Engineering Structures 25 (2003) 755–768www.elsevier.com/locate/engstruct

Column buckling of structural bamboo

W.K. Yu, K.F. Chung∗, S.L. ChanDepartment of Civil and Structural Engineering, the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

Received 7 June 2002; received in revised form 29 November 2002; accepted 4 December 2002

Abstract

Bamboo scaffolding is widely used in construction in the South East Asia, in particular, the Southern China and Hong Kong formany decades. However, bamboo scaffolds are generally erected by scaffolding practitioners based on their intuition and experienceswithout any structural design. In general, column buckling is considered to be one of the critical modes of failure in bambooscaffolds, often leading to their overall collapse.

This paper presents a research and development project for structural bamboo where the column buckling behaviour of twostructural bamboo species, namelyBambusa pervariabilis (or Kao Jue) andPhyllostachys pubescens (or Mao Jue) were investigated.A total of 72 column buckling tests with bamboo culms of typical dimensions and properties were executed to study the columnbuckling behaviour of structural bamboo. Furthermore, a limit state design method against column buckling of structural bamboobased on modified slenderness was established and carefully calibrated against test data. It is shown that for Kao Jue, the averagemodel factors of the proposed design method are 1.63 and 1.86 for natural and wet conditions, respectively. Similarly, the averagemodel factors of the proposed design method for Mao Jue are 1.48 and 1.67 for natural and wet conditions, respectively. Conse-quently, the proposed design method is shown to be adequate.

With the availability of design data on the dimensions and the mechanical properties of structural bamboo together with theproposed column buckling design rule, structural engineers are encouraged to take the advantage offered by bamboo to build lightand strong bamboo structures to achieve enhanced economy and buildability. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Bamboo scaffolds; Green construction; Structural bamboo; Bamboo columns; Column buckling tests

1. Introduction

Timber is regarded as a good natural constructionmaterial, and probably, one of the oldest knownmaterials used in construction. In a modern structuraltimber code [1], ultimate limit state design philosophyis adopted and structural adequacy is assessed withcharacteristic values of both loading and resistance usingappropriate partial safety factors. Bamboo is anothernatural construction material and there are over 1500 dif-ferent botanical species of bamboo across the globe. Ingeneral, it is considered that the mechanical properties ofbamboo are likely to be at least similar, if not superior, tothose of structural timber. In many countries, bambooculms of various species have been used traditionally

∗ Corresponding author. Tel.: 852-2766-6063; fax: 852-2334-6389.E-mail address: cekchung@polyu.edu.hk (K.F. Chung).

0141-0296/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0141-0296(02)00219-5

as structural members in both temporary and permanentstructures under light loads, such as low-rise houses,short span footbridges, roof structures of medium spanand assess scaffolds. Furthermore, as bamboo growsvery fast and usually takes 3–6 years to harvest,depending on the species and the plantation, there is agrowing global interest in developing bamboo as a sub-stitute for structural timber in construction. The effectiveuse of structural bamboo will mitigate the pressures onthe ever-shrinking natural forests in developing coun-tries, and thus, facilitate the conservation of the globalenvironment.

However, a major constraint to the development ofstructural bamboo as a modern construction material isthe lack of design standards on both mechanical proper-ties and structural adequacy. As natural non-homogen-ous organic materials, large variations of physicalproperties along the length of bamboo members areapparent: external and internal diameters, dry density

756 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Nomenclature

De, Di external and internal diameters, respectivelyEc,d, Eb,d design Young’s modulus against compression and bending, respectivelyfc,k, fc,d characteristic and design compressive strengths, respectivelyfb,k, fb,d characteristic and design bending strengths, respectivelyfcr elastic critical buckling strengthfcc,d, fcc,t design and measured compressive buckling strengths, respectively

l modified slenderness ratio defined as �fc,d

fcr

yc, yt design and measured strength reduction factors for compressive buckling defined asfcc, d

fc,d

andfcc, t

fc,d

,

respectively with gm equal to 1.0gm a partial safety factor for material strength

and moisture content. While structural engineers alsoexpect variations in the mechanical properties of struc-tural bamboo, they tend to accept that the mechanicalproperties of bamboo are likely to be more consistentthan those of concrete.

This paper presents a research and development pro-ject for structural bamboo where the column bucklingbehaviour of two structural bamboo species, namelyBambusa pervariabilis (or Kao Jue) and Phyllostachyspubescens (or Mao Jue) were investigated. A total of72 column buckling tests with bamboo culms of typicaldimensions and properties were executed to study thecolumn buckling behaviour of structural bamboo. A limitstate design method against column buckling of struc-tural bamboo based on modified slenderness was thenestablished for general design after careful calibrationagainst test data.

1.1. Bamboo scaffolds

Bamboo scaffolds have been widely used in construc-tion in the South East Asia, in particular, the SouthernChina and Hong Kong for many decades. Figs. 1 and 2illustrate typical applications of bamboo scaffolds inHong Kong [2] as single layered bamboo scaffolds(SLBS) and double layered bamboo scaffolds (DLBS),respectively. In spite of open competition with manymetal scaffolding systems imported from countries allover the world, bamboo scaffold remains one of the mostpreferred access scaffolding systems in building con-struction in Hong Kong and the neighbouring areas. BothKao Jue and Mao Jue are commonly used in access scaf-folds in Hong Kong, and typical dimensions of both KaoJue and Mao Jue are presented in Fig. 3. At present,the typical height of bamboo scaffolds is 15 m and theinstallation of steel bracket supports at regular intervalsallow full coverage of building height.

It should be noted that bamboo scaffolds are generally

Fig. 1. Single Layered Bamboo Scaffolds (SLBS).

erected by scaffolding practitioners based on theirintuition and experiences without any structural design.In general, column buckling is considered to be one ofthe critical modes of failure in bamboo scaffolds, oftenleading to their overall collapse. In order to ensure struc-tural adequacy of bamboo scaffolds and other slenderbamboo structures in building construction, it is highlydesirable to provide rational design rules against columnbuckling of structural bamboo. With a suitable choice ofpartial safety factors, structural engineers are thus able todesign bamboo structures at a known level of confidenceagainst column buckling.

757W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Fig. 2. Double Layered Bamboo Scaffolds (DLBS).

Fig. 3. Typical dimensions of Kao Jue and Mao Jue.

1.2. Recent research in structural bamboo

Structural bamboo have been used traditionally inChina, Philippines, India and Latin America for manyhundreds of years, but little research was reported in thepast. Recent scientific investigations on bamboo as con-struction materials were reported by Au et al. [3] inHong Kong and also by Janssen [4] in Holland. A largeamount of data of the mechanical properties for variousbamboo species all over the world was also reported byJanssen [5]. While these sets of data provide typicalvalues of compressive, bending and shear strengths ofvarious bamboo species, no characteristic strengths formodern structural design were provided.

A series of experimental studies on structural bamboowere reported by Arce-Villalobos [6] and practical con-nection details for bamboo trusses and frames were alsoproposed and tested. Moreover, a recent study on thetraditional design and construction of bamboo in low-rise housing in Latin America was conducted andreported by Gutierrez [7]. It is interesting to note thatbamboo was classified by Amada et al. [8] as a smartnatural composite material with optimized distributionof fibers and matrices, both across cross sections andalong member lengths, in resisting environmental loadsin nature.

In order to promote the effective use of structuralbamboo in building construction, it is essential to pro-

758 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

vide basic design data of mechanical properties anddesign rules against various modes of failure in accord-ance with modern design philosophy. A pilot study wascarried out by Chung and Yu [9] to examine the vari-ation of compressive strength against various physicalproperties along the length of bamboo culms for bothKao Jue and Mao Jue. Moreover, systematic test serieswith a large number of compression and bending testswere also executed [10,11] to establish characteristicvalues of both the strengths and the Young’s moduli ofeach bamboo species for limit state structural design. Asshown in Table 1, both Kao Jue and Mao Jue are goodconstructional materials with excellent mechanicalproperties against both compression and bending.

1.3. Buckling loads for non-prismatic tubular columns

In general, many researchers have studied the buck-ling behaviour of non-prismatic columns such as non-prismatic columns of wide flange I-sections, box sec-tions, and solid sections with different support con-ditions. Gere and Carter [12] presented both exact andapproximate solutions for the critical buckling loads ofnon-prismatic columns, but no solutions for non-pris-matic tubular columns were given. Furthermore, Fogeland Ketter [13] examined the elastic buckling loads of asimply supported column with tapered rectangular cross-section of constant thickness under combined com-pression and bending. The study was then extended tosimilar columns with different support conditions.

For non-prismatic columns with common solid andtubular cross sections with different support conditions,Williams and Aston [14] presented approximate designcurves to assess the lower bounds of elastic bucklingloads. However, no closed-formed analytical solutionswere provided. More recently, Arce-Villalobos evaluatedthe critical buckling loads of bamboo columns through

Table 1Proposed mechanical properties for structural bamboo

Compression Bending

Bamboo species Dry Wet Dry Wet

Bambusa pervariabilis (Kao Jue)Characteristic strength (N/mm2) (at fifth percentile) fc,k 79 35 80 37Design strength (N/mm2) (gm = 1.5) fc,d 53 23 fb,d 53 25Design Young’s modulus (kN/mm2) (average value) Ec,d 10.3 6.8 Eb,d 22.0 16.4

Phyllostachys pubescens (Mao Jue)Characteristic strength (N/mm2) (at fifth percentile) fc,k 117 44 fb,k 51 55Design strength (N/mm2) (gm = 1.5) fc,d 78 29 fb,d 34 37Design Young’s modulus (kN/mm2) (average value) Ec,d 9.4 6.4 Eb,d 13.2 9.6

Dry condition: m.c. �5% for both Kao Jue and Mao Jue. Wet condition: m.c. �20% for Kao Jue and m.c. �30% for Mao Jue. Linear interpolationis permitted for mechanical properties with moisture contents between dry and wet conditions. The shear strengths of both Kao Jue and Mao Jueare conservatively estimated as 0.25 fc,d, but not less than 6 N/mm2 and greater than 15 N/mm2.

classical energy method [6] where bamboo columnswere considered as non-prismatic tubular members withvarying second moment of area and Young’s modulusalong the member length. However, this design pro-cedure was considered to be very lengthy and tedious insome causes, and simple design rule was preferred forpractical design.

2. Objectives and scope of work

In order to promote the effective use of structuralbamboo in building construction, a research and devel-opment program was undertaken by the second and thethird authors from 1999–2001. The program aims to gen-erate scientific design rules and data for the re-engineer-ing of bamboo scaffolds into modern green structuresof high buildability through scientific investigation andtechnology transfer. It is essential to provide not onlydesign data of both physical and mechanical propertiesbut also design rules against various modes of failure inaccordance with modern design philosophy. After estab-lishing the characteristic values of both the strengths andthe Young’s moduli of Kao Jue and Mao Jue [9], it isnecessary to develop design rules against column buck-ling of bamboo culms.

A research and development project was carried outby the authors from 2000–2001 and the column bucklingbehaviour of structural bamboo was examined. A totalof 72 column buckling tests for both Kao Jue and MaoJue over a wide range of practical member lengths wereexecuted to examine their column buckling behaviour.In accordance with existing structural design philosophyon column buckling for both steel and timber structures,a design method based on modified slenderness was thenproposed for general design of both Kao Jue and MaoJue after careful calibration against test data. It should

759W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

be noted that any significant variation on the physicaland the mechanical properties along the member lengthof bamboo columns should be incorporated in assessingtheir axial buckling resistances. Other aspects of thedesign and construction of bamboo scaffolds such asmember configurations, support arrangement, and con-nection design will be reported separately.

3. Experimental investigation

In order to examine the buckling behaviour of bambooculms and to provide test data for the formulation ofdesign rules against column buckling, a set of systematictest series, which is also referred as a qualification testprogramme, for each bamboo species was executed. Ineach test programme, a large number of column bucklingtests were carried out over a practical range of height-to-diameter ratios, diameter variations over memberlength, and also moisture content. In general, the testspecimens were selected and prepared as follows:

� All bamboo culms were about 6 m in length and of3 to 6 years of age. They were air-dried for at least3 months before testing.

� A length of 750 mm from both the top and the bottomends of the bamboo culms was discarded.

� Three specimens were cut out from the top, themiddle and the bottom positions of the culm andmarked with the letters A, B, and C respectively.

� All the specimens were fairly straight with acceptableout-of-straightness under visual inspection. The exter-nal diameters at the top and the bottom ends of eachspecimen did not differ by more than 25 mm.

� Three member lengths were selected, and they were400, 600 and 800 mm for Kao Jue and 1000, 1500and 2000 mm for Mao Jue; the member lengths weredenoted as a, b and c respectively.

Among all the physical properties, moisture contentwas found to be the most important one in governingthe mechanical properties, and hence, the bucklingbehaviour of bamboo culms. Consequently, it is neces-sary to perform the systematic tests with test specimensunder the following moisture conditions:

� The natural condition is denoted as N and it representsthe typical range of moisture contents found in prac-tice.

� The wet condition is denoted as W and it representsthe extreme moisture content approaching water satu-ration in bamboo fibers; this is achieved by immersingthe test specimens under water for 1 week before test-ing.

The designation system for the test specimens aredefined as follows:

�K

M� �A

B

C� �N

W� �a

b

c� �1

2�

where K and M denote Kao Jue and Mao Jue, respect-ively. The total number of tests is equal to 2×3×2×3×2or 72.

It should be noted that after each column bucklingtest, at least two short culms were cut out from thebuckled specimens, and compression tests on the shortculms were carried out to evaluate the compressivestrengths of the bamboo culms. The length of each com-pression test specimen was about twice the externaldiameter of the bamboo culm, but not larger than 75 and150 mm for Kao Jue and Mao Jue, respectively.

3.1. Test set-up

Fig. 4 illustrated the general set-up of the columnbuckling tests. In order to simplify data analysis, smoothball joints were installed to provide simply supportedconditions at the top and the bottom supports. Therefore,the effective length coefficient of all the test specimenswere taken as 1.0 in analysis. The applied load, P, theaxial shortening, w, and the horizontal displacements, uand v, were measured continuously during the test toprovide load-displacement curves for data analysis, andthe maximum applied load and the corresponding dis-placements at failure were obtained for each test.

3.2. Test results

Two failure modes, namely overall buckling and localbuckling, are identified among the tests, and they areshown in Fig. 5. It is found that most Mao Jue membersfail in overall buckling, especially for those long col-umns with high moisture contents. For wet and shortcolumns of Kao Jue, local buckling is critical.

All the measured data of the test specimens includinggeometrical dimensions, moisture contents and failureloads are presented in Tables 2 and 3 while typical load-displacement curves of the test specimens are plotted inFig. 6. It is shown that the load reduction due to columnbuckling of the test specimens are severe, and thus it isnecessary to derive a general design method to assessthe axial buckling resistances of bamboo columns.

4. Design of bamboo column against buckling

Based on modern structural design philosophy, adesign method is proposed for column buckling of bothKao Jue and Mao Jue in a limit state design format. Theproposed design method follows closely to the column

760 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Fig. 4. General set-up of column buckling tests.

buckling design methods of structural steel as given inthe British steel code BS5950 [15]. It should be notedthat the formulation of the proposed design methodadopts the Perry-Robertson interaction formula to evalu-ate the compressive buckling strength of steel columnsafter incorporating the effects of both geometrical andmaterial initial imperfections. In general, the values ofboth the Perry factor and the Robertson constant may bechosen in such a way to fit test data for column bucklingof steel columns with different cross-sections bucklingabout different axes.

Moreover, the same method with a slightly differentformulation is adopted in both the European Steel CodeEurocode 3 [16] and the European Timber Code Euroc-ode 5 [1]. Formulation of the non-dimensionalized col-umn buckling curves with modified slenderness ispresented in the Appendix together with a comparisonon the expressions and the values of the Perry factor, theRobertson constant and the limiting slenderness amongvarious design rules.

As natural non-homogenous organic materials, largevariations of both physical and mechanical propertiesalong the length of bamboo culms are apparent, and it

is important to decide which parameters should be incor-porated in assessing the axial buckling resistance of thebamboo columns. According to the results of the quali-fication test programs for Kao Jue and Mao Jue reportedby Chung and Yu [9], the variation of Young’s modulusalong member length is found to be significant, and,more importantly, also random in pattern. Thus, theaverage value or 50 percentile of the Young’s modulusshould be adopted for the entire member length of bothKao Jue and Mao Jue.

However, the variations of external diameter andthickness in Mao Jue are apparent, and it is essential toallow for the variation of the second moment of areaalong the member length in the column buckling analy-sis. This may be readily achieved by incorporating a non-prismatic parameter, a, to the elastic Euler buckling loadof bamboo columns. The non-prismatic parameter, a,may be evaluated through classical energy method. It isworthwhile to note that the International Standard ISO22156 Bamboo Structural Design [17] recommends atleast a 10% reduction to be applied to the secondmoment of area of bamboo culms whenever the cross-section variation along the member length is significant.

761W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Fig. 5. Typical failure modes of bamboo columns. (a) Overall buckling; (b) Local buckling.

5. Proposed design method

The proposed design method for bamboo columnsagainst buckling is presented as follows:

1. Basic section properties of a bamboo column areevaluated first:

Cross � sectional area: A1 � �p4(D2e�D2

i )� 1

Second moment of area: I1 � � p64(D4

e�D4i )� 1;I2 � � p64

(D4e�D4

i )� 2

Slenderness ratio: l 1 �LE

r 1where r 1 � � I 1

A 1

where subscripts 1 and 2 denote the upper (smaller)cross-section and the lower (larger) cross-section,respectively.

2. The elastic critical buckling strength of the bamboocolumn, fcr, is given by:

fcr � a·p2 Eb,d

l21

where the non-prismatic parameter, a, is the mini-mum root of the following cubic function, g(a) =c3a3 + c2a2 + c1a + c0 = 0 where

c3 � �0.2880

c2 � 2.016 (2 � r)

c1 � �(14.11 � 14.11r � 3.098 r2)

c0 � 10.37 � 15.55r � 7.047r2 � 0.932r3

r �I2 � I1

I1

If the value of r lies between 0 and 3, the value ofa may be evaluated approximately as follows:

a � 1.005 � 0.4751r�0.011r2

where a lies between 1.00 and 2.35.3. The design compressive strength of the bamboo col-

umn, fc,d, is given by:

fc,d �fc,k

gm

4. The design compressive buckling strength of the bam-boo column, fcc,d, is thus given by:

fcc,d �fcr fc,d

f � (f2�fcr fc,d) 1/2

where

f �fc,d � ( 1 � h ) fcr

2

762 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768T

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763W.K. Yu et al. / Engineering Structures 25 (2003) 755–768T

able

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34

f c,k

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ely.

764 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Fig. 6. Typical load displacement curves of column buckling tests. (a) Kao Jue; (b) Mao Jue.

Perry factor, h � 0.001 a (l 1�l 0)

Robertson constant, a � 15 for Mao Jue, or

a � 28 for Kao Jue.

Limiting slenderness, lo � 0.2 p�Eb,d

fc,d

A non-dimensionalized column buckling curve maybe plotted using the following two non-dimensional-ized quantities:

� Modified slenderness ratio, l = �fc,d

fcr

� Strength reduction factor, yc =fcc,d

fc,d

The compressive buckling strength of a bamboo col-umn is thus obtained as a factor of the compressivestrength. For illustration purpose, Figs. 7 and 8 plot theproposed column buckling curves for both Kao Jue andMao Jue, respectively.

5.1. Calibration of design method

In order to calibrate the proposed design method, aback analysis against the test data was carried out withall partial safety factors equal to unity. Moreover, themeasured dimensions of the test specimens were used,and the measured compressive strengths of test speci-mens under natural and wet conditions were adopted. Itshould be noted that the design values of Young’s mod-uli against bending of both Kao Jue and Mao Jue asgiven in Table 1 were used in the back analysis.

765W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

Fig. 7. Column buckling analysis for Kao Jue. (a) Back analysis; (b) Model factor.

Fig. 8. Column buckling analysis for Mao Jue.

The results of the back analysis for both Kao Jue andMao Jue are summarized in Tables 2 and 3 while thetest data is also plotted in Figs. 7 and 8 respectively fordirect comparison with the proposed column bucklingcurves. It should be noted that:

� For Kao Jue, it is found that due to the presence oflarge initial imperfection when compared with theexternal diameter, the Robertson constant is selectedto be 28 in order to give safe design for all test results.The measured modified slenderness ratios are found

to range from 0.44–1.11, while the measured strengthreduction ratios are found to range from 0.31–1.31.

� For Mao Jue, the Robertson constant is selected to be15 due to small initial imperfection when comparedwith the external diameter. The measured modifiedslenderness ratios are found to range from 0.66–2.22and the measured strength reduction ratios are foundto range from 0.23–0.93.

The model factors for the proposed design method ofcolumn buckling against the test data of both Kao Jue

766 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

and Mao Jue are also presented in Tables 2 and 3,respectively. The distributions of the model factors forboth Kao Jue and Mao Jue under different moisture con-ditions are plotted in Figs. 7 and 8, respectively. For KaoJue, the average model factors are found to be 1.63 and1.86 for natural and wet conditions, respectively. Simi-larly, the average model factors for Mao Jue are foundto be 1.48 and 1.67 for natural and wet conditions,respectively. Consequently, the proposed design methodis shown to be adequate.

It should be noted that in the present study, the valueof non-prismatic parameter, a, for Kao Jue is found torange from 1.00–1.28, and thus, the variation of externaldiameter and thickness in bamboo columns of Kao Jueis considered not to be significant in assessing their axialbuckling resistances. However, for Mao Jue, α is foundto range from 1.04–2.11, and thus, it is essential to incor-porate the variation of external diameter and thickness,and hence, second moment of area, in bamboo columnsof Mao Jue in assessing their axial buckling resistance.

6. Practical considerations

Attention should be drawn to the following for practi-cal design and construction of bamboo scaffolds andstructures.

6.1. Basic design data for Kao Jue and Mao Jue

It is important to have statistically corrected engineer-ing data for structural bamboo so that rational design ofbamboo structures is possible. For general application,the following data may be adopted:

� For Kao Jue, the external and the internal diametersare 40 and 30 mm, respectively, and they are constantalong the length of the bamboo culm; the wall thick-ness is 5 mm.

� For Mao Jue, the external and the internal diametersat the top cross-section are 60 and 48 mm, respect-ively, and they are considered to increase linearlydown to the bottom cross-section to 90 and 72 mm,respectively, over a length of 6 m. The wall thicknessincreases linearly from 6 mm at the top cross-sectionto 9 mm at the bottom cross-section.

The dimensions of Kao Jue and Mao Jue are illus-trated in Fig. 3 while the design data of the mechanicalproperties for both Kao Jue and Mao Jue is presented inTable 1. All bamboo culms should be air-dried for threemonths before use and free from visual defects.

6.2. Initial out-of-straightness

As non-homogenous organic materials, most bambooculms are found to have initial out-of straightness with

different magnitudes among individual members. It isimportant to limit the maximum initial out-of-straight-ness, �o,max, found in practice in order to ensure thatthe proposed design method against column buckling forstructural bamboo is valid. Table 4 presents the averagevalues of the initial out-of-straightness measured fromthe column buckling tests. As a rule of thumb, themaximum value of initial out-of-straightness, �o,max,should be limited as follows:For Kao Jue: �o,max � L/100 or 0.15 De, whichever is smaller.

For Mao Jue: �o,max � L/200 or 0.15 De, whichever is smaller.

6.3. Lateral restraints in bamboo scaffolds

In order to achieve overall structural stability of bam-boo scaffolds with high structural efficiency, lateralrestraints should be provided at close intervals. How-ever, in practise, it may not be practical or even imposs-ible to provide lateral restraints in all the main post-mainledger connections of the bamboo scaffolds. In theabsence of sufficient lateral restraints, the effectivelength of bamboo columns will be larger than their mem-ber lengths between ledgers, and the axial bucklingresistance of the bamboo columns may be reduced sig-nificantly. Thus, it is necessary to provide design guid-ance on practical arrangements of lateral restraints.

An advanced non-linear finite element analysis hasbeen carried out [2] to investigate the column bucklingbehaviour of bamboo scaffolds based on high perform-ance beam-column elements using the one-element-per-member formulation [18]. Both the local buckling ofbamboo posts between ledgers and the global instabilityof the entire scaffolds with regular and staggered lateralrestraints were studied carefully. Details of the investi-gation may be found in the literature [2].

Table 4Summary of initial out-of-straightness for structural bamboo

Bamboo Member Average out-of- Ratios on initial out-species length straightness, �0 (mm) of-straightness

�o / L �o / De

Kao Juea 400 3.1 L / 130 0.08b 600 5.9 L / 102 0.15c 800 3.0 L / 264 0.08Maximum value L / 102 0.15

Mao Juea 1000 1.7 L / 595 0.02b 1500 3.1 L / 492 0.04c 2000 8.8 L / 226 0.13Maximum value L / 226 0.13

De = 40 mm for Kao Jue, 70 mm for Mao Jue.

767W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

7. Conclusions

Based on extensive and systematic experimental test-ing on the column buckling behaviour of bamboo culms,a design method against column buckling of structuralbamboo based on modified slenderness is developed andcalibrated successfully against test data of both Kao Jueand Mao Jue. For design purpose, both Kao Jue and MaoJue may be considered to be homogenous in terms ofmechanical properties. Moreover, the cross sectiondimensions of Kao Jue may be considered to be uniformthroughout the length of bamboo culms against columnbuckling. However, the variations of the external diam-eter and the thickness, and hence, the second moment ofarea, in Mao Jue should be incorporated in assessing theaxial buckling resistances of bamboo culms.

The proposed design method is shown to be structur-ally adequate in accordance with modern structuraldesign philosophy, and it may be used effectively todesign against column buckling of structural bamboo inbamboo scaffolds and other bamboo structures. With theavailability of design data on the dimensions and themechanical properties of structural bamboo together withthe proposed column buckling design rule, structuralengineers are encouraged to take the advantage offeredby bamboo to build light and strong bamboo structuresto achieve enhanced economy and buildability.

Acknowledgements

The research and development project leading to thepublication of this paper is partially supported by theInternational Network for Bamboo and Rattan (ProjectNo. ZZ04), and also by the Research Committee of theHong Kong Polytechnic University Research (ProjectNo. G-V849). The authors would like to thank ProfessorJ.M. Ko of the Hong Kong Polytechnic University andProfessor J.J.A. Janssen of the Eindhoven University ofTechnology, co-chairmen of the Steering Committee ofthe INBAR program, for their general guidance andtechnical advice. Moreover, the authors would like toexpress their gratitude to Mr C.M. Ling and Ms K.L.Chan, and also to the technicians of the Heavy StructureLaboratory for the execution of the tests. The test speci-mens were supplied by Wui Loong Scaffolding WorksCo. Ltd.

Appendix. Formulation of non-dimensionalizedcolumn buckling curve

From classical energy method, the critical bucklingload of a column is given by:

Pcr �p2EIL2

Dividing by the cross-sectional area, A, the elasticcritical buckling strength, fcr, is given by:

fcr � p2EIA

1L2

fcr � p2E1

(L / r)2 whereIA

� r2 or fcr � p2E1l2 where l �

Lr

Adopt the following Perry–Robertson interaction for-mula:

(fcr�fcc,d)(fc,d�fcc,d) � h fcrfcc,d

The compressive buckling strength, fcc,d, of the col-umn is given by:

fcc,d �fcrfc,d

f � �f2�fcrfc,d

where f �fc,d � (1 � h)fcr

2

h � Perry factor which is related to initial imperfection of column member � 0.001 a(l�l0)

fc,d � design compressive strength

This is the same expression given in Appendix C ofBS5950: Part 1. For both Eurocodes 3 and 5, a non-dimensionalized column buckling curve is adoptedthrough the use of the following two non-dimensional-ized quantities:

� Modified slenderness ratio, l is defined by:

l � �fc,d

fcr

or � l /l1

where l is the slenderness of the column and

l1 � p� Efc,d

� Reduction factor for axial buckling, c is defined by:

c �fcc,d

fc,d

�fcr

f � (f2�fcr fc,d)1/2 �1

ffcr

� �ffcr2

�fc,d

fcr

Re-writing the formula using different parameters,

c �1

f � �f2�l2

where f = 0.5[ 1 + a (l � 0.2) + l2 ] ; a, is animperfection factor whose value depends on thematerial of the columns, and also the initial out-of-straightness of the column member allowed for. Thevalue is given explicitly in Table 5.5.1 of EC3, or

0.001 a p� Efc,d

.

The expression is similar to the design rule given inClause 5.5.1.2 of ENV 1993-1-1:1992 of Eurocode 3. Inclause 5.2.1 of ENV 1995-1-1:1993 of Eurocode 5, the

768 W.K. Yu et al. / Engineering Structures 25 (2003) 755–768

same formulation is adopted but presented with differentsymbols as follows:

kc �1

k � �k2�l2rel

where kc, reduction factor due to column buckling;

k is 0.5 [ 1 � bc (lrel � 0.5) � l2rel];

lrel is � fc,o,k

sc,crit

� �fc,d

fcr

� l

and bc = a = 0.001 a p� Efc,d

Summary of design rulesApproach Perry factor Robertson Limitingof constant slendernessformulation (ratio)

British Steel h = 0.001 a (l�l0)l0 = 0.2 p� E

fc,dCode BS5950 a = 2.0 for curve a;

= 3.5 for curve b;

= 5.5 for curve c;

= 8.0 for curve d

Buckling strength

European Steel h= a ( l�0.2 ) l0 = 0.2

Code EC3 a = 0.21 for curve a;

= 0.34 for curve b;

= 0.49 for curve c;

= 0.76 for curve d

Modified

slenderness

European Timber h= bc(l�0.5) l0 = 0.5

Code EC5 ßc = 0.1 for glued laminated timber;

= 0.2 for solid timber

Modified

slenderness

References

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[11] Yu WK, Chung KF. Qualification tests on Mao Jue under com-pression and bending. Technical Report, Research Centre forAdvanced Technology in Structural Engineering, Hong KongPolytechnic University, 2000.

[12] Gere JM, Carter WO. Critical buckling loads for tapered columns.Journal of the Structural Division 1963;128(II):736–54.

[13] Fogel CM, Ketter RL. Elastic strength of tapered columns. Jour-nal of Structural Division 1962;88(5):67–106.

[14] Willams FW, Aston G. Exact or lower bound tapered columnbuckling loads. Journal of Structural Engineering1989;115(5):1088–100.

[15] British Standards Institution. BS5950: Structural use of steelworkin building. Part 1: Code of practice for design — Rolled andwelded sections, 2000.

[16] DD ENV 1993-1-1 Eurocode 3: Design of steel structures, Part1.1. General rules and rules for buildings, 1993.

[17] ISO/DIS-22156: Bamboo Structural Design, International Stan-dard, 2001.

[18] Chan SL, Zhou ZH. Second-order analysis of frames using a sin-gle imperfect element per member. Journal of Structural Engin-eering, ASCE 1994;120(6):703–17.