# Constitutive Material Model for the Compressive Behaviour of Engineered Bamboo

^{*}

## Abstract

**:**

## 1. Introduction

_{2}from the atmosphere during its growth stage [9] and hence, bamboo is often considered as a sustainable, environmentally friendly alternative to traditional construction materials. However, the use of natural bamboo, in major construction, is limited due to size constraints as well as the natural variations in material properties across its thickness and along its length [10].

## 2. Stress-Strain Behaviour of Bamboo and Engineered Bamboo

#### 2.1. Natural Full Culm Bamboo

_{f}) on the material behaviour, as shown in Figure 1a, where specimens a-f were extracted from different locations of the culm wall containing 15%, 17%, 22%, 24%, 37% and 46% V

_{f}, respectively. The overall compressive behaviour of bamboo may be divided into three distinct phases before failure—initially a linear elastic response, followed by a nonlinear stage and concluding with a plateau [40]. With the increasing V

_{f}, the initial stiffness and peak stress showed an obvious increase but at the expense of its ductility (as depicted by the length between the red and green dots). This decrease in ductility can be attributed to the decreasing percentage of the parenchyma matrix with the increasing V

_{f}. Shao and Fang [41] investigated the tensile behaviour of four year old Phyllostachys pubescens with a varying V

_{f}. Positive correlations were observed for the initial stiffness and maximum tensile strength with respect to the V

_{f}, as shown in Figure 1b. It is evident that the peak tensile stress is significantly higher than the corresponding compressive stress and stress-strain response under tension is linearly elastic till fracture.

^{3}to 750 kg/m

^{3}and observed a doubling of the strength from 47 MPa to 94 MPa, and the modulus of elasticity increased even higher, from 2067 MPa to 4896 MPa. It was also found that the density increased along the culm from the base to the top. A study [43] was conducted to find the optimum age for the cultivation of bamboo; the trend observed suggests that the mechanical properties were desirable when cultivated between 3–6 years. The moisture content was found to be another contributing factor; a study [44] on Bambusa pervariabilis and Phyllostachys pubescens showed that the compressive strength less than halved after the fibre saturation point (FSP) whereas the compressive modulus of elasticity remained constant.

#### 2.2. Engineered Bamboo

_{f}, which affects the mechanical properties, as was shown in Figure 1. Figure 2 shows the influence of the manufacturing variables on the compressive stress-strain behaviour, which are plotted based on experimental results shown in Table 1 and Table 2.

## 3. Existing Models for the Compression Behaviour of Engineered Bamboo

#### 3.1. Empirical Models

#### 3.1.1. Linear Model (LM) by Li et al. [48]

_{p}) and elastic (E

_{c}) moduli.

_{c}is the compressive modulus, ε

_{cy}is the yield strain, ε

_{c0}is the peak strain/beginning of the plastic plateau, ε

_{cu}is the end of the plastic plateau and f

_{c0}is the corresponding stress for ε

_{c0}.

#### 3.1.2. Quadratic Model (QM1) by Dongsheng et al. [45]

_{1}, λ

_{2}and λ

_{3,}which were determined based on their test results, and the continuum and compatibility conditions of Equation (3).

_{ce}and ε

_{cu}, respectively, as shown in Figure 3b.

_{cu}is the ultimate compressive strain, ε

_{ce}is the strain corresponding to the proportional limit, σ

_{cu}is the ultimate compressive stress and σ

_{ce}is the yield stress.

_{1}, a

_{2}and a

_{3,}were calibrated to maintain continuity and compatibility between the linear and nonlinear regions, as shown in Equation (6a–c).

_{3,}which has been rectified herein; Equation (6c) shows the amended expression for a

_{3}. This amendment made their proposal to be the same as the quadratic model proposed by Dongsheng et al. [45]. Henceforth, both models will be treated as one and will be referred as QM1 for the remainder of this study. Although the quadratic model was shown to replicate the material nonlinearity well, the use of coefficients that do not have any physical significance does not warrant scientific merit. This technique is merely a curve fitting practice, which will require the calibration for each set of experimental data and hence, cannot be accepted as a reliable scientific material model for engineering application.

#### 3.1.3. Quadratic Model (QM2) by Li et al. [19]

_{c0}after the ultimate compressive stain to account for the plateau, as depicted in Figure 4a. The parabolic segment of the model is governed by parameter a, which is a function of the ratio of the elastic and secant moduli, as shown by Equations (7) and (8). This formulation is similar to that of the LM (Section 3.1.1).

#### 3.1.4. Cubic Model (CM) by Li et al. [48]

#### 3.2. Modified Richard-Abbott (RA) Model by Wei et al. [46]

_{c2}) and a reference plastic stress (f

_{0}) that allow to capture material nonlinearity of engineered bamboo in a rational manner. In the RA model, stress is expressed as a function of strain, as shown in Equation (11).

_{c1}is the initial compressive modulus, f

_{cu}is the compressive stress and n is a shape parameter. Wei et al. [46] calibrated this model based on their experimental results obtained from their study. The following values were proposed for bamboo scrimber: E

_{c2}= 0.01E

_{c1}, f

_{0}= 0.97f

_{cu}and n = 2; while these values for LBL were reported to be: E

_{c2}= 0.04E

_{c1}, f

_{0}= 0.82f

_{cu}and n = 2.

#### 3.3. Limitations of the Existing Models

## 4. Material Properties of Bamboo Scrimber and LBL under Compression

## 5. Performance of the Existing Models against All Test Results

_{c2}, f

_{0}and n values proposed by Wei et al. [46] show good agreement with other test results reported in [19,45,46,47,48], but ideally, further calibrations will be required to apply the RA model for other types of engineered bamboo, based on the species and manufacturing technique and other relevant parameters. Whilst the quadratic and cubic models can be fitted for any test results, the empirical nature of the proposed models is not very scientific as the constants do not have any real physical meaning. The RO model has been widely used in modelling various types of materials as it relies on three easily obtainable parameters. The type of nonlinearity observed in engineered clearly shows that the RO equation will be a good option for a suitable constitutive model that can have a general applicability for engineered bamboo products. The advantage of the RO equation over the RA model is the usage of a proof strain (p, discussed in the next section) that has the versatility of being changed according to the unique elastic-plastic transitions in biomaterials, such as bamboo. The corresponding proof stress use in the RO formula has been successfully used in designing structural design rules for stainless steel and aluminium, and hence, the same design philosophy could be used for engineered bamboo.

## 6. Ramberg-Osgood Model for Engineered Bamboo

#### 6.1. Ramberg-Osgood Material Model [37]

_{0}), the proof stress (σ

_{p}), where p is typically taken as 0.2% (strain) and a dimensionless exponent n, which determines the sharpness of the knee of the stress-strain curve. The proof stress (σ

_{p}) is obtained by offsetting a straight line at the corresponding proof strain (p).

_{0.01}is the 0.01% proof stress. Figure 6 shows the key parameters in a typical RO simulation for stainless steel alloy.

_{0.2}, σ

_{0.5}, σ

_{1.0}and n, used in these models were determined from the reported stress-strain response and all of the values are listed in Table 5. It is obvious from Figure 7 that beyond the 0.2% proof stress, the RO curves become inaccurate with the increasing strain in the elastic-plastic region. Use of the 0.5% and 1% proof stresses clearly showed a better performance than the 0.2% proof stress in predicting bamboo scrimber’s nonlinear behaviour. Use of the 1.0% proof stress showed an accurate prediction for the nonlinear response, up to the adopted proof stress i.e., σ

_{1.0}. Similar discrepancies i.e., the over or under prediction of stress values, were also significant for both the 0.2% and 0.5% proof stress. This observation clearly shows that the complete stress-strain behaviour may not be accurately predicted by using a single equation. The full-range RO modelling technique has been adopted by several researchers to replicate such behaviour for various nonlinear metallic materials, including high strength steel and stainless steel. It is worth noting that the RO model with a 0.5% proof stress was used by Zhou et al. [55] to compare against the compressive stress-strain behaviour of large-scale bamboo scrimber under local compression. Zhu et al. [56] also proposed the RO model with a 0.3% proof stress for compression parallel to the grain of bamboo scrimber and a 0.1% proof stress for compression transverse to the grain. However, these studies were limited to their own specific test results and did not look at devising any general design guidance for the accurate modelling of engineered bamboo, which is essential for developing reliable numerical models as well as the design rules for engineered bamboo products.

#### 6.2. Full-Range Ramberg-Osgood Model

_{1.0}, and the second part is from σ

_{1.0}up to the ultimate stress σ

_{u}. Equation (18) shows the proposed full-range RO model for engineered bamboo.

_{u}is the peak stress, m is an exponent dependent on the peak stress in relation to the proof stress, e is the non-dimensional proof stress and ε

_{1.0}is the corresponding strain at a 1.0% proof stress.

#### 6.3. Sensitivity Analysis

## 7. Proposed Model for Laminated Bamboo Lumber

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Bansal, A.K.; Prasad, T.R.N. Manufacturing laminates from sympodial bamboos—An Indian experience. J. Bamboo Ratt.
**2004**, 3, 13–22. [Google Scholar] [CrossRef] - Liese, W. Research on bamboo. Wood Sci. Technol.
**1987**, 21, 189–209. [Google Scholar] [CrossRef] - Li, Z.; Liu, C.-P.; Yu, T. Laminate of Reformed Bamboo and Extruded Fiber-Reinforced Cementitious Plate. J. Mater. Civ. Eng.
**2002**, 14, 359–365. [Google Scholar] [CrossRef] - Wong, K.M. The Bamboos of Peninsular Malaysia; Forest Research Institute Malaysia: Kuala Lumpur, Malaysia, 1995. [Google Scholar]
- Awalluddin, D.; Ariffin, M.A.M.; Osman, M.H.; Hussin, M.W.; Ismail, M.A.; Lee, H.-S.; Lim, N.H.A.S. Mechanical properties of different bamboo species. MATEC Web Conf.
**2017**, 138, 01024. [Google Scholar] [CrossRef] - Naik, N.K. Mechanical and Physico-Chemical Properties of Bamboos; Indian Institute of Technology: Bombay, India, 2009. [Google Scholar]
- Molari, L.; Mentrasti, L.; Fabiani, M. Mechanical characterization of five species of Italian bamboo. Structures
**2020**, 24, 59–72. [Google Scholar] [CrossRef] - Abdullah, A.; Karlina, N.; Rahmatiya, W.; Mudaim, S.; Fajrin, A. (Eds.) Physical and Mechanical Properties of Five Indonesian Bamboos; IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2017. [Google Scholar]
- Archila-Santos, H.F.; Ansell, M.P.; Walker, P. Low Carbon Construction Using Guadua Bamboo in Colombia. Key Eng. Mater.
**2012**, 517, 127–134. [Google Scholar] [CrossRef] - Sharma, B.; Gatoo, A.; Bock, M.; Mulligan, H.; Ramage, M. Engineered bamboo: State of the art. Proc. Inst. Civ. Eng.-Constr. Mater.
**2015**, 168, 57–67. [Google Scholar] [CrossRef] - Andy, W.C.L.; Xuesong, B.; Audimar, P.B. Selected Properties of Laboratory-Made Laminated-Bamboo Lumber. Holzforschung
**1998**, 52, 207–210. [Google Scholar] - Nugroho, N.; Ando, N. Development of structural composite products made from bamboo I: Fundamental properties of bamboo zephyr board. J. Wood Sci.
**2000**, 46, 68–74. [Google Scholar] [CrossRef] - Sulastiningsih, I.; Nurwati. Physical and mechanical properties of laminated bamboo board. J. Trop. For. Sci.
**2009**, 21, 246–251. [Google Scholar] - Mahdavi, M.; Clouston, P.; Arwade, S. A low-technology approach toward fabrication of Laminated Bamboo Lumber. Constr. Build. Mater.
**2012**, 29, 257–262. [Google Scholar] [CrossRef] - Chen, F.; Jiang, Z.; Deng, J.; Wang, G.; Zhang, D.; Zhao, Q.; Cai, L.; Shi, S.Q. Evaluation of the Uniformity of Density and Mechanical Properties of Bamboo-Bundle Laminated Veneer Lumber (BLVL). BioResources
**2014**, 9, 554–565. [Google Scholar] [CrossRef] [Green Version] - Lugt, P. Design interventions for stimulating bamboo commercialization—Dutch design meets bamboo as a replicable model. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlans, 2008. [Google Scholar]
- Huang, Y.; Ji, Y.; Yu, W. Development of bamboo scrimber: A literature review. J. Wood Sci.
**2019**, 65, 25. [Google Scholar] [CrossRef] - Li, H.-T.; Zhang, Q.-S.; Huang, D.-S.; Deeks, A.J. Compressive performance of laminated bamboo. Compos. Part B Eng.
**2013**, 54, 319–328. [Google Scholar] [CrossRef] - Li, H.; Qiu, Z.; Wu, G.; Wei, D.; Lorenzo, R.; Yuan, C.; Zhang, H.; Liu, R. Compression Behaviors of Parallel Bamboo Strand Lumber Under Static Loading. J. Renew. Mater.
**2019**, 7, 583–600. [Google Scholar] [CrossRef] - Zhao, P.; Zhang, X. Size effect of section on ultimate compressive strength parallel to grain of structural bamboo scrimber. Constr. Build. Mater.
**2019**, 200, 586–590. [Google Scholar] [CrossRef] - Sharma, B.; Gatóo, A.; Ramage, M.H. Effect of processing methods on the mechanical properties of engineered bamboo. Constr. Build. Mater.
**2015**, 83, 95–101. [Google Scholar] [CrossRef] - Penellum, M.; Sharma, B.; Shah, D.U.; Foster, R.M.; Ramage, M.H. Relationship of structure and stiffness in laminated bamboo composites. Constr. Build. Mater.
**2018**, 165, 241–246. [Google Scholar] [CrossRef] - Qiu, Z.; Wang, J.; Fan, H.; Li, T. Anisotropic mechanical properties and composite model of parallel bamboo strand lumbers. Mater. Today Commun.
**2020**, 24, 101250. [Google Scholar] [CrossRef] - Norris, C.; McKinnon, P. Compression, Tension, and Shear Tests on Yellow-Poplar Plywood Panels of Sizes that Do Not Buckle with Tests Made at Various Angles to the Face Grain; USDA. 1956. Available online: https://ir.library.oregonstate.edu/concern/defaults/4q77fw59k (accessed on 23 August 2022).
- Tsai, S.W.; Hahn, H.T. Introduction to Composite Materials; Routledge: London, UK, 2018. [Google Scholar]
- Li, X.; Ashraf, M.; Li, H.; Zheng, X.; Wang, H.; Al-Deen, S.; Hazell, P.J. An experimental investigation on Parallel Bamboo Strand Lumber specimens under quasi static and impact loading. Constr. Build. Mater.
**2019**, 228, 116724. [Google Scholar] [CrossRef] - Li, X.; Ashraf, M.; Li, H.; Zheng, X.; Al-Deen, S.; Wang, H.; Hazell, P.J. Experimental study on the deformation and failure mechanism of parallel bamboo Strand Lumber under drop-weight penetration impact. Constr. Build. Mater.
**2020**, 242, 118135. [Google Scholar] [CrossRef] - Li, H.; Su, J.; Xiong, Z.; Ashraf, M.; Corbi, I.; Corbi, O. Evaluation on the ultimate bearing capacity for laminated bamboo lumber columns under eccentric compression. Structures
**2020**, 28, 1572–1579. [Google Scholar] [CrossRef] - Li, H.-T.; Chen, G.; Zhang, Q.; Ashraf, M.; Xu, B.; Li, Y. Mechanical properties of laminated bamboo lumber column under radial eccentric compression. Constr. Build. Mater.
**2016**, 121, 644–652. [Google Scholar] [CrossRef] - Li, H.-T.; Wu, G.; Zhang, Q.-S.; Su, J.-W. Mechanical evaluation for laminated bamboo lumber along two eccentric compression di-rections. J. Wood Sci.
**2016**, 62, 503–517. [Google Scholar] [CrossRef] - Su, J.-W.; Deeks, A.; Zhang, Q.-S.; Wei, D.D.; Yuan, C.G. Eccentric Compression Performance of Parallel Bamboo Strand Lumber Columns. Bioresources
**2015**, 10, 7065–7080. [Google Scholar] - Tan, C.; Li, H.; Wei, D.; Lorenzo, R.; Yuan, C. Mechanical performance of parallel bamboo strand lumber columns under axial compression: Experimental and numerical investigation. Constr. Build. Mater.
**2020**, 231, 117168. [Google Scholar] [CrossRef] - Tan, C.; Li, H.; Ashraf, M.; Corbi, I.; Corbi, O.; Lorenzo, R. Evaluation of axial capacity of engineered bamboo columns. J. Build. Eng.
**2021**, 34, 102039. [Google Scholar] [CrossRef] - Li, H.-T.; Su, J.-W.; Zhang, Q.-S.; Deeks, A.J.; Hui, D. Mechanical performance of laminated bamboo column under axial compression. Compos. Part B Eng.
**2015**, 79, 374–382. [Google Scholar] [CrossRef] - IT. ISO 22156; 2021 Bamboo Structures—Bamboo Culms—Structural Design. BSI Standards Limited: London, UK, 2021.
- Harries, K.A.; Sharma, B.; Richard, M. Structural Use of Full Culm Bamboo: The Path to Standardization. Int. J. Arch. Eng. Constr.
**2012**, 1, 66–75. [Google Scholar] [CrossRef] - Ramberg, W.; Osgood, W.R. Description of Stress-Strain Curves by Three Parameters; NASA Scientific and Technical Information: Washington, DC, USA, 1943. [Google Scholar]
- Ochi, S. Mechanical Properties of Uni-Directional Long Bamboo Fiber/Bamboo Powder Composite Materials. Mater. Sci. Appl.
**2014**, 05, 1011–1019. [Google Scholar] [CrossRef] - Xie, J.; Qi, J.; Hu, T.; De Hoop, C.F.; Hse, C.Y.; Shupe, T.F. Effect of fabricated density and bamboo species on physical–mechanical properties of bamboo fiber bundle reinforced composites. J. Mater. Sci.
**2016**, 51, 7480–7490. [Google Scholar] [CrossRef] - Zhang, X.; Li, J.; Yu, Z.; Yu, Y.; Wang, H. Compressive failure mechanism and buckling analysis of the graded hierarchical bamboo structure. J. Mater. Sci.
**2017**, 52, 6999–7007. [Google Scholar] [CrossRef] - Shao, Z.-P.; Fang, C.-H.; Huang, S.-X.; Tian, G.-L. Tensile properties of Moso bamboo (Phyllostachys pubescens) and its components with respect to its fiber-reinforced composite structure. Wood Sci. Technol.
**2010**, 44, 655–666. [Google Scholar] [CrossRef] - Li, X. Physical, Chemical, and Mechanical Properties of Bamboo and Its Utilization Potential for Fiberboard Manufacturing. Master’s Thesis, Louisiana State University, Baton Rouge, LA, USA, 2004. [Google Scholar] [CrossRef]
- Sanchez, L. Bamboo as a Sustainable Engineering Material: Mechanical Properties, Safety Factors, and Experimental Testing; University of South Florida: Tampa, FL, USA, 2019. [Google Scholar]
- Chung, K.; Yu, W. Mechanical properties of structural bamboo for bamboo scaffoldings. Eng. Struct.
**2002**, 24, 429–442. [Google Scholar] [CrossRef] - Dongsheng, H.; Aiping, Z.; Yuling, B. Experimental and analytical study on the nonlinear bending of parallel strand bamboo beams. Constr. Build. Mater.
**2013**, 44, 585–592. [Google Scholar] [CrossRef] - Wei, Y.; Zhou, M.; Zhao, K.; Zhao, K.; Li, G. Stress–strain relationship model of glulam bamboo under axial loading. Adv. Compos. Lett.
**2020**, 29, 2633366X20958726. [Google Scholar] [CrossRef] - Sheng, B.; Bian, Y.; Liu, Y.; Chui, Y.-H. Experimental Study of the Uniaxial Stress-strain Relationships of Parallel Strand Bamboo in the Longitudinal Direction. BioResources
**2019**, 14, 13. [Google Scholar] - Li, H.; Zhang, H.; Qiu, Z.; Su, J.; Wei, D.; Lorenzo, R.; Yuan, C.; Liu, H.; Zhou, C. Mechanical Properties and Stress Strain Relationship Models for Bamboo Scrimber. J. Renew. Mater.
**2020**, 8, 13–27. [Google Scholar] [CrossRef] - Chen, G.; Yu, Y.; Li, X.; He, B. Mechanical behavior of laminated bamboo lumber for structural application: An experimental investigation. Eur. J. Wood Wood Prod.
**2020**, 78, 53–63. [Google Scholar] [CrossRef] - ASTM International. Standard Test Methods of Static Tests of Lumber in Structural Sizes; ASTM DASTM International: West Conshohocken, PA, USA, 2009. [Google Scholar]
- Gardner, L.; Ashraf, M. Structural design for non-linear metallic materials. Eng. Struct.
**2006**, 28, 926–934. [Google Scholar] [CrossRef] - Richard, R.M.; Abbott, B.J. Versatile Elastic-Plastic Stress-Strain Formula. J. Eng. Mech. Div.
**1975**, 101, 511–515. [Google Scholar] [CrossRef] - Rohatgi, A. WebPlotDigitizer; Austin, Texas, USA. 2017. Available online: https://automeris.io/WebPlotDigitizer/ (accessed on 23 August 2022).
- Ashraf, M.; Gardner, L.; Nethercot, D.A. Structural Stainless Steel Design: Resistance Based on Deformation Capacity. J. Struct. Eng.
**2008**, 134, 402–411. [Google Scholar] [CrossRef] - Zhou, K.; Li, H.; Hong, C.; Ashraf, M.; Sayed, U.; Lorenzo, R.; Corbi, I.; Corbi, O.; Yang, D.; Zuo, Y. Mechanical properties of large-scale parallel bamboo strand lumber under local compression. Constr. Build. Mater.
**2020**, 271, 121572. [Google Scholar] [CrossRef] - Zhu, W.; Qiu, Z.; Zhou, J.; Jin, F.; Fan, H. Size design and nonlinear stress-strain relationship of parallel bamboo strand lumbers in tension and compression. Eng. Fail. Anal.
**2022**, 140, 106587. [Google Scholar] [CrossRef] - Mirambell, E.; Real, E. On the calculation of deflections in structural stainless steel beams: An experimental and numerical investigation. J. Constr. Steel Res.
**2000**, 54, 109–133. [Google Scholar] [CrossRef] - Rasmussen, K.J. Full-range stress–strain curves for stainless steel alloys. J. Constr. Steel Res.
**2003**, 59, 47–61. [Google Scholar] [CrossRef]

**Figure 6.**A schematic showing the typical RO stress-strain approximation for stainless steel alloys.

Ref. | Species | Age (Years) | Growth-Height | Laminate Method | Thermal Treatment | Resin | Resin Content (%) | Density (kg/m ^{3}) | Final MC (%) |
---|---|---|---|---|---|---|---|---|---|

Li et al. [19] | Phyllostachys pubescens | 3–4 | Upper | Hot-pressed | SST | PF | - | 1250 | - |

Dongsheng et al. [45] | - | 5 | Upper | Hot-pressed | - | - | - | ||

Wei et al. [46] | - | - | - | Cold-pressed /heat curing | - | PF | - | ||

Sheng et al. [47] | Phyllostachys pubescens | 5 | - | - | - | - | - | ||

Li et al. [48] | Phyllostachys pubescens | 3–4 | - | Hot-pressed | SST | PF | - | 1254 | 8.2 |

Ref. | Species | Age (Years) | Growth-Height | Strip Dimensions (mm) | Resin | Density (kg/m ^{3}) | Final MC (%) | |
---|---|---|---|---|---|---|---|---|

Width | Thickness | |||||||

Li et al. [18] | Phyllostachys pubescens | 3–4 | Upper | 17 | 4 | PF | 647 | 8.3 |

Li et al. [34] | Phyllostachys pubescens | 3–4 | Lower | 21 | 8 | PF | 635 | 7.6 |

Chen et al. [49] | Phyllostachys pubescens | 4 | All three heights | - | - | PF | 780 | 10.6 |

Product Type | Experimental Study | E_{c}(MPa) | ε_{cy}(µε) | ε_{c0}(µε) | f_{cy}(MPa) | f_{c0}(MPa) | Selection Criteria for Representative Stress-Strain Response |
---|---|---|---|---|---|---|---|

Bamboo scrimber | Li et al. [19] | 14,275 | 4380 | 32,320 | 62.94 | 105.79 | A sample was chosen with the lowest slenderness ratio which is optimal for failure by compression. |

Dongsheng et al. [45] | 11,600 | 2690 | 30,010 | 33.04 | 60.20 | This curve pertains to a sample which lies well within the upper and lower curve of all samples examined. | |

Wei et al. [46] | 12,100 | 4074 | 61,352 | 49.71 | 89.52 | The average curve of the five samples was taken. | |

Sheng et al. [47] | 11,440 | 3060 | 28,620 | 35.46 | 58.81 | The average curve of the compressive samples was taken. | |

Li et al. [48] | 11,320 | 2900 | 33,210 | 33.29 | 87.08 | A stress-strain curve that was well within the upper and lower curves was chosen. | |

LBL | Li et al. [18] | 9200 | 4220 | 31,820 | 36.66 | 59.36 | A sample with the upper growth portion was selected. |

Li et al. [34] | 9930 | 3860 | 16,760 | 36.77 | 59.43 | A sample was chosen with the lowest slenderness ratio which is optimal for failure by compression. | |

Chen et al. [49] | 10,880 | 3140 | 32,500 | 32.24 | 54.48 | The source has not specified which test result was taken in the results comparison, however 20 samples were tested. |

_{c}—Compressive elastic modulus; ε

_{cy}—Compressive yield strain; ε

_{c0}—Compressive ultimate strain; f

_{cy}—Compressive yield stress; f

_{c0}—Compressive ultimate stress.

**Table 4.**WLS values obtained for the existing models against all reported experiments of bamboo scrimber.

Experimental Study | QM 1 | QM 2 | RA | LM | CM |
---|---|---|---|---|---|

Li et al. [19] | 0.0591 | 0.0603 | 0.0697 | 0.1040 | 0.1728 |

Dongsheng et al. [45] | 0.0750 | 0.0698 | 0.0855 | 0.1273 | 0.3211 |

Wei et al. [46] | 0.1093 | 0.1110 | 0.0626 | 0.1669 | 0.4034 |

Sheng et al. [47] | 0.0389 | 0.0414 | 0.0420 | 0.0976 | 0.2860 |

Li et al. [48] | 0.0678 | 0.0700 | 0.0302 | 0.1491 | 0.1397 |

σ_{0.01}(MPa) | σ_{0.2}(MPa) | σ_{0.5}(MPa) | σ_{1.0}(MPa) | e | n | m | E_{0}(GPa) | σ_{u}(MPa) | ε_{p} | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.2% | 0.5% | 1.0% | 0.2% | 0.5% | 1.0% | 0.2% | 0.5% | 1.0% | ||||||||

Li et al. [19] | 49 | 75 | 85 | 97 | 0.00468 | 0.00530 | 0.00605 | 7.04 | 7.10 | 6.74 | 3.48 | 3.81 | 4.21 | 14.28 | 105.79 | 0.03231 |

Dongsheng et al. [45] | 35 | 43 | 47.5 | 53 | 0.00317 | 0.00350 | 0.00390 | 14.55 | 12.81 | 11.10 | 3.50 | 3.76 | 4.08 | 11.60 | 60.20 | 0.03001 |

Wei et al. [46] | 49 | 63 | 71 | 81 | 0.00492 | 0.00555 | 0.00633 | 11.92 | 10.55 | 9.16 | 3.51 | 3.82 | 4.22 | 12.10 | 89.52 | 0.03082 |

Sheng et al. [47] | 37 | 45.5 | 49 | 54 | 0.00389 | 0.00420 | 0.00463 | 14.49 | 13.93 | 12.18 | 3.71 | 3.92 | 4.21 | 11.44 | 58.81 | 0.02862 |

Li et al. [48] | 33 | 55 | 68 | 78.5 | 0.00453 | 0.00560 | 0.00647 | 5.86 | 5.41 | 5.31 | 3.21 | 3.73 | 4.16 | 11.32 | 87.08 | 0.03321 |

‘n’ for Full-Range RO with 1.0% Proof Stress | Existing Models | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

5 | 7 | 9 | 11 | 13 | QM 1 | QM 2 | RA | LM | CM | |

Li et al. [19] | 0.0519 | 0.0385 | 0.0369 | 0.0394 | 0.0420 | 0.0591 | 0.0603 | 0.0697 | 0.1040 | 0.1728 |

Dongsheng et al. [45] | 0.0702 | 0.0531 | 0.0506 | 0.0546 | 0.0604 | 0.0750 | 0.0698 | 0.0855 | 0.1273 | 0.3211 |

Wei et al. [46] | 0.0556 | 0.0459 | 0.0462 | 0.0496 | 0.0529 | 0.1093 | 0.1110 | 0.0626 | 0.1669 | 0.4034 |

Sheng et al. [47] | 0.0429 | 0.0227 | 0.0175 | 0.0185 | 0.0227 | 0.0389 | 0.0414 | 0.0420 | 0.0976 | 0.2860 |

Li et al. [48] | 0.0248 | 0.0353 | 0.0450 | 0.0518 | 0.0570 | 0.0678 | 0.0700 | 0.0302 | 0.1491 | 0.1397 |

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Goonewardena, J.; Ashraf, M.; Reiner, J.; Kafle, B.; Subhani, M.
Constitutive Material Model for the Compressive Behaviour of Engineered Bamboo. *Buildings* **2022**, *12*, 1490.
https://doi.org/10.3390/buildings12091490

**AMA Style**

Goonewardena J, Ashraf M, Reiner J, Kafle B, Subhani M.
Constitutive Material Model for the Compressive Behaviour of Engineered Bamboo. *Buildings*. 2022; 12(9):1490.
https://doi.org/10.3390/buildings12091490

**Chicago/Turabian Style**

Goonewardena, Janeshka, Mahmud Ashraf, Johannes Reiner, Bidur Kafle, and Mahbube Subhani.
2022. "Constitutive Material Model for the Compressive Behaviour of Engineered Bamboo" *Buildings* 12, no. 9: 1490.
https://doi.org/10.3390/buildings12091490