Limit State Design of Steel Pin Connections for Bamboo Truss Structures

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1 Limit State Design of Steel Pin Connections for Bamboo Truss Structures Luís Eustáquio Moreira 1,a, Khosrow Ghavami 2,b 1 Federal University of Minas Gerais, Structures Department, Brazil 2 Pontifical Catholic University of Rio de Janeiro, RJ, Brazil a luis@dees.ufmg.br, b ghavami@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

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

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

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

2 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 k4 N/mm k5( N/mm k6( N/mm k7( N/mm k8( N/mm 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 δ i (θ) is given by Eq. 1.

E2=E3 = 1,2 GPa. Likewise the transversal modulus are G12=G13=G23= 1,2 GPa, Figure 3.

In this element an axial tension force equal to 20 kn is the limit to compression stress σ 11 = 80 MPa, Figure 4 a, which

Two interesting results had been observed in the axial tension tests with one and two pin by ends.

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

In the case of this lot of bamboos, the shear stresses which can cause the failure of bamboo between the pins are in the

3 Thus, the elastic local displacement below the pin δ i (θ) is given by Eq. 1. δ i (θ) = R p 1 R ρ 2 R p 2 sen 2 θ + δ + i R h cosθ (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 σ 11 = 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

The forces between pins are of neglegible difference; the pin

can be considered to occur between pins in the structure.

the 3 different specimens tested in compression.

pins with 12,7 mm in diameter with an internal angle of 90 0.

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

Loads P(N) 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.

10000 8000 6000 4000 2000 0 0 20 40 60 80 Vertical displacements (mm) a) b) Figure 8 : a) Load P versus vertical displacement δ; b) Moments M vs angular displacement γ

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

5 Loads P(N) 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 Vertical displacements (mm) 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

In this modeling, Figure 10, bamboos were

Then the element 4 was modeled as follows.

resultant R of forces in the hole, as seen in

Figure 11: Position of springs in pin- hole

shows the results for simultaneous actions of

results for tension axial force and moments

6 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

7 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

8 Figure 18: S13 BASE ZOOM Figure 19: S33 BASE ZOOM Figure 20: S13 TOP ZOOM Figure 21: S33 TOP ZOOM

9 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

10 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/m 3 ) 7.5± ± ±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 = M r t t r τ τ r d p d pr σ 90 σ r90 (2) F = 2 F r t t r d p d r τ τ r σ σ r (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.

Figure 28 a) shows the axial force F = -4 kn and Figure 28 b) the bending moment M = 0,15 knm.

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

11 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 σ r90 is considered equal a quarter of the normal stress parallel to fibers σ r, or σ r90 = 0,25σ r. 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.

b) a) 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

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

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.

12 b) a) 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

13 ] References [1] Moreira, L.E and Ghavami, K. Limits States Analysis for Bamboo Pin Connections. Key Engineering Materials (on line), v. 517, 2012, p [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.

Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CE-NEW)/SEM-3/CE-301/ SOLID MECHANICS

Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CE-NEW)/SEM-3/CE-301/ SOLID MECHANICS Name :. Roll No. :..... Invigilator s Signature :.. 2011 SOLID MECHANICS Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers