The rule of Young’s moduli in longitudinal direction of bamboo variated to the volume friction of fibers is shown in Fig. 7. It can be observed that the Young’s moduli in longitudinal direction
Fig. 7. Longitudinal Young’s modulus connected to the volume fraction variation of fibers.
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A. Zhou et al. / Construction and Building Materials 35 (2012) 191–196
4.2. Poisson’s ratio The values of Poisson’s ratio calculated by Eq. (13) are given in Table 3. It suggests that the Poisson’s parameters of bamboo approximately remain in constant in despite of the volume fraction of the fibers keeping variation. As shown in Table 3, the mean value of 0.28 may be taken as the Poisson’s parameters of bamboo. Considering Eq. (10), the Poisson’s ratios of fibers and matrixes should be taken mf = mm = 0.28. Substituting Eq. (8a) and (8b) into Eq. (9b) and considering mf = mm = m and Eq. (9a), the shear moduli in transverse direction can be calculated by
G12 ¼ G13 ¼
E12 2ð1 þ mÞ
ð18Þ
4.3. Tensile strength of the phases
Table 4 Comparing of calculating and testing results of tension strength in longitudinal direction. Specimen number
1 2 3 4 5 6 7 8 9 10 11 12
Volume fraction of fibers
0.1612 0.2033 0.2334 0.3418 0.3353 0.3522 0.3716 0.3887 0.3911 0.4050 0.4255 0.4625
Tension strength in longitudinal direction (MPa) Calculating value
Test value
85.0 104.8 118.6 156.0 165.4 173.2 182.1 190.0 191.1 197.5 206.9 223.9
92.5 106.6 134.2 154.3 153.2 176.2 190.4 191.3 195.0 193.6 206.0 231.1
Tensile strength tests for the bamboo were implemented by the approach described above. The volume fractions of fibers in broken section were surveyed. The longitudinal strength related to the variation of volume fraction of fibers is shown in Fig. 8 according to the data analysis of 77 specimens. It can be concluded that the tension strength in longitudinal direction is linearly related to the volume fraction of fibers, which can be numerically expressed as
ft ¼ 448r f þ 11:4
ð19Þ
Therefore, the tensile strength of fibers and matrixes can be determined by comparing Eq. (19) with Eq. (11), i.e. ftf ¼ 459:4 MPa and ftm ¼ 11:4 MPa. Validation experiments of 12 specimens were carried out and the results were compared to that
Table 2 Comparing of calculating and testing results of Young’s moduli in longitudinal direction. Specimen number
Volume fraction of fibers
1 2 3 4 5 6 7 8 9 10 11 12
Young’s moduli in longitudinal direction (MPa)
0.1685 0.1923 0.2206 0.2991 0.3126 0.3254 0.3551 0.4015 0.4153 0.4231 0.4321 0.4924
Calculating value
Test value
5144.6 5853.5 6696.4 9034.4 9436.5 9817.7 10702.3 12084.3 12495.3 12727.6 12995.7 14791.6
5012.1 5793.6 9055.6 8824.1 9541.6 10539.0 10999.1 11864.4 12411.2 12344.4 12867.8 14994.9
Table 3 Poisson’s ratio of bamboo. Specimen number
Poisson’s ratio
Specimen number
Poisson’s ratio
Specimen number
Poisson’s ratio
1 2 3 4 5 6 7 8 9 10 11
0.28 0.30 0.29 0.29 0.29 0.31 0.29 0.27 0.30 0.30 0.27
12 13 14 15 16 17 18 19 20 21 22
0.29 0.29 0.29 0.27 0.27 0.29 0.24 0.24 0.27 0.29 0.31
23 24 25 26 27 28 29 30 31 32 33
0.29 0.30 0.28 0.29 0.29 0.30 0.29 0.27 0.24 0.27 0.27
Fig. 8. Longitudinal strength of bamboo related to the volume fraction variation of fibers.
predicted by Eq. (11) in Table 4. Well agreements of the two results can be observed. This indicates that the hybrid method to determinate the mechanical parameters of fibers and matrixes is feasible. 5. Conclusions The Young’s moduli and the tensile strength of natural bamboo in longitudinal direction are quantitatively determined by that of fibers and matrixes as well as the volume fraction of fibers, and are linearly related to volume fraction of fibers. The Young’s moduli of fibers and matrixes and of bamboo in transverse direction are rested with the longitudinal moduli of fibers and matrixes, the volume fraction of fibers, and the Poisson’s ratio of them. The Young’s moduli of fibers and matrixes of the samples along the longitudinal direction studied in this paper are approximately reached 30,000 MPa and 120 MPa, and the tensile strengths of them are about 450 MPa and 11 MPa respectively. The Poisson’s ratio of bamboo and its fiber and matrix may be taken as 0.28. Acknowledgment The research is supported by the Provincial Fund of Science, Jiangsu province (No. BK2009394). References [1] van der Lugt P, van den Dobbelsteen AAJF, Jassen JJA. An environmental, economic and practical assessment of bamboo as a building material for supporting structures. Constr Build Mater 2006;20:648–56. [2] Ghavami Khosrow. Bamboo as reinforcement in structural concrete elements. Cem Concr Compos 2005;27:637–49.
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