Availableonlineatwww.sciencedirect.com °scⅰ ence Direct Journal of Bionic Engineering 5(2008)231-238 Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo Jian-feng Ma, Wu-yi Chen, Ling Zhao, Da-hai Zhao 1. School of Mechanical Engineering and automatio Bejing 1000g Beijing University of Aeronautics and Astronautics, 2. Shenyang Aircraft Design Institute, Shenyang 110035, P. R. China Abstract High load-bearing efficiency is one of the advantages of biological structures after the evolution of billions of years iomimicking from nature may offer the potential for lightweight design. In the viewpoint of mechanics properties, the culm of bamboo comprises of two types of cells and the number of the vascular bundles takes a gradient of distribution. A three-point bending test was carried out to measure the elastic modulus. Results show that the elastic modulus of bamboo decreases adually from the periphery towards the centre. Based on the structural characteristics of bamboo, a bionic cylindrical structure as designed to mimic the gradient distribution of vascular bundles and parenchyma cells. The buckling resistance of the bionic structure was compared with that of a traditional shell of equal mass under axial pressure by finite element simulations Results show that the load-bearing capacity of bionic shell is increased by 124.8%. The buckling mode of bionic structure is global buckling while that of the conventional shell is local buckling Keywords: bionic design, bamboo culm, thin-walled cylindrical structure, buckling, load-carrying efficiency Copyright o 2008, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. shell, supported by a compliant core, can bear a higher 1 Introduction buckling load than an equivalent hollow shell of the Material-efficient and lightweight structures are same weight and radius under uniaxial compression or critical requirements in the technical fields of aeronau- pure bending 2.1 Axial and circumferential stiffeners tics and astronautics. One of the essential objectives is to could improve the load-carrying capacity and change the improve load-carrying efficiency(the ratio of load bear- buckling mode of the cylindrical shells ng to weight) in the structure design. So the form, shape Numerical examples of cylindrical shell structures and size of structures need to be optimized to achieve the were presented to throw light on the structural analysis most appropriate configuration for special loadings. and optimization for better load-carrying efficiency. The Thin-walled cylindrical shells as efficient load-carrying buckling and post-buckling of cylindrical shells under structures are widely used in the fields of aeronautics, combined loading of external pressure and axial com- astronautics and shipbuilding. But due to high pression were investigated by Shen and ChenI.Using a length-to-radius ratio, thin-walled cylindrical shells are linear eigenvalue finite element analysis, Spagnoli prone to buckling, i.e., a cross-section flattening may studied different modes of instabil ity in stiffened conical occur which significantly reduces the resistance of the shells under axial compression sI. Sridharan and Zeg structure to bending. To avoid buckling, the thickness gane studied the interaction between local and overall and/or the radius of the hollow beam section are often buckling in stiffened plates and cylindrical shells with a increased to improve its bending stiffness. However, this novel finite elements model. The stabilization of a will reduce the load-carrying efficiency of the structures functionally graded cylindrical shell under axial har because of the increase in weight. Some studies of elas- monic loading was investigated by ng et al. 7) tic buckling suggested that a thin-walled cylindrical Nature is the largest laboratory from where many E-mail: zhiim. g author: Jian-feng Ma C un.ma@gmail.com
Corresponding author: Jian-feng Ma E-mail: zhijun.ma@gmail.com Journal of Bionic Engineering 5 (2008) 231–238 Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo Jian-feng Ma1 , Wu-yi Chen1 , Ling Zhao1 , Da-hai Zhao2 1. School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China 2. Shenyang Aircraft Design Institute, Shenyang 110035, P. R. China Abstract High load-bearing efficiency is one of the advantages of biological structures after the evolution of billions of years. Biomimicking from nature may offer the potential for lightweight design. In the viewpoint of mechanics properties, the culm of bamboo comprises of two types of cells and the number of the vascular bundles takes a gradient of distribution. A three-point bending test was carried out to measure the elastic modulus. Results show that the elastic modulus of bamboo decreases gradually from the periphery towards the centre. Based on the structural characteristics of bamboo, a bionic cylindrical structure was designed to mimic the gradient distribution of vascular bundles and parenchyma cells. The buckling resistance of the bionic structure was compared with that of a traditional shell of equal mass under axial pressure by finite element simulations. Results show that the load-bearing capacity of bionic shell is increased by 124.8%. The buckling mode of bionic structure is global buckling while that of the conventional shell is local buckling. Keywords: bionic design, bamboo culm, thin-walled cylindrical structure, buckling, load-carrying efficiency Copyright © 2008, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. 1 Introduction Material-efficient and lightweight structures are critical requirements in the technical fields of aeronautics and astronautics. One of the essential objectives is to improve load-carrying efficiency (the ratio of load bearing to weight) in the structure design. So the form, shape and size of structures need to be optimized to achieve the most appropriate configuration for special loadings. Thin-walled cylindrical shells as efficient load-carrying structures are widely used in the fields of aeronautics, astronautics and shipbuilding. But due to high length-to-radius ratio, thin-walled cylindrical shells are prone to buckling, i.e., a cross-section flattening may occur which significantly reduces the resistance of the structure to bending[1]. To avoid buckling, the thickness and/or the radius of the hollow beam section are often increased to improve its bending stiffness. However, this will reduce the load-carrying efficiency of the structures because of the increase in weight. Some studies of elastic buckling suggested that a thin-walled cylindrical shell, supported by a compliant core, can bear a higher buckling load than an equivalent hollow shell of the same weight and radius under uniaxial compression or pure bending[2,3]. Axial and circumferential stiffeners could improve the load-carrying capacity and change the buckling mode of the cylindrical shells. Numerical examples of cylindrical shell structures were presented to throw light on the structural analysis and optimization for better load-carrying efficiency. The buckling and post-buckling of cylindrical shells under combined loading of external pressure and axial compression were investigated by Shen and Chen[4]. Using a linear eigenvalue finite element analysis, Spagnoli studied different modes of instability in stiffened conical shells under axial compression[5]. Sridharan and Zeggane[6] studied the interaction between local and overall buckling in stiffened plates and cylindrical shells with a novel finite elements model. The stabilization of a functionally graded cylindrical shell under axial harmonic loading was investigated by Ng et al. [7]. Nature is the largest laboratory from where many
Journal of Bionic Engineering(2008)Vol 5 No. 3 scientific and engineering inventions and innovations organisms, could be extracted and applied to optimize are obtained. After several billion years of evolution, the the mechanical design process, and material-efficient structures of organisms have developed excellent prop- structures could be derived to improve the load-bearing erties and ingenious frames, which provide innovative behavior of the structural component prototypes approaches for solvin neering problems and improving design. Successful bionic designs in other fields can be used for further research on characteristics of biological structures and their application in mechanical design. This new method breaks through the traditional design mode and will create products with better performance and lighter structure 2 Cylindrical structures in nature and Fig. 2 Cross-section of plants: (a) Brazilian Giant Horsetail;(b) engineering Dutch Rush 9] Thin-walled cylindrical structures are found widely 3 Buckling of cylindrical shells in both engineering components and nature. The typical ratios of shell radius to thickness of a variety of cylin- 3.1 Buckling problems of cylindrical shells drical engineering structures can be seen in Fig. 1. In A cylindrical shell is shown in Fig. 3 with one end some applications, such as space shuttle fuel tanks, air- simply supported, which bears an axial uniform pressure craft fuselages, and offshore oil platforms, the ratio of P, and the other end clamped. The axially compressed load bearing to weight is an essential element of design. cylindrical shell can fail either by global buckling with a In nature, thin-walled cylindrical structures are often wavelength related to its length, or by local buckling supported by a honeycomb or foam-like cellular core with a wavelength related to the shell thickness, or by the that increases the resistance to buckling, for example, in yielding of the material of the shell. So the ratio of radius plant stems(Fig. 2), porcupine quills, or hedgehog to thickness determines the instability mode of the cy spines a) lindrical shell. In particular, when the thickness is rela- By analysis of macro and micro characteristics of tively small, the cylinder fails by local buckling( shown organisms, load-bearing structures'principles, which in Fig. 4 buckling shape B and C); while global buckling determine the excellent mechanical properties of occurs with a larger thickness(shown in Fig. 4 buckling Silos and,tanks nautical structures ofFshore oil structuref Biological structur ⊥⊥LLL LLL The ratio of radius to thickness g. 1 The ratio of radius to thickness a/t for typical engineering ylindrical structures/8 Fig 3 Cylindrical shell under axial compression
232 Journal of Bionic Engineering (2008) Vol.5 No.3 scientific and engineering inventions and innovations are obtained. After several billion years of evolution, the structures of organisms have developed excellent properties and ingenious frames, which provide innovative prototypes and creative approaches for solving engineering problems and improving design. Successful bionic designs in other fields can be used for further research on characteristics of biological structures and their application in mechanical design. This new method breaks through the traditional design mode and will create products with better performance and lighter structure. 2 Cylindrical structures in nature and engineering Thin-walled cylindrical structures are found widely in both engineering components and nature. The typical ratios of shell radius to thickness of a variety of cylindrical engineering structures can be seen in Fig. 1. In some applications, such as space shuttle fuel tanks, aircraft fuselages, and offshore oil platforms, the ratio of load bearing to weight is an essential element of design. In nature, thin-walled cylindrical structures are often supported by a honeycomb or foam-like cellular core that increases the resistance to buckling, for example, in plant stems (Fig. 2), porcupine quills, or hedgehog spines[8]. By analysis of macro and micro characteristics of organisms, load-bearing structures’ principles, which determine the excellent mechanical properties of Fig. 1 The ratio of radius to thickness a/t for typical engineering cylindrical structures[8]. organisms, could be extracted and applied to optimize the mechanical design process, and material-efficient structures could be derived to improve the load-bearing behavior of the structural component. Fig. 2 Cross-section of plants: (a) Brazilian Giant Horsetail; (b) Dutch Rush[9]. 3 Buckling of cylindrical shells 3.1 Buckling problems of cylindrical shells A cylindrical shell is shown in Fig. 3 with one end simply supported, which bears an axial uniform pressure P, and the other end clamped. The axially compressed cylindrical shell can fail either by global buckling with a wavelength related to its length, or by local buckling with a wavelength related to the shell thickness, or by the yielding of the material of the shell. So the ratio of radius to thickness determines the instability mode of the cylindrical shell. In particular, when the thickness is relatively small, the cylinder fails by local buckling (shown in Fig. 4 buckling shape B and C); while global buckling occurs with a larger thickness (shown in Fig. 4 buckling shape A). Fig. 3 Cylindrical shell under axial compression
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo 33 optimization of the cylindrical shell is necessary 3.2 Solution In order to improve the load-bearing capacity, two measures can be applied. One is to change the section dimension to gain a larger slenderness ratio which w however. increase the weight of the structure. The other is lateral bracing to improve the critical Euler column Buckling shape A buckling stre Some lateral bracing such as honeycomb-cored io/o sandwich panels or stringer stiffeners can improve the load-bearing capacity(Fig. 5). Typically the stiffener sh ilat, rolled T- or L-profiles, cold-formed L-profl elded T- or box profiles hat-stiffened plates. However, all the above mentioned Buckling shape B ( Local buckling) cylindrical shells are empirically designed and might not be the opti structure for the particular load condi- Fig. 4 Buckling shapes of a cylindrical shell. tion. If sandwich structures of the cylindrical shell could Actually, when global buckling occurs, the axial imitate the structures of organisms, better load-bearing stress a produced by the design pressure P must satisfy efficiency would probably be obtained the classic Euler buckling inequality σ≤σ (1) here u is the length coefficient, oo is the critical Euler column buckling stress, and I is cylindrical shells minimum second moment of area, a[D'-(D (a) Honeycomb core shell 64 Fig 5 Conventional lightweight cylindrical shell and the mass of the shell is 4 Structure and mechanical properties of dH「D3 D-21 A structure with a large slenderness ratio is very where d is the diameter of the cylinder, H is the leng easy to lead instability. Although bamboo has a diameter and t is the thickness of the shell. So the load-carrying to length ratio of 1/150-1/250(shown in Fig. 6)it also efficiency Is has high strength, flexibility, and stability under wind TED2+(D and snow loads. Compared to conventional materials 162H3p (4) such as timber. steel and concrete. bamboo exhibits higher specific strength. Outstanding structure deter- It is clear that a thicker shell can lead to lower mines the excellent mechanical properties, so it is nec oad-carrying efficiency when global buckling occurs In essary to make an in-depth analysis of the structure of order to improve the load-carrying efficiency, structural bamboo
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo 233 Fig. 4 Buckling shapes of a cylindrical shell. Actually, when global buckling occurs, the axial stress σ produced by the design pressure P must satisfy the classic Euler buckling inequality[10]: ( ) 2 0 2 π EI H σ σ μ ≤ = , (1) where μ is the length coefficient, σ0 is the critical Euler column buckling stress, and I is cylindrical shell’s minimum second moment of area, ( ) 4 4 π 2 64 D D t I ⎡ ⎤ − − ⎣ ⎦ = , (2) and the mass of the shell is ( )2 π 2 2 4 H m D Dt ρ = −− ⎡ ⎤ ⎣ ⎦ , (3) where D is the diameter of the cylinder, H is the length and t is the thickness of the shell. So the load-carrying efficiency is ( ) 2 2 2 0 2 3 π 2 16 E D Dt m H σ η μ ρ ⎡ ⎤ + − ⎣ ⎦ = = . (4) It is clear that a thicker shell can lead to lower load-carrying efficiency when global buckling occurs. In order to improve the load-carrying efficiency, structural optimization of the cylindrical shell is necessary. 3.2 Solution In order to improve the load-bearing capacity, two measures can be applied. One is to change the section dimension to gain a larger slenderness ratio which will, however, increase the weight of the structure. The other is lateral bracing to improve the critical Euler column buckling stress. Some lateral bracing such as honeycomb-cored sandwich panels or stringer stiffeners can improve the load-bearing capacity (Fig. 5). Typically the stiffener shapes are flat, rolled T- or L-profiles, cold-formed L-profiles, welded T- or box profiles, or trapezoidal, hat-stiffened plates. However, all the above mentioned cylindrical shells are empirically designed and might not be the optimum structure for the particular load condition. If sandwich structures of the cylindrical shell could imitate the structures of organisms, better load-bearing efficiency would probably be obtained. (a) Honeycomb core shell (b) Longitudinal hat-stiffened shell Fig. 5 Conventional lightweight cylindrical shells.. 4 Structure and mechanical properties of bamboo A structure with a large slenderness ratio is very easy to lead instability. Although bamboo has a diameter to length ratio of 1/150~1/250 (shown in Fig. 6) it also has high strength, flexibility, and stability under wind and snow loads. Compared to conventional materials such as timber, steel and concrete, bamboo exhibits higher specific strength. Outstanding structure determines the excellent mechanical properties, so it is necessary to make an in-depth analysis of the structure of bamboo
34 Journal of Bionic Engineering(2008)Vol 5 No. 3 only a few parenchyma cells between them. Towards the of the culm, the vascular and more widely distributed. The sequence of vascular bundle types from the periphery towards the centre fo distinct pattern which reflects the whole trum of variation across the culm-wall. So the bamboo structure can be generally viewed as a functionally graded structure made of long and aligned vascular bundles embedded in a parenchyma matrix 4.2 Structures in relation to mechanical properties Experiments were carried out to see the variation of mechanical properties at different position in bamboo structure. The observations were focused on the elasti modulus of the material Fig. 6 Bamboo with large slenderness ratio 4.2.1 Specimens preparation 4.1 Microstructure of bamboo The following methods were used to prepare the In brief the culm of bamboo comprises of about specimens. Four fan-shaped bamboo blocks were cut 52% parenchyma, 40% fibers and 8% conducting tissue from a segment between two bamboo nodes along the (vessels and sieve tubes) From the viewpoint of longitudinal direction. Each block was split into five mechanical properties, these cells can be divided into pieces along the radial direction. The thickness of the two types, the first type is parenchyma cells(Fig. 7b), specimens was about 1. 2 mm, as shown in Fig 8. Each hich are circular thin-walled structures and connected specimen has a length of 160 mm and width of 15 mm to each other by numerous simple pits, playing the role of load transmission, and the second type is vascular bundles, which are surrounded by parenchyma cells(Fig 7b), acting as a load bearing structure Parenchyma cells Vascular bundle Fan-shaped (a) Fig. 8 Specimen fabrication. Fig 7 Cross-section of bamboo A location parameter p was used to indicate the The form, size, number and concentration of the position of a specimen layer relative to the outer edge of bundles change continuously from the periphery of the segment. If the thickness of bamboo segment is t, and the culm towards the centre(Fig. 7a). Near the periphery the distance of specimen layer to the outer edge of the wall is bundles are smaller and more numerous. so that there are then p is defined as p=r/t
234 Journal of Bionic Engineering (2008) Vol.5 No.3 Fig. 6 Bamboo with large slenderness ratio. 4.1 Microstructure of bamboo In brief the culm of bamboo comprises of about 52% parenchyma, 40% fibers and 8% conducting tissue (vessels and sieve tubes)[11]. From the viewpoint of mechanical properties, these cells can be divided into two types, the first type is parenchyma cells (Fig. 7b), which are circular thin-walled structures and connected to each other by numerous simple pits, playing the role of load transmission, and the second type is vascular bundles, which are surrounded by parenchyma cells (Fig. 7b), acting as a load bearing structure. (a) (b) Fig. 7 Cross-section of bamboo. The form, size, number and concentration of the bundles change continuously from the periphery of the culm towards the centre (Fig. 7a). Near the periphery the bundles are smaller and more numerous, so that there are only a few parenchyma cells between them. Towards the centre of the culm, the vascular bundles become bigger and more widely distributed. The sequence of vascular bundle types from the periphery towards the centre forms a distinct pattern which reflects the whole spectrum of variation across the culm-wall. So the bamboo structure can be generally viewed as a functionally graded structure made of long and aligned vascular bundles embedded in a parenchyma matrix. 4.2 Structures in relation to mechanical properties Experiments were carried out to see the variation of mechanical properties at different position in bamboo structure. The observations were focused on the elastic modulus of the material. 4.2.1 Specimens preparation The following methods were used to prepare the specimens. Four fan-shaped bamboo blocks were cut from a segment between two bamboo nodes along the longitudinal direction. Each block was split into five pieces along the radial direction. The thickness of the specimens was about 1.2 mm, as shown in Fig. 8. Each specimen has a length of 160 mm and width of 15 mm. Fig. 8 Specimen fabrication. A location parameter ρ was used to indicate the position of a specimen layer relative to the outer edge of segment. If the thickness of bamboo segment is t, and the distance of specimen layer to the outer edge of the wall is r, then ρ is defined as ρ = r/t
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo 35 42.2 Graded elastic modulus of bamboo gion and sparse in the inner surface region(Fig. 7a) A three-point flexural test of the bamboo specimens Actually, the strength distribution in the cross-section of was carried out on an Instron 5565 at 5 mm min using culm is proportional to the volume fraction of fibers, and a 5 kN load cell. An extensometer was used to precisely the culm has a higher strength in the outer surface region monitor the deformation of the specimens(shown in Fig. than in the inner region[ 2 9). The initial linear portion of the load-displacement curve was used to calculate the elastic modulus. as 5 Bionic design of cylindrical shell shown in Fig. 10, the graded elastic moduli displayed a 5.1 Bionic design trend of decrease with the increase in location parameter To a great extent, the structures of organisms de- p. Each value reported is the average of four specimens pend on the environmental loads, so it is necessary to at the same radial location from different fan-shaped analyze the growth environment in order to understand block. the mechanical advantage of the structures of organisms The similarities between the biological and engi neering structures, including structural, functional and loading comparabilities, should be studied extensively Then the structural superiority of the selected organism could be applied in the engineering structural optimiza- tion. Biological structures are often more complicated than the engineering ones, so the manufacturability of the designed bionic structures must be taken into ac count. After full research on the structures of organisms, the bionic structure would be modeled with 3D model- ing software and a structural analysis would be run with FEA software to verify the effect of the optimization. If this approach fails, repeating the similarity analysis and structural characteristics to further the design criteria Fig 9 Test set-up for testing bamboo specimens and methods until the bionic structure obtains excellent mechanical properties, as shown in Fig. 1 Based on the relationship between ture and the mechanical properties of bamboo, a bio cylindrical shell structure was designed(shown in Fi 14 12), where the pipes and the ribs have the same function as the parenchyma cells, and the thin-walled cylindrical shells have the same function as the vascular bundles aggregated at the same radial location of the bamboo The graded thickness of the shells, the variational pipes and the ribs show that the bionic cylindrical shell is functionally graded structure In order to show the superiority of the bionic 0.5 Location parameter ructure,an equal-mass conventional shell witl Fig 10 Elastic modulus of bamboo specimens Fig 5b)was compare the load-carrying efficiency. Both of the The fiber distribution of sclerenchyma tissue in the structures have the same outer diameter D of 200 mm cross-section of bamboo culm is dense in the outer re- and the same length H of 1000 mm
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo 235 4.2.2 Graded elastic modulus of bamboo A three-point flexural test of the bamboo specimens was carried out on an Instron 5565 at 5 mm·min−1 using a 5 kN load cell. An extensometer was used to precisely monitor the deformation of the specimens (shown in Fig. 9). The initial linear portion of the load-displacement curve was used to calculate the elastic modulus. As shown in Fig. 10, the graded elastic moduli displayed a trend of decrease with the increase in location parameter ρ. Each value reported is the average of four specimens at the same radial location from different fan-shaped blocks. Fig. 9 Test set-up for testing bamboo specimens. Fig. 10 Elastic modulus of bamboo specimens. The fiber distribution of sclerenchyma tissue in the cross-section of bamboo culm is dense in the outer region and sparse in the inner surface region (Fig. 7a). Actually, the strength distribution in the cross-section of culm is proportional to the volume fraction of fibers, and the culm has a higher strength in the outer surface region than in the inner region[12]. 5 Bionic design of cylindrical shell 5.1 Bionic design To a great extent, the structures of organisms depend on the environmental loads, so it is necessary to analyze the growth environment in order to understand the mechanical advantage of the structures of organisms. The similarities between the biological and engineering structures, including structural, functional and loading comparabilities, should be studied extensively. Then the structural superiority of the selected organisms could be applied in the engineering structural optimization. Biological structures are often more complicated than the engineering ones, so the manufacturability of the designed bionic structures must be taken into account. After full research on the structures of organisms, the bionic structure would be modeled with 3D modeling software and a structural analysis would be run with FEA software to verify the effect of the optimization. If this approach fails, repeating the similarity analysis and structural characteristics to further the design criteria and methods until the bionic structure obtains excellent mechanical properties, as shown in Fig. 11. Based on the relationship between the microstructure and the mechanical properties of bamboo, a bionic cylindrical shell structure was designed (shown in Fig. 12), where the pipes and the ribs have the same function as the parenchyma cells, and the thin-walled cylindrical shells have the same function as the vascular bundles aggregated at the same radial location of the bamboo. The graded thickness of the shells, the variational pipes and the ribs show that the bionic cylindrical shell is a functionally graded structure. In order to show the superiority of the bionic structure, an equal-mass conventional shell with hat-stiffened plates (shown in Fig. 5b) was used to compare the load-carrying efficiency. Both of the structures have the same outer diameter D of 200 mm and the same length H of 1000 mm
236 Journal of Bionic Engineering(2008)Vol 5 No. 3 5. 2. 1 Nonlinear buckling analy Structural bionic design A nonlinear buckling analysis is a static analys with large deflection action, extended to a point where the structure reaches its limited load or maximum load Organism structure Mechanical structu Before a nonlinear buckling analysis, an eigenvalue buckling is needed to achieve the upper limit of the bearing load. Eigenvalue buckling analysis predicts the Similarity analysis theoretical buckling strength of an ideal elastic structure lt the structural eigenvalue for the tem load and constraints This is known as classical imilarity Euler buckling analysis Nonlinear buckling analysis employs non-linear Extract structural characteristics with excellent and large-deflection static analysis to predict buckling mechanical properties of organism loads. Its mode of operation is very simple: it gradually increases the applied load until a load level is found at Structural bionic design which the structure becomes unstable (i.e. suddenly a (Satisfy machining requests) very small increase in the load will cause very large deflection). The true non-linear nature of this analysis al property analysis thus permits the modeling of geometric imperfections, load perturbations, material nonlinearities and gaps 5.2.2 Results and discussion 1)Load-bearing efficiency Finish The load-bearing efficiency defined by Eq (4) Fig. 1l The flow chart of bionic design. alculated for the bionic structure and the conventional structrure(shown in Table 2 ). It can be seen that the load-bearing efficiency of the bionic structure is sig nificantly improved that is about 124.8% higher than that of the conventional one Table 2 Comparison of load-bearing efficiency onventional aprovement 8.237e+4 1.852e+5 124.8% Fig. 12 Bionic cylindrical shell based on bamboo (2)Failure mode a buckling mode represents a secondary deformed 5.2 FEM analysis shape that has the same potential energy as the primary The material parameters of steel for both cylindr deformation. There are two main buckling modes the cal shells are shown in Table 1 global buckling and the local bucklin Table 1 Material parameters of alloy steel Besides higher load efficiency, the failure modes of the bionic structure are also favorable. which N Ultimate compressive strength Elastic modulus was proved by the FME analysis result as shown in 7.8 210 0 Fig. 13. The conventional structure buckles locally while the bionic structure buckles globally
236 Journal of Bionic Engineering (2008) Vol.5 No.3 Fig. 11 The flow chart of bionic design. Fig. 12 Bionic cylindrical shell based on bamboo. 5.2 FEM analysis The material parameters of steel for both cylindrical shells are shown in Table 1. Table 1 Material parameters of alloy steel Density (g·cm−3 ) Ultimate compressive strength (GPa) Elastic modulus (GPa) Poisson ratio 7.8 2.5 210 0.3 5.2.1 Nonlinear buckling analysis A nonlinear buckling analysis is a static analysis with large deflection action, extended to a point where the structure reaches its limited load or maximum load. Before a nonlinear buckling analysis, an eigenvalue buckling is needed to achieve the upper limit of the bearing load. Eigenvalue buckling analysis predicts the theoretical buckling strength of an ideal elastic structure. It computes the structural eigenvalue for the given system load and constraints. This is known as classical Euler buckling analysis. Nonlinear buckling analysis employs non-linear and large-deflection static analysis to predict buckling loads. Its mode of operation is very simple: it gradually increases the applied load until a load level is found at which the structure becomes unstable (i.e. suddenly a very small increase in the load will cause very large deflection). The true non-linear nature of this analysis thus permits the modeling of geometric imperfections, load perturbations, material nonlinearities and gaps. 5.2.2 Results and discussion (1) Load-bearing efficiency The load-bearing efficiency defined by Eq. (4) was calculated for the bionic structure and the conventional structrure (shown in Table 2). It can be seen that the load-bearing efficiency of the bionic structure is significantly improved that is about 124.8% higher than that of the conventional one. Table 2 Comparison of load-bearing efficiency Conventional Structure Bionic structure Percentage of improvement Load-bearing efficiency (kN·kg−1 ) 8.237e+4 1.852e+5 124.8% (2) Failure modes A buckling mode represents a secondary deformed shape that has the same potential energy as the primary deformation. There are two main buckling modes: the global buckling and the local buckling. Besides higher load-bearing efficiency, the failure modes of the bionic structure are also favorable, which was proved by the FME analysis result as shown in Fig. 13. The conventional structure buckles locally while the bionic structure buckles globally
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo optimization based on bamboo is feasible AN 6 Co uSIon 3 (1) Bamboo resembles a long cylinder, reinforced by strong fibers, with a hollow core that reduces its 0 weight. The geometry of bamboos longitudinal profile has a macroscopically functionally graded structure whose compositions and mechanical properties vary continuously from the outer to the inner. The fiber den sity of sclerenchyma tissues within the bamboo is a good indicator of the strength capacity of bamboo 2) Three-point flexural test shows that the elastic (a)Conventional structure modulus in the outer surface region of bamboo culm is AN higher than that in the inner region, which shows the graded distribution of the mechanical properties of (3) Nonlinear analysis of cylindrical shells sub- jected to axial compression shows that the bionic shell based on bamboo takes advantages both in load-bearing efficiency and failure modes. In particular, load-bearing efficiency of the bionic structure is over two times of that of the conventional structure Failure mode of the conventional shell is local buckling while that of the bionic structure is global buckling (b) Bionic structure Acknowledgement Fig. 13 Buckling modes of the two cylindrical structures This research was supported by National Natural Before Brazier buckling occurs, the cross section of Science Foundation of China(Grant No. 50575008)and the cylindrical shell will become elliptical. The second the Aeronautical Science Foundation of China(Grant ment of area of the elliptical ring is No.05B01004) 1=42-(a-)(- (5) References [1] Fuchs H P, Hyer M W. The nonlinear prebuckling response where ay, b1 are the outer semi-major axis and the short of short thin-walled laminated composite cylinders in half axis of the elliptical ring bending Composite Structures, 1996, 34, 309-324 It is easy to calculate that the second moment of [2] Barlga S, Rothert H An idealization concept for the stability area of a cylindrical shell is larger than that of an ellip- analysis of ring-reinforced cylindrical shells under external tical ring with the same outer circumference. So the pressure. International Journal of Non-Linear Mechanics, load-carrying capacity of the deformed structure will 2002,37,745-756. 3] Hutchinson J w, He M Y Buckling of cylindrical sandwic Under the environment load. one of evolution shells with metal foams cores. International Journal of principles of organisms is that the least material bears Solids and Structures. 2000.37. 6777-6794 the greatest external force. The superior load-I aring [4] Shen H S, Chen T Y Buckling and postbuckling behaviour behavior of the bionic structure shows that the structural of cylindrical shells under combined external pressure and axial compression. Thin-Walled Structures, 1991, 1
Ma et al.: Elastic Buckling of Bionic Cylindrical Shells Based on Bamboo 237 (a) Conventional structure (b) Bionic structure Fig. 13 Buckling modes of the two cylindrical structures. Before Brazier buckling occurs, the cross section of the cylindrical shell will become elliptical. The second moment of area of the elliptical ring is ( )( ) 3 3 11 1 1 π 4 I ab a t b t = −− − ⎡ ⎤ ⎣ ⎦, (5) where a1, b1 are the outer semi-major axis and the short half axis of the elliptical ring. It is easy to calculate that the second moment of area of a cylindrical shell is larger than that of an elliptical ring with the same outer circumference. So the load-carrying capacity of the deformed structure will be weakened. Under the environment load, one of evolution principles of organisms is that the least material bears the greatest external force. The superior load-bearing behavior of the bionic structure shows that the structural optimization based on bamboo is feasible. 6 Conclusion (1) Bamboo resembles a long cylinder, reinforced by strong fibers, with a hollow core that reduces its weight. The geometry of bamboo’s longitudinal profile has a macroscopically functionally graded structure whose compositions and mechanical properties vary continuously from the outer to the inner. The fiber density of sclerenchyma tissues within the bamboo is a good indicator of the strength capacity of bamboo. (2) Three-point flexural test shows that the elastic modulus in the outer surface region of bamboo culm is higher than that in the inner region, which shows the graded distribution of the mechanical properties of bamboo. (3) Nonlinear analysis of cylindrical shells subjected to axial compression shows that the bionic shell based on bamboo takes advantages both in load-bearing efficiency and failure modes. In particular, load-bearing efficiency of the bionic structure is over two times of that of the conventional structure. Failure mode of the conventional shell is local buckling while that of the bionic structure is global buckling. Acknowledgement This research was supported by National Natural Science Foundation of China (Grant No. 50575008) and the Aeronautical Science Foundation of China (Grant No. 05B01004). References [1] Fuchs H P, Hyer M W. The nonlinear prebuckling response of short thin-walled laminated composite cylinders in bending. Composite Structures, 1996, 34, 309–324. [2] Barlga S, Rothert H. An idealization concept for the stability analysis of ring-reinforced cylindrical shells under external pressure. International Journal of Non-Linear Mechanics, 2002, 37, 745–756. [3] Hutchinson J W, He M Y. Buckling of cylindrical sandwich shells with metal foams cores. International Journal of Solids and Structures, 2000, 37, 6777–6794. [4] Shen H S, Chen T Y. Buckling and postbuckling behaviour of cylindrical shells under combined external pressure and axial compression. Thin-Walled Structures, 1991, 12
238 Journal of Bionic Engineering(2008)Vol 5 No. 3 9 5 Spagnoli A. Different buckling modes in axially stiffened S. Plants as concept generators for biomimetic lightweight conical shells. Engineering Structures, 2001, 23, 957-965 structures with variable stiffness and self-repair mechanisms [6 Sridharan S, Zeggane M. Stiffened plates and cylindrical Journal of Bionics Engineering, 2004, 1, 199-205 shells under interactive buckling Finite Elements in Analy- [10 Wu Q. Mechanical Properties and Optimization ofUl- sis and Design, 2001, 38, 155-178 tra-Light Weight Sandwich Structures, Master Thesis, [7 Ng T Y, Lam KY, Liew K M, Reddy JN. Dynamic stability Northwestern Polytechnical University, 2006.(in Chinese) analysis of functionally graded cylindrical shells under pe- [11 Liese w. The Anatomy of Bamboo Culms, Technical Report, riodic axial loading. International Journal of solids and 1998,2690 structures,2001,38,1295-1309 [12 Tommy Y L, Cui hz, Leung H C. The effect of fiber density [8] Dawson M A, Gibson L J. Optimization of cylindrical shells on strength capacity of bamboo. Materials Letters, 2004, 58 with compliant cores. International Journal of Solids and 2595-2598 Structures,2007,14,1145-1160
238 Journal of Bionic Engineering (2008) Vol.5 No.3 321–334. [5] Spagnoli A. Different buckling modes in axially stiffened conical shells. Engineering Structures, 2001, 23, 957–965. [6] Sridharan S, Zeggane M. Stiffened plates and cylindrical shells under interactive buckling. Finite Elements in Analysis and Design, 2001, 38, 155–178. [7] Ng T Y, Lam K Y, Liew K M, Reddy J N. Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading. International Journal of Solids and Structures, 2001, 38, 1295–1309. [8] Dawson M A, Gibson L J. Optimization of cylindrical shells with compliant cores. International Journal of Solids and Structures, 2007, 14, 1145–1160. [9] Speck T, Masselter T, Prüm B, Speck O, Luchsinger R, Fink S. Plants as concept generators for biomimetic lightweight structures with variable stiffness and self-repair mechanisms. Journal of Bionics Engineering, 2004, 1, 199–205. [10] Wu Q. Mechanical Properties and Optimization of Ultra-Light Weight Sandwich Structures, Master Thesis, Northwestern Polytechnical University, 2006. (in Chinese) [11] Liese W. The Anatomy of Bamboo Culms, Technical Report, 1998, 26–90. [12] Tommy Y L, Cui H Z, Leung H C. The effect of fiber density on strength capacity of bamboo. Materials Letters, 2004, 58, 2595–2598