Limit State Design of Steel Pin Connections for Bamboo Truss Structures

Luís Eustáquio Moreira1,a, Khosrow Ghavami2,b

1Federal University of Minas Gerais, Structures Department, Brazil

2Pontifical Catholic University of Rio de Janeiro, RJ, Brazil

aluis@dees.ufmg.br, bghavami@puc-rio.br

Keywords: bamboo, pin connections, mechanical tests, limit diagram

Abstract. One of the most important parts of any structure, independent of the material used, is

the joint of its elements. In the steel pin connections used in bamboo tubes to assemble truss

structures, stress concentration occurs at the contact point between steel pin and bamboo. The

stress concentration causes local crushing and cracking along the fibers. Bamboo is a

Functionally Graded Material (FGM), which consists of a composite anatomical structure

divided in three different microstructures: - sieve vessels surrounded by strong high resistance

longitudinal fibers stretching along de culm, both inner the aired parenchyma tissue with its

hollow cells. This aligned anatomy could contribute to initiate small cracks close to the contact

area between the steel pin and hole. This is due to the combined action of the shear along and

the tension perpendicular to the fibers. These cracks oriented in the direction of the fibers could

propagate or not, according to the stress fields in the neighborhood of the contact area and along

the possible crack path. In cases where two steel pins are used, the optimum space between

them and between the first pin and bamboo board should be established. If the axial forces

between connected bars do not converge exactly to the same point, the local eccentricities cause

moments and the superposition of these moments and axial forces acting simultaneously over

the two pins should be considered establishing the local stress field in the bamboo wall. Based

on mechanical tests of steel pin connections of PP bamboo tubes, where moments and axial

forces are simultaneously applied and establishing the local stresses field using Finite Element

Method (FEM), Limit State Diagrams are generated for the design of pin ended joints.

Introduction

It is a fact that bamboo in its natural form has pin connection problems due to the linear

distribution along the culm of the resistant fibers imersed in the porous parenchima tissue. This

particular anatomy contributes to facilitate the crack propagation if a crack has started and the

connection can fail under relatively low loadings. Nevertheless, this type of connections has been

used all over the world making beautiful and useful bamboo structures. During many years the

authors have investigated this type of connection, mainly under axial tension forces, Figure 1.

In this paper, the connections were also subjected to bending moments to represent the real

situation in many cases where the bars of a truss do not converge exactly to the same point.

Considering all previous investigations, these new tests made it possible to develop a theory to

verify the safety of these type of connections. A limit state diagram was idealized and a safety

region was also created. Therefore, the problem can be easily solved through this diagram if the

axial force and the bending moment were known, which are solicitated in the joint.

Materials and Methods

The theory suggested here was based on many tension tests of connections such as the one in

Figure 1 using one or two pins at the end and tests as shown in Figure 7, where there is a

combination of axial force and bending moments. The first case was analised by Moreira &

Ghavami [1] and it is used to calibrate the modeling of bamboo 3 and 4 of the specimen CP2,

Figure 7, to estimate the limit resistance of the bamboo when subjected only to tension forces.

a) b) c)

Figure 1 : a) failure mode I (squashing of fibers); b) mode II (shear of fibers); c) mode III

(splitting of fibers)

Moreira & Ghavami [1] have concluded that the failure mode III is in consequence of failure

mode I. If the squashing is not interrupted the pin pushes the fibers laterally and splits the bamboo.

Figure 1c) is an exemple where the pin caused a progressive squashing and consequent splitting of

fibers.

Table 1: spring constants

k1(

N/mm

k2(

N/mm

k3(

N/mm

k4N/mm

k5(

N/mm

k6(

N/mm

k7(

N/mm

k8(

N/mm

1950 1345 1270 1190 1075 845 500 100

In the present study, the same concepts of local displacements, previouslly investigated, was

used as shown in Figure 2a. But differently, instead of applied forces directly in the hole, as done

before, constant springs are positioned inside the hole in the contact zone, Figure 2b. A maximum

displacement δ = 0.5 mm was assumed and the springs constants were calculated proportionally to

the local displacements, given in Table 1.

. . a) b)

Figure 2: a) contact displacement pin-hole; b) positions and numbering of springs

Thus, the elastic local displacement below the pin 𝛿𝑖(𝜃) is given by Eq. 1.

𝛿𝑖(𝜃) = 𝑅𝑝√1 −𝑅𝜌

2

𝑅𝑝2 𝑠𝑒𝑛2𝜃 + 𝛿 + 𝑖 − 𝑅ℎ𝑐𝑜𝑠𝜃 (1)

Bamboo 4, Figure 7c has a diameter equal to d=12.7mm and mean wall thickness t = 8.2 mm.

It was modeled as a thick shell and the material was considered orthotropic, with modulus of

elasticity E1 = 10 GPa and E2=E3 = 1,2 GPa. Likewise the transversal modulus are

G12=G13=G23= 1,2 GPa, Figure 3. The concepts of symmetry were used to decrease computing

time. In this element an axial tension force equal to 20 kN is the limit to compression stress 𝜎11=

80 MPa, Figure 4 a, which starts the squashing of the fibers. Two interesting results had been

observed in the axial tension tests with one and two pin by ends. Independently of the occurrence of

failure mode I or II, the contact compression stress stays next the limit and the possibility of the

occurrence of one or another mode depends on the local thickness wall and on the moisture content

of the specimen. In the case of this lot of bamboos, the shear stresses which can cause the failure of

bamboo between the pins are in the order of the stresses shown in Figure 4c,d; Figures 5 and 6. Pin

connections have been tested with pins of 12.7; 15.9 and 19 mm in bamboos with diameters varying

from 80 to 110 mm and wall thickness from 7 to 13 mm. All presented the same mechanical

performance. In the specific case of bamboo 4, Figure 7 c) , the bolts were 12.7 mm and the test is

considered to be the reference test. Base and Top refer to invisible and visible surface respectively.

Figure 3: limit axial force F1 = - 20 kN; dp = 12,7 mm

a) b)

c) d)

e) f) Figure 4: a)S11 BASE; b)S11 TOP; c)S13 BASE; d)S13 TOP; e)S33 BASE; f)S33 TOP

Figure 5: Shear S13 BASE zoom and possible shear failure line

Figure 6: Shear S13 TOP zoom and possible shear failure line

The forces between pins are of neglegible difference; the pin of the left carries 52 % of the

applied load and equal forces can be considered to occur between pins in the structure.

But when axial forces were combined with bending moments, the things changed completely.

Figure 7 shows the specimen CP2, Moreira & Gaspar [2], one of the 3 different specimens tested in

compression. Two bamboos are pin connected to 4 bamboos through 4 steel pins with 12,7 mm in

diameter with an internal angle of 900 .

a) b)

c) d)

Figure 7 : a), b)Connection under combined actions; c)Cracks over bars 3 and 4 ; d) Crack on bar 6

The first cracks occur on bar 4 and bar 3, simultaneously with the load of 7 kN, Figure 8 a) , before

the ultimate load equal 8,8 kN. Figure 8 b) shows the relation between applied moment M versus

distortion 𝛾 .

a) b)

Figure 8 : a) Load P versus vertical displacement δ; b) Moments M vs angular displacement 𝛾

Figure 9 a) shows the scheme of the tests; Figure 9 b) the numeration of the bars and Figure 9

c) shows the geometric relations.

Bar 3 was thouroughly investigated beforehand because it was also under tension force. Bar 4,

subjected to bigger forces than bar 3, was also investigated and is presented here in this paper.

a) b) c)

Figure 9: a) Test squeme; b) Numeration of bars; c) Test structural system

The results of numerical modeling through FEM, Moreira e Gaspar [2 ], with the load P = 7 kN,

limit of proportionality, are shown in Figure 10 a) and b).

Figure 10 : a) bending moments on bars 3 and 4; b) axial forces on bars 3 and 4

0

2000

4000

6000

8000

10000

0 20 40 60 80

Lo

ad

s P

(N)

Vertical displacements (mm)

In this modeling, Figure 10, bamboos were considered as pipe bar elements. Then the element 4

was modeled as follows. The strings were positioned according to the resultant R of forces in the

hole, as seen in Figure 11 .

Figure 11: Position of springs in pin- hole contact according to local forces

Figure 12 shows the results for simultaneous actions of moments and compression forces. Figure 13

shows the results for the application of only moment and Figure 14 the results for tension axial

force and moments simultaneously.

a) b)

c) d)

e) f)

Figure 12: a)LOADING; b) S11 BASE; c) S13BASE; d) S13 TOP; e) S33 BASE; f) S33 TOP

Figure 13: S13 BASE – ZOOM

Figure 14: S33 BASE - ZOOM

Figure 15: S13 TOP – ZOOM

Figure 16: S33 TOP – ZOOM

a)

b) c)

d) e)

Figure 17 : a) FAILURE MOMENT b)S13 BASE; c)S13 TOP; d)S33 BASE; e)S33 TOP

Figure 18: S13 BASE ZOOM

Figure 19: S33 BASE ZOOM

Figure 20: S13 TOP ZOOM

Figure 21: S33 TOP ZOOM

a) b)

c) d)

e) f)

Figure 22 : a)LOADING; b) S11 BASE; c) S13 BASE; d) S13 TOP; e) S33 BASE; f)S33 TOP

Figure 23: S13 BASE ZOOM

Figure 24: S33 BASE ZOOM

Figure 25: S13 TOP ZOOM

Figure 26: S33 TOP ZOOM

The heterogeneity of bamboo anatomy shows clearly that the simultaneous application of

moments and axial forces give results completely different of the superposition of the isolated

effects. This is because bamboo is fibre oriented and the inclination of the contact zone between pin

and hole increases the local resistance when the bar is compressed and decreases it when tensioned.

The curves of stress distribution along the possible failure lines is relatively easy to make but in

this paper was done preference to images and color scale. The Saint Vennant Principle is visible all

over pictures.

The comparison of the stress fields in each case, including the analysis of the bar 3 under

simultaneous actions of tension and bending moments, not presented in this paper, permits consider

the Limit State Diagram Design to verify connection resistance, Figure 27. In this Figure, P1 and

P2 are the solicitations of bar 4 and 3 in the instant of the appearance of first cracks on the

mechanical test, under the load of 7 kN.

To use the proposed diagram for different species and geometries, it is necessary to determine

the new values of M´ and F´ to the selected lot of bamboos, according to Eq. 2 and Eq.3, and then to

make the particular diagram to the lot. So, this diagram refers to the reference lot caracterized in

our laboratories which have the mechanical properties of reference given in Table 2.

Table 2 : Physical and mechanical reference properties (Phyllostachys pubescens species)

Moisture content % 𝜎r (MPa) 𝜏r (MPa) Density (kN/m3)

7 .5±0.4 80.0±6.4 7.9 ±0.9 7.,7

It was used also as reference, the diameter of the pin dpr = 12,7 mm; bamboo thickness wall

equal tr = 8,2 mm; moment in the connection M´=Mr =0,4 kNm; axial force F´= Fr = 20 kN.

𝑀′ = 𝑀𝑟 ×𝑡̅

𝑡𝑟×

𝜏̅

𝜏𝑟×

𝑑𝑝

𝑑𝑝𝑟×

𝜎90

𝜎𝑟90 (2)

𝐹´ = 2 × 𝐹𝑟 ×𝑡̅

𝑡𝑟×

𝑑𝑝

𝑑𝑟×

�̅�

𝜏𝑟×

�̅�

𝜎𝑟 (3)

Figure 27: Limit State Design Diagram and Safety Region

As can be seen in Figure 27, the safety region in green color was obtained to a safety factor

equal 2 applied on M´and F´. This procedure was used to calculate the structure of the Figure 29,

where the bar indicated in Figure 29 a), fixed by one bolt and 2 bolts at ends, is evaluated as an

example. Figure 28 a) shows the axial force F = -4 kN and Figure 28 b) the bending moment M =

0,15 kNm . This point is inside the safety region of the Figure 27 and show us that only one bar can

absorb the solicitation if the bar is selected with thickness wall greater than dpr = 8,2 mm, although

there is two parallel bars in this position due to buckling problems of the same stick. The limit

normal stress perpendicular to fibers 𝜎𝑟90 is considered equal a quarter of the normal stress parallel

to fibers 𝜎𝑟, or 𝜎𝑟90= 0,25𝜎𝑟.

a) b)

Figure 28: a) axial forces; b) bending moments

The structure was used for a Chapel for the novel Araguaia of the Television Globo Net. Figure

29 c) shows the chapel which was designed through engineering concepts and a tower on the left

which was constructed intuitively. In this second case there is a large and visible eccentricity on the

connections, what is not a good solution.

a)

b)

c)

Figure 29 a,b,c: Connections investigated through the LIMIT STATE DESIGN DIAGRAM

Conclusions

The resistance of bolt connections decreases when axial forces are superimposed to bending

moments especially when combined with tension forces. In this case, the Limit State Diagram

Design and Safety Region developed here is a practical way to verify connections in this common

type of structures and can be improved if a major number of specimens are tested in the future.

After many investigations of the stress field it was concluded that the maximum axial force can

be limited by the squashing of the fibres in the contact area, or mode failure I. The occurrence of

failure mode I or II depends on the local thickness wall and on the moisture content of the

specimen, but in both cases the contact stress attain the squashing of the fibres. Moisture content

equal 15% or more tend to failure by progressive squashing of the fibres followed by splitting

mode III.

Acknowledgements

The authors thank to the Brazilian Council of Scientific and Technological Development - CNPq

by research funding. The authors also thank Ana Paula Nardy Nascimento for executing part of the

analysis developed here and to the designer Pedro Orlando Botelho by Chapel photographs

presented in this paper

References

[1] Moreira, L.E and Ghavami, K. Limits States Analysis for Bamboo Pin Connections. Key

Engineering Materials (on line), v. 517, 2012, p. 3-12.

[2] Moreira, L. E. and Gaspar, I. C.P. Functioning of Bolted Connections for Bamboo Structures

under Bending Moments, Shear and Axial Forces. Proceedings of the Non Conventional Materials

and Technologies International Congress XIV – XIV NOCMAT, João Pessoa, PB, Brazil, 2013.

]