STRUCTURAL RESPONSE OF K AND T TUBULAR JOINTS UNDER STATIC LOADING Luciano R. O. de Lima, Pedro C. G. da S. Vellasco, Sebastião A. L. de Andrade Structural Engineering Department, UERJ, Rio de Janeiro, Brazil lucianolima@uerj.br; vellasco@uerj.br; seb.andrade@uol.com.br José G. S. da Silva Mechanical Engineering Department, UERJ, Rio de Janeiro, Brazil jgss@uerj.br Luís F. da C. Neves Civil Engineering Department, Coimbra, Portugal luis@dec.uc.pt Mateus C. Bittencourt Civil Engineering Post-Graduate Program, UERJ, Rio de Janeiro, Brazil matebittencourt@yahoo.com.br Abstract: The intensive worldwide use of tubular structural elements, mainly due to its associated aesthetical and structural advantages, led designers to be focused on the technologic and design issues. Consequently, their design methods accuracy plays a fundamental role when economical and safety points of view are considered. Additionally, recent tubular joint studies indicate further research needs, especially for some joint geometries. In this work, a nonlinear numerical analysis based on a parametric study is presented, for K and T tubular joints where both chords and braces are made of hollow tubular sections. Starting from test results available in the literature, a model has been derived, taking into account the weld geometry, material and geometric nonlinearities. The proposed model was validated by comparison to the experiments, analytical results suggested on the Eurocode 3 (23) and to the classic deformation limits proposed in literature. INTRODUCTION Structural hollow sections (Figure 1) are widely used by designers, due to their aesthetical and structural advantages (Packer et al., 1992). On the other hand, the adoption of tubular sections frequently leads to more expensive and complex connections, since there is no access to the interior of the connected parts. This problem can be solved by special blind bolted connections or, more frequently, by the extensive use of welded joints. In addition to the fabrication costs, a proper connection design has to be performed since their behaviour frequently governs the overall structural response. This paper deals with the structural behaviour of SHS T joints and CHS K joints widely used in trusses under static loading (Figure 2). The effects of shear, punching shear and bending are considered to predict the possible joint failure mechanisms.
The circular hollow section (CHS) K-joint configuration is commonly adopted in steel offshore platforms (e.g. jackets and jack-ups) which are designed for extreme environmental conditions during their operational life. The ultimate and service strengths of such structures significantly depend on the component (member and joint) responses. Consequently, in the past few years various research programmes on tubular joints funded by oil and gas companies and national governments were initiated. c) T tubular joint detail a) footbridge in Coimbra, Portugal b) footbridge in Vila Nova de Gaia, Portugal d) K tubular joint detail Figure 1. Examples of tubular structures with T and K tubular joints Traditionally, design rules for hollow sections joints are based on either plastic analysis or on a deformation limit criteria. The use of plastic analysis to define the joint ultimate limit state is based on a plastic mechanism corresponding to the assumed yield line pattern. Typical examples of these approaches can be found on Packer et al (1992), Cao et al (1998), Packer (1993), Choo et al. (26) and Kosteski et al (23). Each plastic mechanism is associated to a unique ultimate load that is directly related to this particular failure mechanism. The typically adopted yield lines were: straight, circular, or a combination of those patterns. b β = b 1 b γ = 2t b μ = t d + d 2d 1 2 i, β = 1 5 a) T joint b) K joint Figure 2. Joint geometry and governing parameters, Eurocode 3 (25) d t i, γ = d 2t Deformation limits criteria usually associate the ultimate limit state of the chord face to a maximum out of plane deformation of this component. The justification for a deformation limit criterion instead of the use of plastic analysis for the prediction of the ultimate limit state is that, for slender chord faces, the joint stiffness is not exhausted after the complete onset of yielding, and can assume quite large values
due to membrane effects. This phenomenon is clearly shown in the curves obtained from the material and geometrical nonlinear finite element analysis performed in the present study. It is evident that, if the maximum load is obtained from experimental curves, the absence of a knee in the curve could complicate the identification of this ultimate limit state point. Additionally, there is still the need of further comparisons to experimental and plastic analysis results based on a deformation criteria. For T tubular joints, Korol and Mirza (1982) proposed that the ultimate limit state should be associated to a chord face displacement of 1.2 times its thickness. This value is approximately equal to 25 times the chord face elastic deformation. Lu et al (1994) proposed that the joint ultimate limit state should be associated to an out of plane deformation equal to 3% of the face width, corresponding to the maximum load reached in their experimental study. This 3% limit was proposed as well by Zhao (1991), and is actually adopted by the International Institute of Welding to define this particular ultimate limit state. Similarly, for K tubular joints, the deformation limit proposed by Lu et al. (1994) and reported by Choo et al. (23) may be used to evaluate the axial and/or rotational capacity of a joint subjected to the corresponding brace axial or moment loads. The joint strength is based on a comparison of the deformation at the bracechord intersection for two strength levels: the ultimate strength, N u which corresponds to a chord indentation, Δ u =.3d, and the serviceability strength, Ns that is related to Δ s =.1d. Lu et al. (1994) stated that the first peak in the load-deformation diagram should be used if it corresponds to a deformation smaller than the limit Δ u =.3d. According to Lu et al. (1994), if the ratio of N u /N s is greater than 1.5, the joint strength should be based on the ultimate limit state, and if N u /N s < 1.5, the serviceability limit state controls the design. In the case of CHS joints, N u /N s > 1.5 and the appropriate deformation limit to be used to determine the ultimate joint strength should be equal to.3d. EUROCODE 3 PROVISIONS (Eurocode 3, 25) For connections between CHS joints, such as the ones represented in Fig. 1 and Fig. 2, the methodology proposed by the Eurocode 3 (23) part 1-8 is based on the assumption that these joints are pinned. Therefore the relevant design characteristic (in addition to the deformation capacity) is the chord and braces strength, primarily subjected to axial forces according to Eurocode 3 (23) provisions. Equation (1) and (2) define, according to Eurocode 3 (23), the chord face plastic load for the investigated T and K joint, respectively, with the geometric parameters defined in Figure 2. N 1,Rd is the brace axial load related to the development of the chord face yielding or punching limit states. 2 = knfyt 2β ( ) N + 4 β 1 β senθ senθ 1 1,Rd (1) 1 1 where k n is 1, for tensioned members, f y is the chord yield stress, t the chord thickness, β is a geometrical parameter defined in Figure 2 and θ 1 the angle between the chord and the brace. 2 kgkpfyt d 1 sin θ1 N1,Rd = 1.8 1.2 / γm5 sen + 1 d and N2,Rd = N1, Rd (2) θ sin θ2
where f y is the chord yield stress, t the chord thickness, θ 1 and θ 2 are the angle between the chord and the braces, k p and k g can be obtained from eq. (3). 1.2.2.24γ Np,Sd M,Sd k = γ + g 1 ; kp = 1.3 np(1+ np ) 1. ; n p = + (3) 1+ exp(.5g/ t 1.33) Afy Wfy NUMERICAL MODEL CALIBRATION AND RESULTS Tubular joints are most commonly modelled by shell elements that represent the mid-surfaces of the joint member walls. The welds are usually represented by shell (see Figure 3) or three-dimensional solid elements, may be included or not in the model. It is common practice to analyse this type of joints without an explicit consideration of the welds. This is made simply modelling the mid-surfaces of the member walls using shell elements, Lee (1999) and Lie et al. (26). Despite this fact, some authors stated that this effect may be significant especially for K-joints with a gap, since the weld does not have a negligible size when compared to the gap size, Lie et al. (26). In the present investigation the weld for T joint was firstly modelled by using a ring of shell elements (SHELL 181 - four nodes with six degree of freedom per node), Figure 3, similarly to the configuration proposed by Lee (1999) and Van der Vegte et al. (27). Afterwards, solid elements (SOLID45 - eight nodes with three d.o.f. at each node) were used to considerate the joint welds to properly assess its influence. a) shell elements (after Lee, 1999) b) shell elements c) solid elements Figure 3. Modelling of the welds For an ultimate strength analysis, this approach is generally acknowledged as sufficiently accurate to simulate the overall joint structural response. The decisions on the choice of element and the better strategy to represent the welds (if included), should be made in advance since it determines the model layout and the required mesh density. The finite element models in the present study were generated using automatic mesh generation procedures. A finite element model adopted four-node thick shell elements, therefore considering bending, shear and membrane deformations. For the T joint model calibration, the numerical results found in Lie et al. (26) (T1 & T2 models) were used. Their mechanical and geometrical properties are depicted in Table 1. For the T1 joint, the parameters of Figure 2 assumes values of β =.57, μ = 23.3, μ 1 = 12.5 and γ = 11.67. It should be noted that a value of β =.57 is not critical.
The complete model was made of 9482 nodes and 9284 elements (see Figure 4) and the analysis was performed using the Ansys 1. (25) program. The model calibration was performed on a RHS T-joint considering material and geometric nonlinearities (see Figure 5). The material non-linearity was considered by using a Von Mises yield criterion associated to a three-linear stress-strain relationship to incorporate strain hardening of 5% and 1%, respectively. The geometrical nonlinearity was introduced in the model by using a Updated Lagrangean formulation. A refined mesh was used near the weld, where a stress concentration is likely to occur. An effort was made to create a regular mesh with well proportioned elements to avoid numerical problems. The load was applied in terms of force in the extremity of the brace. Table 1. Mechanical and geometrical properties T1 and T2 models (Lie et al.,26) Specimen b h t b 1 h 1 t 1 t w f y f u f w (MPa) (MPa) (MPa) T1 35 35 15 2 2 16 12 38.3 529. 6 T2 35 35 15 2 2 12 12 38.3 529. 6 a) T tubular joint b) K tubular joint Figure 4. Finite element model performed in Ansys 1. software (25) 14 12 1 14 12 1 8 6 T1 - Lie et al. (26) Solid w eld 4 Without w eld Shell w eld 2 Deformation limit 5 1 15 2 25 3 35 8 6 T2 - Lie et al. (26) Solid w eld 4 Without w eld Shell weld 2 Deformation limit 5 1 15 2 25 3 35 4 a) T1 tubular joint b) T2 tubular joint Figure 5. Results comparison T tubular joints Considering the K joint model based on Choo et al. (26), different boundary conditions may impose significant effects on the joint strength, altering chord axial stress magnitudes, chord bending stress magnitudes and introducing additional brace bending loads on the brace versus chord intersection. At present,
there are insufficient data from experiments to provide a good basis to characterise the ideal boundary conditions that could represent the effects imposed by adjacent structural members on the particular investigated joint, (Choo et al., 26). Lee (1999), states that the best way to model the boundary conditions of a K joint to simplify the test layout procedures is to consider the pinned brace ends with the translations in all coordinate directions fixed at the nodes. The load was applied by means of displacements at the nodes present at the right end of the chord while the left end was left unrestrained in the horizontal direction (see Figure 6). NULL T Figure 6. Applied boundary conditions on the K tubular joint numerical model The geometrical and mechanical properties of the K joint model are presented in Table 2. These parameters lead to values of β = / d. 4, γ = 18 < 25, d1 =.2 < di / d =.4 < 1., 1 < d / t = 18 < 5 and 1 < di / ti = 14.4 < 5. It must be emphasized that these parameters satisfy the Eurocode 3 limits (Eurocode 3, 25). For this numerical model a full material (a bilinear material model was considered with a 5% strain hardening) and a geometric nonlinear analysis was also performed. Table 2. Mechanical and geometrical properties K joint d t d 1 = d 2 t 1 = t 2 θ 1 =θ 2 (º) e g f y (MPa) f u (MPa) f w (MPa) 46 11.28 162.4 11.28 3 378.4 355 43 6. The results allow the assessment of the Eurocode 3 (23) performance not only in terms of maximum load (however the maximum numerical load is compared to the plastic load calculated from the Eurocode 3, 23), but also in terms of the load versus displacement curve. This may lead to the derivation of conclusions in terms of the stiffness and post-limit behaviour of the chord face, namely for the assessment of the performance of deformation limits criteria for the chord face resistance, or for the evaluation of the available joint over-strength achieved by membrane action. Figure 7(a) presents the load versus axial displacement curves for the brace members. From this figure it may be observed that, in the elastic range, an excellent agreement of the curves was obtained. Figure 7(b) presents the load versus axial displacement curve for the chord member. C T Δ 2 16 12 1 - right diagonal (compression) 8 2 - left diagonal (tension) 4 ultimate load Lu et al - ultimate strength 5 1 15 2 25 3 Axial displacement 35 3 25 2 15 1 5 5 1 15 2 25 Axial displacement a) chord members (1) and (2) b) brace member (3) Figure 7. Load versus displacement curves
According to the deformation limit proposed by Lu et al. (24), and reported by Choo et al. (26), N s = 155kN and N u = 165kN. Using Eurocode 3 (Eurocode 3, 25) provisions, the joint ultimate load, also represented in Fig. 7 is equal to 946kN being an inferior limit to the numerical model results. The joint ultimate load was controlled by the chord local buckling at the compression brace member region (see Figure 8, where the Von Mises stress distribution of the model that did not explicitly considered the welds are presented). F1 = 1527.4kN; F2 = 1523.2kN; F3 = 1742.7kN F1 = 1485.7kN F2 = 1886.8kN F3 = 2996.4kN Figure 8. Von Mises stress distribution (in MPa) deformed scale factor equal to 2 PARAMETRICAL ANALYSIS To evaluate the influence of the parameter β on the T joint global behaviour, five models were used in a parametric analysis, keeping the same chord for all models (35x35x15). The same mechanical properties early used were adopted i.e.: the braces width were: 9, 18, 26 and 3 mm, that correspond to values of β of.25,.5,.75,.8 and.857, respectively. The results are presented in Figure 9. As expected, increasing the value of β leads to a strong increase in the strength of the connection specially if β, 75. However, if β, 75, an increase of this parameter leads to an increase of strength with a magnitude much smaller than expected. This is due to the fact that for large values of β, the limit state related to bending does not control, while shear and punching shear begins to be the governing limit states. The individual load versus displacement curves are presented in Figure 9. Through the observation of these curves, it may be concluded that the numerical results have in general a good agreement with the Eurocode 3 (23) provisions. The joint resistance was derived at a load magnitude corresponding to a limit deformation of the chord face deformation of 3% of the chord width, i.e., 1.5 mm according to the proposal of Lu et al. (1994). However, the last model where β =.857, presented different results when compared to Eurocode 3 (23) provisions. Three additional models were included in the parametric analysis to evaluate the influence of the ratio diameter by thickness on chord of K joint global behaviour, more, keeping the same brace characteristics for all models (φ46x11.28). The same mechanical properties early used were also adopted. The chord diameters were 125,
25 and 245 mm, corresponding to β values of 36, 31.2 and 27, respectively. The results are presented in Figure 1. As expected, increasing the value of β leads to a substantial increase in the connection load carrying capacity. 2 16 b1 = 9 (beta =.25) Deformation Limit 2 16 b1 = 18 (beta =.5) Deformation Limit 12 8 12 8 4 4 2 16 5 1 15 2 25 3 35 4 5 1 15 2 25 3 35 4 a) beta =.25 a) beta =.5 b1 = 26 (beta =.75) Deformation Limit 2 16 b1 = 28 (beta =.8) Deformation Limit 12 8 12 8 4 4 2 16 12 8 4 5 1 15 2 25 3 35 4 5 1 15 2 25 3 35 4 c) beta =.75 d) beta =.8 b1 = 3 (beta =.857) Deformation Limit 5 1 15 2 25 3 35 4 2 16 12 8 4 b1 = 9 (beta =.25) b1 = 18 (beta =.5) b1 = 26 (beta =.75) b1 = 28 (beta =.8) b1 = 3 (beta =.857) Deformation Limit (3%b) 5 1 15 2 25 3 35 4 e) beta =.857 f) all analysed models Figure 9. Load versus displacement curves T tubular joints Figure 1, depicts the individual curves for brace members (left and right) where it is possible to assess the joint resistance according to Choo et al. (26). It may be observed that the joint resistance evaluated according to Eurocode 3 (23) provisions represents a lower limit for the analysed joints.
FINAL REMARKS A finite element geometrical and material non linear model was developed to simulate the T and K joints behaviour using four-node thick shell elements. This strategy enable the assessment of the proper influence of bending, shear and membrane deformations. Deformation limits criteria were used to obtain the joint ultimate load. This criterion usually associates the ultimate limit state of the chord face to a maximum out of plane deformation of this particular component. The reason for using a deformation limit criterion instead of the use of plastic analysis for the prediction of the ultimate limit state is that, for slender chord faces, the joint stiffness is not exhausted after the complete yielding onset due to membrane effects. The results of the analysis were used to assess the EN 1993-1-8 [4] performance not only in terms of maximum load, but also in terms of the global load versus displacement curves to fully characterise the joint structural response in terms of stiffness and ductility capacity. Through the observation of the analytical curves, it could be concluded that the numerical results achieved a good agreement with the Eurocode 3 [4] provisions for the joint resistance combined with a serviceability limit criterion associated to the joint chord face deformation. 3 3 25 25 2 15 d1 = 125 (beta =.31) 1 d1 = 162 (beta =.4) d1 = 25 (beta =.5) 5 d1 = 245 (beta =.6) Lu et al - ultimate strength Lu et al - serviciability strength 5 1 15 2 25 3 5 2 15 d1 = 125 (beta =.31) 1 d1 = 162 (beta =.4) d1 = 25 (beta =.5) d1 = 245 (beta =.6) 5 Lu et al - ultimate strength Lu et al - serviciability strength 5 1 15 2 25 3 35 a) left brace (2) b) right brace (1) 4 3 2 1 d1 = 125 (beta =.31) d1 = 162 (beta =.4) d1 = 25 (beta =.5) d1 = 245 (beta =.6) 5 1 15 2 25 beta N 1,Rd Lu et al.(1%) Lu et al.(3%) N 1,Rd N 1,Rd.31 795 1316 1419.4 946 16 182.5 1118 21 236.6 128 2341 2537 c) chord d) joint resistance comparison (in kn) Figure 1. Load versus displacement curves K tubular joints Acknowledgements: The authors would like to thank the Brazilian and Rio de Janeiro State Science and Technology Developing Agencies CNPq, CAPES and FAPERJ for the financial support provided to enable the development of this research program.
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