Mohamadou Aminou Sambo,Guy Richard Kol and Gambo Betchewe

Abstract In this paper,the influence of geometric parameters on the stress concentration factors due to three different types of axial loading on 81 TY tubular structures is studied.Our results reveal that,geometric parameters have a considerable impact on the variation of stress concentration factors on tubular TY-joints under axial loads. Thus, the highest stress concentration factor values are observed on the vertical brace than on the inclined one.The finite element results of the tubular structures were verified by parametric equations and experimental data.A parametric study was carried out by analyses using the nonlinear regression method to obtain parametric equations.These equations are used to calculate stress concentration factors and to analyse the fatigue resistance of TY-joints due to axial loads.

Keywords Offshore structure;Tubular TY-joint;Stress Concentration Factor;Fatigue;Axial load

1 Introduction

Welded tubular assemblies are present in several fields of steel construction including some offshore constructions(Visser, 1974). Namely, handling cranes, bridges, metal supports for platforms and especially marine structures made for the offshore oil industry (Arsem, 1987). These assemblies are formed by welding one or more braces to the chord and are constantly subjected to the combined dy‐namic forces of waves,wind,current,and even seismic ac‐tivity (API, 1993). Such loading results in a large number of stress cycles leading to fatigue damage(Bellagh, 2001).In addition, the complex intersections of these tubular joints represent structural discontinuities leading to high stress concentrations near the welds (Mohamad, 2007).Therefore, an accurate assessment of the magnitude of stress concentrations is necessary to properly address the prob‐lem of fatigue damage and to make more reliable and resil‐ient tubular joints(N’Diaye et al.,2007).

Extensive research has been conducted on various as‐pects of fatigue performance evaluation of tubular joints,in particular the determination of the stress concentration factor on the weld toe of tubular joints (Efthymiou, 1988;Lloyds,1992;Shao,2004).The main objective of these re‐search efforts is thus the proposal of parametric equations to determine the stress concentration factor (Chang and Dover, 1996). Kuang et al. (1975), Wordsworth (1981),Efthymiou and Durkin (1985), Hellier et al. (1990) and Smedley and Fisher(1991)worked on analytical studies to determine the stress concentration factor for different con‐figurations of tubular joints. These studies were carried out through experimental studies.In their work the authors showed that the distribution of the stress concentration fac‐tor on the weld toe of the tubular T,Y, X, K and KT-joints under various loads is at specific points.Shao et al.(2009)have studied the influence of geometrical parameters on the distribution of the SCF through numerical and experi‐mental studies. They demonstrated the dimensionless pa‐rameters have an effect on the positioning of the hot spot along the weld edge of the tubular K and T-joints under balanced axial loads. In the same line, Lotfollahi-Yaghin and Ahmadi (2009, 2010) have numerically indicated di‐mensionless parameters have an effect on the variation of the stress concentration factors along the weld edge of the tubular KT-joint under balanced axial loads. They pro‐posed a parametric equation to predict the stress concentra‐tions along the weld edge of the tubular KT-joint.Ahmadi et al. (2011) and Ahmadi and Lotfollahi-Yaghin (2012) in their work stated that there are very remarkable differenc‐es between the values of the stress concentration factor in a three-plane tubular KT-joint, two-plane and single-plane tubular joints having the same geometrical properties un‐der axial loads.It provides a parametric equation for deter‐mining stress concentration factors at various positions.Ahmadi and Lotfollahi-yaghin (2015), Ahmadi and Es‐maeil (2015) and Ahmadi and Ali (2016) in their work have shown that the stress concentration factors play a cru‐cial role in evaluating the fatigue performance of tubular K and KT-joints under in-plane and out-of-plane bending loads. Iberahin (2018) and Iberahin and Talal (2019) in their work on the analysis of stress concentration factors under axial loading and out-of-plane bending on the tubu‐lar K-joint, show that the effect of dimensionless parame‐ters on stress concentration factors depends on the type of load case undergone by the tubular joint. Lalitesh et al.(2019) in their work on stress concentration factors on the weld toe of T and Y-joints subjected to in-plane bending loads,using numerical analysers,the authors show that the stress concentration factors are high for tubular joints with a larger brace-chord tilt angle. Gho and Gao (2004) in their work proposed a set of parametric formulae for deter‐mining the stress concentration factor (SCF) in the tubular K(N)-joints under various axial, in-plane bending and outof-plane bending loads. In-Gyu et al. (2014) in their work on the distribution of stress concentration factors in hollow and concrete-filled tubular TY(N)-joints for bridges using a numerical study,reported dimensionless parameters have a negligible effect on concrete-filled tubular TY(N)-joints than on hollow tubular TY(N)-joints. They proposed a parametric equation to determine the sensitivity of the di‐mensionless parameters.

In the majority of the works presented in the literature,it can be said that stress concentration factors in particular are one of the most influential factors in the fatigue strength of tubular joints than other factors such as frequency, tem‐perature, corrosion, etc (Vincent, 2011).This justifies a lot of work on stress concentration factors on different shapes of tubular joints. Although tubular TY-joints are commonly used in offshore jacket structures, stress concentration fac‐tors on TY-joints under axial loads have not been addressed in this case.

In this work, the influence of dimensionless parameters on the stress concentration factor of tubular TY-joints are studied. This analysis is performed by the finite element method on 81 models of TY tubular structures under axial loading conditions using Comsol Multiphysics software.The results of the analysis are used to examine the evalua‐tion of the influence on the fatigue strength of tubular joints, which plays an important role in safeguarding the integrity of structures. In addition, a new set of SCF para‐metric equations has been developed,based on the non-linear regression method, for the fatigue analysis and design of TY-joints subjected to axial loads.

2 Finite element modeling

In parallel with the various experimental works, numer‐ous numerical simulation studies have been carried out to determine the behavior of tubular joints under loads that best reflect the operating conditions of offshore platforms.

Due to the high cost of materials and the restrictions of its facilities,the experimental method is difficult to use for a thorough study of tubular joints with various parameters without real dimensions. Nevertheless, the finite element method has been successfully used to examine the behav‐ior of tubular joints with fairly realistic geometrical dimen‐sions (Shao, 2004). In this work, we apply the finite ele‐ment method to perform our study. Better accuracy of the results of a finite element stress analysis in tubular joints depends on the elements used,the fineness of the mesh es‐pecially near areas of high stress concentrations, whether the weld toe is modelled and the boundary conditions adopted.

2.1 Parametric study

In this work,we analyze the behavior of tubular gap TYjoints (illustrated in Figure 1) subjected to axial loads.This tubular joint has a circular cross-section often found in offshore marine platforms. This is one of the tubular joints found in many of the offshore structures. It is com‐posed of two secondary tubes (brace) connected to a main tube (chord) by welding. One of the brace forms an angle ofθ= 90° and the other one inclined at an angle ofθ ＜90°respectively with chord.

To avoid a very large gap between the braces, we have studied the effect of seven models with different values of the length of the gap between the braces on the hot spot stress shown in Table 1.Then,from Table 1,it appears that the length of the braces gap(g)has no significant effect on the hot spot stress values.Therefore,a typical value of gap parameterξ=0.15 is assigned to all our models.

Figure 1 Geometric representation of a tubular gap TY-joint

Table 1 Value of hot spot stress according to gap length (γ = 24,θ=45°,τ=0:7;load condition 3)

Efthymiou (1988), in his work have explained that, to avoid an impact on stresses at the chord/brace intersection that may be affected by the boundary conditions applied at the chord levels, the length of the chord must be really long. In order to choose a value for the chord length pa‐rameterαfor our type of tubular joint, we have performed a study of the effect of this parameter on the SCF. There‐fore,the results of this partial study have been presented in Table 2. From this analysis on the six models with differ‐ent values ofαgiven in Table 2, it appears that the value of this parameter has no effect on the values of the SCF.A realistic value ofα= 12 is assigned in all models in the current study.

Table 2 SCF value obtained by variation of α(β=0:5,γ=24,θ=45°,τ=0∶7;load condition 3)

In previous works, Chang and Dover (1999) showed that the brace length parameter has no influence on the hot spot stress when the value ofαBis greater than the thresh‐old value. In the present study, a realistic valueαB= 8 is used for all models in the current study to avoid of short length brace. Steel with a Young’s modulus of 207 GPa and a Poisson’s ratio of 0.3 was used as material for our model.

The geometric parameters considered for our models are summarized in the Table 3. These geometric parameters cover actual practical values that are commonly found for making tubular joints in offshore platform structures. The geometric values of all braces are the same for different specific design models similar to the one proposed by In-Gyu et al.(2014).Depending on the values ofγ,τandβof each joint, the diameter and wall thickness parameters of the brace are changed from one study model to another.

Table 3 Range of values defined for geometric parameters

2.2 Mesh generation procedure and defined boundary conditions

The mesh used to discretise our tubular gap TYjoints(see Figure 2a) are Lagrangian and the element type used is triangular. This type of element has 6 nodes. It favours the representation of affine stress fields per element and are isoparametric because they are derived from these ref‐erence elements by a transformation that is also quadratic.However, they also have richer basis functions and there‐fore lead to smaller deviations from reality.In practice,ex‐perience shows that these elements generally lead to a bet‐ter accuracy/cost ratio than their first order counterparts.The chord, brace and weld toe have acceptable displace‐ments and are well suited to the curved boundary profile.By applying this type of 3D element, the weld toe profile can be modelled as a clean notch. This approach provides a more accurate and detailed stress distribution near the in‐tersection than a single envelope analysis.It should be not‐ed that the size of the weld toe connecting the chord and the brace is in accordance with the AWS(2002).

Figure 2 Meshed TY-joint in Comsol multiphisi

The mesh generated for the weld toe profile and the area near the intersection is illustrated in Figure 2b. Our struc‐tures are meshed in such a way as to achieve a balance be‐tween the accuracy of the results and the computer by eval‐uating the time,software,file size produced,etc.However,in finite element calculations, numerical blocking in the convergence of solutions for large shear stiffnesses is en‐countered in some problems. It is possible to reduce or eliminate this phenomenon by lowering the level of the in‐terpolation functions or by introducing elements with a larger number of degrees of freedom (Bharat et al. 2020).In order to improve the performance of the element and to avoid the phenomenon of shear locking, reduced integra‐tion has been employed. To check the convergence of the finite element analysis,a compliance test is performed and meshes of different qualities are used in this framework,before generating our 81 models of the tubular TY-joints.

A convergence study is performed to determine the re‐quired mesh division at which the displacement values converge as presented in the work of Ajay et al. (2015)and Bharat et al.(2021).Furthermore,a mesh convergence study is necessary to have more reliable results,we refined the mesh down to 3 mm on the intersection area and as CPU increases, the result converges to a constant value of 10.1 MPa (Table 4) as the maximum stress obtained under a loading condition 3 (Figure 3).Thus, the convergence of the mesh is carried out on one of the tubular joints (Joint Ref.10/1)in UK HSE(1997).

Table 4 Mesh convergence study

As shown in Figure 3 in this work, three different axial loading conditions are considered in order to study stress concentration factors in tubular TY-joints. In the load con‐dition 1, the inclined brace is subjected to axial tensile loading and the straight brace is assumed to be fixed (see Figure 3a). In the load condition 2, the straight brace is subjected to an axial compressive load and the inclined brace is assumed to be fixed (see Figure 3b). In the load condition 3,which is the operating condition of tube joints in practice,the brace(straight and inclined)is subjected to balanced axial loads (see Figure 3c). The boundary condi‐tions considered for all our joints designs are the fixing of the both ends of the chord.The modification of the bound‐ary conditions will change the SCF values accordingly as they are very specific values (Anish et al., 2019;Abhay et al.,2019).Thus,this boundary condition allows for reduced displacements and rotations in all directions.

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2.3 Stress concentration factor

The tubular joint has a specific geometry, which results in high stress concentrations near the welds toe. Under re‐peated loading,these concentrations lead to the creation of cracks, which can become large enough to cause the joint to fail. The location of the maximum stress concentration is called a hot spot. The location of the hot spots, where the stress concentration factor values are highest and which are preferred sites for fatigue crack initiation,is im‐portant in assessing the service life.Therefore, it is impor‐tant to accurately determine the distributions of stress con‐centration factor values near the intersection lines of the tu‐bular joints.

Figure 3 Representation of the different types of axial load applied

The hot spot stress along the edge of the weld toe is de‐termined by the extrapolation method.As shown in Figure 4,the stresses at points A and B, which lie within a specified extrapolation region, are previously obtained from the fi‐nite element analysis (IIW-XV-E, 1999). The hot spot stress is then calculated by extrapolating the stresses at points A and B linearly to the end of the weld. The distri‐bution of the hot spot stress along the edge of the weld toe is absolutely identical to the distribution of the SCF be‐cause the nominal stress is determined by applying the beam theory on the brace according to the type of load.However,the Von Mises stress is a better parameter for the analysis(Lalitesh et al.2018).Therefore,for all FEA anal‐yses in this work, the hot spot stress is given by the value of the Von Mises stress in the results.

Figure 4 Extrapolation method to obtain the hot spot stress

2.4 Validation of the FEM

The accuracy of the predictions of the finite element analysis must be verified against the results of the experi‐mental tests.Therefore,to validate the finite element struc‐ture, we will compare our results with those of the tests carried out in the UK HSE(1997).The geometrical charac‐teristics (Table 5) of the finite element model (FEM) were chosen on the basis of the information given by the UK HSE (1997), for a tested steel structure, in order to obtain the maximum value of the SCFs. The results of the pro‐cess verification are presented in Table 6.A good correla‐tion can be observed on the results of our current model with those of the experimental results. Thus, we see that the maximum difference between the results in the Table 6 is less than 10%. Thus, we observe a slight variation in SCF which could be due to a modelling technique. So, the finite element models generated in this work can be con‐sidered as so accurate that they offer reliable results.

Table 5 Geometrical parameter of the tubular TY-joint used for the verification of present FE model

Table 6 Results of the FE model verification based on experimental data

3 Results and discussion

In this section, we present the influence of geometrical parameters on the SCF,which is one of the most important factors influencing the fatigue strength of tubular joints.This study is carried out on the tubular TY-joint subjected to axial loads.

3.1 Analysis of the effects of parameters β on the SCF under axial loading

This subsection presents the effects of the brace-tochord diameter ratio parameterβon the SCFs and its inter‐action with the chord thickness ratio parameterγ. Thus,the influence of the parametersτ,γandθon the effect ofβon the SCF is also studied. Figure 5 shows the values of the SCF due to the effect of the variations of the parameterβand its interaction with the parameterγ.The effect of the parameterβon the SCF values under load conditions 1 and 3 (Figures 5a and 5d) forβvarying from 0.4 to 0.6,leads to a reduction of the SCF values at the position of the tip of the crown of the inclined brace of the tubular TY‐joint. On the other hand, under load conditions 2 and 3(Figures 5b and 5c), the increasing variation ofβalso leads to a reduction of the SCF values at the position of the sad‐dle of the vertical brace of the tubular TY-joint. Our re‐sults thus reveal that the maximum values of the SCF are located at the level of the saddle position. They represent the critical values,which can be responsible for the fatigue failure phenomena. However, the tubular gap TY-joint is more resistant for values of 0.5＜β ＜1 and the parameterβis a determining factor in the stress distribution due to the way the load transfer is accomplished.Thus,this result does not depend on the values of the other geometric pa‐rameters or on the type of axial loading applied. It is also evident that the parameterγis more effective in increasing the SCF values compared to the parameterβ.

3.2 Analysis of the effects of parameters γ on the SCF under axial loading

This subsection presents the influence of the chord thickness ratio parameterγon the SCF and its interaction with the wall thickness ratio parameterτ.The influence of the parametersβ,τandθon the effect ofγon the SCF is also studied. Figure 6 shows the values of the SCF due to the effect of the parameterγand its interaction with the pa‐rameterτ.The effect of the parameterγon the SCF values under load conditions 1 and 3 (Figures 6a and 6d) forγvarying from 12 to 24,leads to the increase of the SCF val‐ues at the position of the tip of the inclined bracing crown.On the other hand,for loading conditions 2 and 3(Figures 6b and 6c),the increasing variation ofγleads instead to a signif‐icant increase of the SCF values at the position of the sad‐dle of the vertical brace. Thus, for all axial loading condi‐tions studied in Figure 3, the effect of increasing variation ofγincreases the hot spot values at the crown and saddle position.Thus, our results show that, the maximum values of the SCF are located at the saddle position towards the side of the chord and thus correspond to the critical point.Therefore,for small variations ofγ,by fixingτ,i.e.,by in‐creasing the thickness of the chord (T), we obtain an in‐crease in the wall stiffness of the chord of the tubular TYjoint with a small stress distribution. Therefore, this result does not depend on the values of the other geometrical pa‐rameters or on the type of axial loading applied.This coun‐terpart, the parameterτ, is more effective in increasing the values of the SCF compared to the parameterγ.

3.3 Analysis of the effects of parameters τ on the SCF under axial loading

The influence of the parameter of the wall thickness ra‐tioτand its interaction with the angleθof inclination of the brace on the SCF is illustrated in this subsection. The influence of the parametersβ,γandθon the effect of the parameterτon the SCF is also studied.Figure 7 shows the values of the stress concentration factor (SCF) due to the variation of the parameterτand its interaction with the pa‐rameterθ.Consequently,an increasing variation of the pa‐rameterτsignificantly increases the values of the SCF.Thus, Figures 7a and 7d show the increase of the SCF for variations of the parameterτduring the loading condition 1 and 3 at the crown. However, we observe a low value of SCF forθ= 30° and 45°. Furthermore, Figures 7b and 7c show that the effect of the parameterτfor load conditions 2 and 3 reduces the SCF values at the saddle position. This result does not depend on the other values of the dimen‐sionless parameters. Moreover, we note that the variation of the parameterθ, weakly influences the SCFs values at the saddle position and increases at the toe position.Thus,the critical SCF values are observed in the vicinity of the saddle position. However, the parameterτis an indication of the relative bending stiffness of the brace and chord.The effect of the variation of the parameterτon the SCF values is less important than the effect of the parameterβ.

Figure 6 Analysis of the effects of parameters γ on the SCF

Figure 7 Analysis of the effects of parameters τ on the SCF

3.4 Analysis of the effects of parameters θ on the SCF under axial loading

This subsection shows the effects of the brace inclina‐tion angleθon the SCFs and its interaction with the pa‐rameterβ. Thus, Figure 8 shows the effect of the parame‐terθon the FSCs under three axial loading conditions(Figure 3).Therefore,Figures 8a and 8d show that for vari‐ations ofθf(wàn)orβ=0.4 an increase and decrease in SCF val‐ues at the saddle. This is due to a significant increase in the SCF value forθ=45°.In contrast,for changes inθf(wàn)orβ= 0.5 and 0.6, there is a slight increase in SCF values.Figures 8b and 8c show us a decrease in SCF values for different increasing variations of the parameterθat the saddle position. Thus, a significant increase in the SCF value forθ= 45° (see Figure 8c). This result does not de‐pend on the type of axial load studied or on the other val‐ues of the geometric parameters. Thus, the maximum val‐ues of the SCFs are located in the vicinity at the saddle po‐sition towards the side of the chord. These values repre‐sent critical values in the vicinity of the saddle on the weld toe. However, the parameterβis more effective in signifi‐cantly varying the SCF values compared to the parameterθ.

4 Establishing the parametric equations to calculate the SCF

In this paper, four new parametric equations have been developed to determine the stress concentration factor at the crown toe and the saddle point of tubular TY-joints subjected to three types of axial loads. These parametric equations expressed as dimensionless parameters are use‐ful for the fatigue design of tubular joints. The parametric equations to calculate the SCFs were established by the non-linear regression method using the statistical package(SPSS).The SCF values represent the dependent variables and the geometric parametersβ,τ,γ,θthe independent variables. These different variables constitute the input in‐formation imported in the form of a matrix. The rows of this matrix give information about each value of the SCF on the weld toe in tubular TY-joints with specific geometri‐cal characteristics. Furthermore, when the dependent and independent variables are defined, the expression of the equation has to be elaborated with given parameters, as presented by equation(3).

Figure 8 Analysis of the effects of parameters θ on the SCF

These parameters are unknown coefficients and expo‐nents (c1,c2,c3,c4andc5the coefficients defined in Eq. (3)).The researcher should designate an initial value for each factor as close as possible to the expected final answer.In‐correct initial values can lead to the failure of a solution(local or global) to converge. Several expressions of the equation must be developed to derive a parametric equa‐tion with a high coefficient of determination ofR2.In addi‐tion, when a large number of non-linear calculations are performed, the following parametric equations are used to obtain the value of the SCF at the Crown toe and Saddle of tubular TY-joints subjected to the three types of axial loads proposed.

The commonly used UK DoE(1995),recommends vali‐dation criteria for using the established parametric equa‐tions calculating the SCF.The validation results compared by the UK DoE(1995)criteria are presented in Table 7.

According to Table 7, all the proposed equations satisfy the UK DoE (1995) criteria and can therefore be reliably used to study the fatigue strength of offshore jacket struc‐tures.P/Rexpressed as a percentage(%)in Table 7,repre‐sents the ratio between the SCF predicted by the estab‐lished parametric equation and the SCF recorded during a test or analysis.

Thus, we compare the equations established with respect to the experimental data and the parametric equations from the literature. For this purpose, we compare Eq. (7) which is one of the equations obtained by the balanced axial load at the Saddle with these different studies published in the litera‐ture.According to the UK HSE(1992),a detailed experimen‐tal database of SCFs is presented for acrylic complex joints,including uni-planar,multi-planar and K and TY-joints.

Table 7 Validation of parametric equations based on UK DoE(1995)

This paper deals only with the SCF values at the posi‐tion of specific points of the weld toe. Experimental data determined from two acrylic specimens (designated 8.A1 and 8.A2)in this paper were used to compare the proposed equations. The geometric parameters of these specimens are shown in Figure 9 and Table 8.In addition to the exper‐imental data, our proposed equations were also compared with the equations proposed by Wordsworth, Efthymiou and Kuang which are specifically set up to predict the val‐ue of the SCF at the Saddle position.

Figure 9 Representation geometric of the acrylic specimens (UK HSE 1992)

Table 8 Geometrical parameter of the acrylic specimens 8.A1 and 8.A2

The values obtained by these equations are presented,respectively in UK HSE (1992). In addition, these differ‐ent equations are used to predict the SCF at the position of the Saddle and Crown points.The validation results for the saddle point position were summarized in Table 9.Howev‐er, possible reasons for the variation in results from the present work could be due to a different modelling tech‐nique. In addition, some authors use the principal con‐straints while others use the Von Mises constraint as the hot spot stress. Thus, Table 9 presents a good agreement between the results of the proposed equation and the para‐metric equations proposed in the literature.

Table 9 Results of validating the proposed equation at the saddle position

5 Conclusions

In this work,the influence of geometrical parameters on the SCF of tubular TY-joints has been studied under three axial loading conditions. 81 models of TY tubular struc‐tures were evaluated in this work.The analysis results will be used to show the influence of the geometric parameters brace-to-chord diameter ratio, chord wall slenderness ra‐tio, brace-to-chord thickness ratio and outer brace inclina‐tion angle on the values of the SCF at the different saddle and crown toe position. Based on the finite element re‐sults, a set of parametric equations calculating the SCF was established using non-linear regression analysis. In the axial load condition 1, the SCF value at the toe posi‐tion is slightly lower than the corresponding value at the toe position in the load condition 3. Furthermore, we can say that the highest SCF value is at the toe position in the load condition 3. On the other hand, in the axial load con‐dition 2, the SCF value at the position of the saddle point is slightly lower than the corresponding value at this posi‐tion in the load condition 3. Thus, we obtain the highest SCF value at the saddle point in the case of the load condi‐tion 3. Furthermore, among the four parametric equations found, the SCF values at the toe and saddle points are the minimum and maximum values, respectively. The four equations established can be reliably applied to the fatigue strength of TY tubular structures of offshore jacket platforms.