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Weld root magnification factors for semi-elliptical cracks in T-butt joints
Acta Mechanica Solida Sinica, Vol. 26, No. 3, June, 2013 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
WELD ROOT MAGNIFICATION FACTORS FOR SEMI-ELLIPTICAL CRACKS IN T-BUTT JOINTS⋆⋆ Zhanxun Song1,2
Yeping Xiong2⋆
Jilong Xie1
Jing Tang Xing2
(1 School of Mechanical, Electronic & Control Engineering, Beijing Jiaotong University, Beijing 100044, China) (2 Faculty of Engineering & the Environments, Fluid-Structure Interaction Research Group, University of Southampton, Highfield, Southampton SO17 1BJ, UK) Received 20 July 2011, revision received 29 December 2012
ABSTRACT Many researchers have focused their efforts on fatigue failures occurring on weld toes. In recent years, more and more fatigue failures occur on weld roots. Therefore, it is important to explore the behaviour of weld root fatigues. This paper investigates numerically the Magnification factors (Mk) for types of semi-elliptical cracks on the weld root of a T-butt joint. The geometry of the joint is determined by four important parameters: crack depth ratio, crack shape ratio, weld leg ratio and weld angle. A singular element approach is used to generate the corresponding finite element meshes. For each set of given four parameters of the semi-elliptical root crack, the corresponding T-butt joint is numerically simulated and its Mk at the deepest point of the weld root crack is obtained for the respective tension and shear loads. The variation range of the four parameters covers 750 cases for each load, totaling 1500 simulations are completed. The numerical results obtained are then represented by the curve to explore the effects of four parameters on the Mk. To obtain an approximate equation representing Mk as a function of the four parameters for each load, a multiple regression method is adopted and the related regression analysis is performed. The error distributions of the two approximate equations are compared with the finite element data. It is confirmed that the obtained approximate functions fit very well to the database from which they are derived. Therefore, these two equations present a valuable reference for engineering applications in T-butt joint designs.
KEY WORDS welded root cracks, singular element method, stress intensity factors, magnification factors, multiple regressions
I. INTRODUCTION T-butt joints are widely used in weld metallic structures, for examples, offshore platforms and highspeed heavy-load trains. Fatigue failures of this type of joints often occur on its stress concentration region. Sharp changes in the section geometry of T-butt weld structures cause fatigue cracks initiating from two important stress concentration regions: weld toe and root regions[1]. With initial cracks propagation, fatigue failures often occur in these regions. As reported in references[2, 3], for a double fillet welded T joint, fatigue failures usually occur on its weld toe region, however, for single fillet welded T joint there exist fatigue failures occurred on both the weld toe and weld root regions. Since fatigue failures are one of the extremely important issues affecting the safety operation of weld structures, Corresponding author. E-mail: [email protected] Project supported by the National Basic Research Program (973 Program) of China (No. 2011CB711102). The authors would like to acknowledge Professor M.M.K. Lee for providing information and deep discussions. ⋆
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scientists and engineers have been developing many techniques and analysis approaches to improve the fatigue life of weld structures. In a well-written review paper by Kirkhope[4] , a quite wide range of successful techniques were described in detailed. Fricke[5] discussed four approaches to analyze the fatigue life of welded joints. These four methods are based on the nominal stress, hot-spot stress, notch stress, and stress intensity factor (SIF) using fracture mechanics. It has been accepted that the method in base of fracture mechanics is more accurate than the other methods. Therefore, the method of fracture mechanics is often used to verify some new methods proposed for the fatigue analysis of weld structures[6] . A T-butt with a crack is a 3-dimensional structure, for which to find an analytical solution of SIF is very difficult. Experimental investigations could provide an accurate result but costs are too high. Therefore, numerical methods for SIF estimations of weld structures are often used[7–9] . Weld toe and root cracks normally have an approximate shape of a semi-ellipse, so that a semi-elliptical crack model of weld toe and root cracks is adopted in many publications[10–13] to calculate SIF. An accurate estimation of SIF of a weld structure affected by different geometrical parameters is necessary and important to reliably predict the fatigue life of structures operated in complex environments. As reported in reference[14] , SIF of T-butt weld joints is mainly affected by the four geometric parameters: weld leg, weld angle, crack depth and crack ratio. Brennan et al.[15] investigated the SIF of a T-butt weld toe considering different geometric parameters. They reported the obtained SIF in the forms of parametric equations. For a convenience of practical applications, Maddox[16] proposed a magnification factor, denoted by Mk, which is defined as the ratio of calculated SIF for a 3-dimensional T-butt weld structure crack and the plain plate crack. SIF of the plate crack has been investigated by Newman & Raju[17] , and the wellknown Newman’s equation provides SIF of semi-elliptical surface crack. The concept of Mk has widely accepted, and the previous works[18–20] have provided several sets of Mk results. More importantly, Mk of weld toe cracks obtained by Bowness et al.[18, 19] using a finite element method have been included in the new British Standard BS7910[21]. The evaluation of Mk for welded structure cracks could be performed by using: (a) stress strain method and (b) J-integral method. For two-dimensional weld cracks, these two methods have been confirmed to solve weld crack problems[22–24] and have been used to investigate three-dimensional weld cracks[25–27]. In these approaches, the strain field of crack tip was obtained by a singular element method using a crack virtual technology[28] . As experimentally demonstrated by Gurney[29] and further investigated by Balasubramanian and Guha[14] that weld root failures cannot be prevented unless the weld dimensions are appropriate to the plate thickness. In recent years, more and more fatigue failures on weld root region occurred and therefore much effort have been undertaken to explore the behavior of weld root fatigues by using numerical and experiment approaches[30–34]. Wolfgang[30] investigated this problem numerically and experimentally using the notch stress theory. Zhou[31] analyzed the effect of root flaws on the fatigue property of friction stir welds based on the experimental investigations using normal stress method. Kanvinde[32] presented the results on the strength and ductility of weld root notch obtained by the twenty-four cruciform weld experiments in association with the complementary finite element simulations. Chung[33] designed an experimental method to measure the crack opening displacements by using a digital camera to determine SIF. Fatigue crack propagation behavior and fatigue life of weld root cracks under mixed mode I and II were studied through many experiments[34] . However, to the best knowledge of the authors, there are no available publications that provide a set of complete integrated results on Mk for weld root cracks considering the related geometric parameters. This paper intends to address this issue to provide the required results for engineering designs. In this research, the strain field of a crack tip is obtained by a singular element method with the crack virtual technology as discussed in Ref.[28]. The software ABAQUS is used in the numerical simulations. To choose a suitable mesh structure, a convergence test of element size on the crack front region is performed, and a comparison of the result by the singular element method with Newman’s equation is carried out to confirm the element size around the crack front region. The numerical investigations for the 3-dimensional T-butt with semi-elliptical crack at weld root region take into account of the effects of the four geometrical parameters on SIF and Mk. The related curves affected by the four parameters are provided. The approximate equations of Mk are obtained by a multiple regressions method based
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on the numerical results. The results obtained for weld root cracks are similar to the ones for weld toe cracks given by Ref.[18,19], and it is expected that the results in this paper might be included in the British Standard[21] to provide a more complete data for engineering design applications.
II. PROBLEM DESCRIPTION Figure 1 shows the model of T-butt joint studied in this paper. The T-butt consists of a horizontal base plate of thickness B = 20 mm and length Wb = 160 mm and a vertical attachment plate of thickness A = 20 mm and height H = 80 mm. The two plates of same width Wa = 100 mm are single-fillet welded with a weld leg L and a weld angle θ to form the T butt joint. It is assumed that there exists a semi-elliptical crack of depth a and length c, as shown in Fig.1, on the horizontal melded interface in the weld root region of two plates. The depth and length axes of the semi-elliptical crack are parallel to the length and width directions of the base plate, respectively. The T-butt joint is fixed at the two sides of the base plate and is subject to a normal stress σyy = 100 MPa or a shear stress σyz = 100 MPa which are uniformly applied on the top section of the vertical attachment plate. The materials of both plates are constructed by the same isotropic steel of Young’s modulus E = 210 kNmm−2 and Poisson’s ratio µ = 0.3.
Fig. 1. Half-model of T-butt joint.
In the initial investigation, we will not consider the details of different weld materials but only assume that the weld material is same as the one of the two plates. Furthermore, we assume that the residual stress produced in the weld process is released. Also, for high computation efficiency in numerical analysis, the T-butt shown in Fig.1 is treated as a half-model cut out along the vertical symmetry plane of the full T-butt of double width 2Wa = 200 mm. Therefore, the condition of symmetry constraints on the symmetry plane is used in the analysis. To reveal the characteristics of weld root Mk affected by different geometrical sizes, we define the following three non-dimensional parameters: (1) the crack depth ratio a/A, which is the ratio of the crack depth and the thickness of attachment plate; (2) the weld leg ratio L/A, which defines the ratio of the weld leg and the thickness of attachment plate; (3) the crack shape ratio a/c, which is determined by the two axes of the semi-elliptical crack. For each loading case, i.e., a normal or a shear load, we choose different values of the three non-dimensional parameters and the weld angle θ, to numerically investigate 750 cases to obtain the Mk values. The considerations on how to choose the parameters and the ones used in our calculations are as follows: (a) The crack depth ratio a/A = 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5. We have noticed the fact that Mk decreases rapidly for shallow cracks as reported by Ref.[18], so that the five values of a/A are chosen below 0.1. As demonstrated in practices, weld joints fail if crack is very deep. Therefore, an investigation of cases with higher values of the crack depth ratio has no practical interests, so that the maximum crack depth 0.5 is considered in this paper.
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(b) The weld leg ratio L/A = 0.5, 0.8, 1, 1.2 and 1.5. Here, we consider that the fatigue intensity of a weld joint with a too short weld leg is very low. Therefore the minimum weld leg ratio chosen in this study is 0.5. On the other hand, the fatigue intensity of heat-affected zone would decreases if weld leg ratios are too big. This is why the maximum weld leg ratio 1.5 is used in this paper. Three ratios (0.8, 1 and 1.2) between the chosen minimum and maximum ratios are investigated to reveal the effects of the weld leg ratios on Mk. (c) The crack shape ratio a/c = 0.2, 0.4, 0.6, 0.8 and 1. The effect of crack shape ratios on SIF is very important demonstrated by Newman’s Equation[17] . However, the change rate of SIF with respect to the crack shape ratio is not high. We choose the above 5 crack shape ratios in the calculations. (d) The weld angle θ = 30◦ , 45◦ and 60◦ . As usually observed in weld products, the weld angle for joints is 45◦ , and other different weld angles are hardly found. In this research we consider the above three cases to obtain the information for calculations of Mk affected by weld angles. To provide useful results for engineering, we choose the related geometric parameters of T-butt joints investigated in the paper according to the previous works[18–20] which have been included in BS7910[21].
III. FINITE ELEMENT MODEL OF T-BUTT JOINT 3.1. Mesh Structure √ As it is well known that there exists the 1/ r singularity of the strain field in the crack front region according to linear elastic theory[35] , where r represents the distance from crack tip. To model this singularity, Henshell[36] ] and Barsoum[37] proposed a quadrilateral eight-node element and Barosum[37] proposed a triangle six-node element. For both of the two elements, the nodes located at a distance of 1/4 element side length from the crack tip are chosen to calculate the strain and the displacement. The Software ABAQUS[38] designed a 15 node three-dimensional singular element C3D20R based on the idea developed by Henshell[36] and Barsoum[37] to deal with the singularity problems around threedimensional crack tip regions. This 15 node three-dimensional singular element is used in this research. Figure 2(a) shows a global mesh structure of the T-butt joint simulated in this paper. To build a suitable mesh structure, we consider the following two characteristics of the studied problem. The first one is that the singularity locates in the region near to the crack tip curve. To address this, there are 80 3-D wedge singular elements attached to the crack tip curve. Figure 2(b) shows the details of the mesh structure in the region near to the 3-D semi-elliptical crack of depth a, length c and open distance b = 0.005 mm. In this figure, all elements contacting to the tip curve of the crack are the singular elements. The second one involves the boundary effects caused by the crack. According to the Saint-Venant’s principle[39] , at a point sufficiently distant from the crack the stress distribution is almost not affected by the boundary effects of the crack. Therefore, for our mesh structure, the points inside the internal domain with a fine finite element mesh structure are less than 3∼4c distance away from the crack tip, as shown in Fig.2(c). In the outside domain, the large size elements are adopted. All of elements except the singular elements used in the simulation are generated using the conventional element C3D20R of the software ABAQUS. 3.2. Boundary Conditions As shown in Fig.2(a), on the top surface, the uniform averaged node forces are added to model the normal and shear stresses on the top surface shown in Fig.1. On both ends, the fixed boundary conditions are used, and therefore the six degrees of freedom of each node are constrained. On the symmetrical surface, the symmetry condition is added, so that the displacement in X direction and the rotations about Y and Z axes are constrained. 3.3. Convergence Check to Determine Element Sizes Considering that the displacement element method based on conventional elements is well developed and coded, the size of conventional elements in outside domain can be determined by experience. A corresponding suitable element size for the crack domain is determined based on a convergence test. We choose a square of side length 0.2 mm of which the centre is located at the crack tip point on the symmetrical surface of the T-butt as shown in Fig.2(a). This square is divided into 16 triangles with a
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Fig. 2. The finite element mesh structure for the Half T-butt joint.
same centre angle. Drawing different number n = 1, 2, 3, 4, 5, 6 of circles, which are centered at the centre of the square and have an interval distance d = 0.1/(n + 1) between any two neighboring circles, we generate the different mesh lines of this square. Moving this square with its mesh division along the crack tip arch of the semi-elliptical crack produces a 3-D mesh structure around the crack tip arch. The convergence test is carried out by adjusting the number of the circles in the square but keeping an unchanged element number and mesh structure outside this 3-D domain around crack arch. For example, for all test cases the half arch of the semi-elliptical crack is divided into 5 parts as shown in Fig.2(b). Since the distance between the singular node and the √ crack tip is different when the element size changes, the ratio R of singular node’s displacement and r is used as a measure to choose the suitable element size. As shown in Fig.3, for both load cases: tension and shear, the ratio R tends to a constant when 5 circle mesh is adopted, which determines the suitable distance 0.0167 mm between two circles of the meshes used in this paper. With the parameters changing, the number of total elements used is different. In our calculations, the minimum element number is 5130 with 23187 nodes for the case of crack depth ratio a/A = 0.01, weld leg ratio L/A = 0.5, crack shape ratio a/c = 1 and weld angle θ = 30◦ as well as the maximum element number 44837 with 187796 nodes for the case of crack depth ratio a/A = 0.5, weld leg ratio L/A = 1.5, crack shape ratio a/c = 0.2 and weld angle θ = 60◦ .
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Fig. 3. Convergence test curves.
3.4. Comparison with Newman Equation To confirm the mesh size determined in §3.3, we study the same problem reported in Ref.[17] using the similar size mesh structure of singular element method. The obtained results are compared with Newman’s Equation as shown in Fig.4. Here, SE indicates the results of singular element method and New for the results of Newman Equation. As the crack depth ratio increasing, non-dimensional SIFs at the deepest point of the crack are increasing under tension load and decreasing under shear load. The results of singular element method show a good agreement with the ones from Newman’s Equation; therefore the chosen element size in the front region of the crack is further confirmed for the following numerical analysis.
Fig. 4. A comparison of the singular element method and Newman equation.
IV. SOLUTION OF SIF AND Mk For a 3-D weld toe crack of T-Butt joints, the Ref.[19] obtained the SIF at the deepest point of the crack using a two-dimensional formulation and the results have been included in BS7910[21]. The same two-dimensional formulation is used to obtain the SIF at the deepest point on the symmetrical plane of the weld root crack in this paper. The opening displacement vz(1/4) between the singular nodes is obtained form the finite element analysis with singular element method[37] and the SIF is calculated by using the following equation: s Evz(1/4) 2π K= (1) 4 (1 − µ2 ) r(1/4) where K is stress intensity factor (SIF), E and µ are the Young’s modulus and the Poisson’s ratio of the material, respectively. Since the symmetrical plane with the crack is approximately considered as a plain strain state, the Poisson’s ratio is chosen for plane strain cases.
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Mk[16] is defined as a ratio of the SIF obtained by Eq.(1) over the SIF of a plate crack with the same crack depth and crack shape ratio, i.e. Mk =
Kweld structure Kplate
(2)
where the SIF of the plate crack based on the Newman equation is obtained by the singular element method with the same FEA model used in Ref.[17].
V. RESULTS AND DISCUSSION In this paper, the 1500 values of Mk of the cracks are obtained, of which the 750 data for the case of tension load and another 750 data for the case of shear load. Some selected results are shown in the following figures in this section to illustrate the relationships between Mk and the four parameters: crack depth ratio a/A, crack shape ratio a/c, weld leg ratio L/A and weld angle θ. 5.1. Effect of the Crack Depth Ratio a/A As shown in Fig.5, considering the weld leg ratio L/A = 1.5 and weld angle θ = 60◦ in association with each case of crack shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the crack depth ratio a/A. At the deepest point under tension load and shear load, an increasing crack depth ratio a/A causes a decreasing in Mk. This decrease is large for shallow crack a/A < 0.1, and it becomes smaller as the crack depth ratio a/A is deeper for a/A > 0.1.
Fig. 5. The curves of Mk affected by the crack depth ratio a/A and the crack shape ratio a/c.
The notch stress of weld root region increases since the attachment plate stiffens the weld root region, so that the SIF of weld structure crack is larger than the one of plate crack and therefore Mk > 1. The notch stress effect on SIF of weld structure crack sharply decreases as crack depth ratio a/A increases for the shallow crack of a/A < 0.1. However, the SIF of the plate crack decreases not so sharply[17] , so that Mk at the deepest point of crack sharply decreases based on Eq.(2). The notch stress effect on weld structure crack is weakened as the crack depth ratio a/A increasing in the range of a/A > 0.1. The SIF of weld structure crack tends to the SIF of plate crack as a/A > 0.1 further increases, so that Mk at the deepest point of crack tends to a horizontal line. 5.2. Effect of the Crack Shape Ratio a/c As shown in Fig.6, considering the weld leg ratio L/A = 1.5 and crack depth ratio a/A = 0.1 in association with each case of weld angle θ (30◦ , 45◦ , 60◦ ), we obtain the curves of Mk affected by the crack shape ratio a/c. At the deepest point under tension load and shear load, an increase in crack shape ratio a/c causes a decrease in Mk. This decrease is large for tension load and marginal for shear load as the crack shape ratio a/c becomes larger. The slopes of the curves for both loading cases are approximately negative constants, which is not affected by the values of weld angle θ. An increase of the weld angle θ reduces the slopes of the curves.
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Fig. 6. The curves of Mk affected by the crack shape ratio a/c and the weld angle θ.
In the case of tension load, as shown in Fig.6(a), the increase of the Mk value due to the weld angle θ changed from 30◦ to 45◦ is larger than that when θ changed from 45◦ to 60◦ . However, for the shear load case as shown in Fig.6(b), the Mk increase for the weld angle changed from 30◦ to 45◦ is smaller than the one from 45◦ to 60◦ . 5.3. Effect of the Weld Leg Ratio L/A As shown in Fig.7, considering the crack depth ratio a/A = 0.1 and weld angle θ = 45◦ in association with each case of crack shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the weld leg ratio L/A. At the deepest point under tension load, an increase of weld leg ratio L/A leads to an increase in Mk (Fig.7(a)). This change becomes smaller as the weld leg ratio L/A becomes larger. At the deepest point under shear load, an increase of weld leg ratio L/A causes a reduction in Mk when L/A < 1 but an increase when L/A > 1 (Fig.7(b)). Therefore the curves behave a parabolic form. Furthermore, an increase of the crack shape ratio a/c results in a decrease in Mk. When the weld leg ratio L/A = 1, there is an interesting trend in Mk for tension load while a/c = 1. For the increase of crack shape ratio from a/c = 0.8 to 1, there is marginal decrease in Mk for shear load as shown in Fig.7(b).
Fig. 7. The curves of Mk affected by the weld leg ratio L/A and the crack shape ratio a/c.
5.4. Effect of the Weld Angle θ As shown in Fig.8, considering the crack depth ratio a/A = 0.1 and weld leg ratio L/A = 0.5 in association with each case of weld shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the weld angle θ. Again, at the deepest point, an increase in weld angle θ causes an increase in Mk for tension load but a decrease for shear load. This change becomes small as weld angle θ becomes larger for both tension and shear loads. In addition, an increase in crack shape ratio a/c
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causes a decrease in Mk. For the larger crack shape ratios a/c = 0.8 and 1, there is a marginal change in Mk for shear load.
Fig. 8. The curves of Mk affected by the weld angle θ and the crack shape ratio a/c.
VI. REGRESSION EQUATIONS OF Mk 6.1. Regression Analyses The multiple regression[40] function of MATLAB[41] is used to generate the approximate analytical equation based on the data obtained in our parametric study. As discussed in §V, the Mk is a function of the four parameters: the crack depth ratio a/A, the crack shape ratio a/c, the weld leg ratio L/A and the weld angle θ, as represented in the general form: a a L Mk = f , , ,θ (3) A c A For each load case, the total calculated Mk values obtained by simulating 750 cases covers the following parameter range: a/A = 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5; a/c = 0.2, 0.4, 0.6, 0.8, 1; L/A = 0.5, 0.8, 1, 1.2, 1.5 and θ = 30◦ , 45◦ , 60◦ . According to the curves obtained in §V, we can assume the corresponding forms of the functions for the tension load and the shear load as follows. 6.1.1. Tension load For the case of tension load, we assume that the Mk function takes the following form: Mk = f1 + f2 + f3
(4)
where f1 = A1
a A2 A
A1 = B1 θB2 + B3 A2 = B4 θB5 + B6 A3 = B7 θB8 + B9
+ A3 ,
(5) C1 = D1
f2 = C1 exp
f3 = E1
a A
a E2 A
+ C2
+ E3 ,
a 2 A
+ C3
a A
,
C2 = D4
a 2 ac 2 ac 2
C3 = D7 c 2 L L E1 = F1 + F2 + F3 A A 2 L L E2 = F4 + F5 + F6 A A 2 L L E3 = F7 + F8 + F9 A A
+ D2 + D5 + D8
a
+ D3
ac
+ D9
ac c
+ D6
(6)
(7)
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The aim of the regression is to determine the suitable constants B1 ∼ B9 , D1 ∼ D9 and F1 ∼ F9 to obtain a minimum residual between the calculated data and the Mk function values. This is completed by the following three steps: (a) Determination of function Mk1 = f1 In the first development stage of the equation, we consider only the function f1 as shown in Eq.(5) to determine the related 9 constants B1 ∼ B9 . The total 750 values of calculated Mk are divided into three subsets according to different values θ = (30◦ , 45◦ , 60◦ ). For each set data with the given θ values, the corresponding constants A1 ∼ A3 can be obtained by the regression analysis considering different values of parameter a/A. As a result of this, we obtain the three sets of values of constants A1 ∼ A3 from which the 9 contents B1 ∼ B9 are calculated by the following regression process. Finally, we determine the function f1 . (b) Determination of function Mk2 = Mk1 + f2 In the second stage, it is aimed to determine the function f2 based on the obtained function f1 in the first stage. The 750 Mk data are split into the five subsets according to different values a/c = (0.2, 0.4, 0.6, 0.8, 1). Each set data with a given value a/c and different crack depth ratio a/A are used to determine a set of constants C1 ∼ C3 . Therefore, after obtained the five sets of data of constants C1 ∼ C3 , the constants D1 ∼ D9 and the function f2 are further obtained by regression method. (c) Determination of function Mk = Mk1 + Mk2 + f3 In the third stage, we aim to determine the function f3 . The 750 Mk data are split into the five subsets according to different values L/A = (0.5, 0.8, 1, 1.2, 1.5). Each set data with a given value L/A results in a set of coefficients, E1 ∼ E3 , based on which the constants F1 ∼ F9 are calculated and the function f3 is obtained. After completing these calculations, the suitable constants, as listed in Table 1, are obtained. Table 1. The suitable coefficients of the regression Eq.(4) for the tension load
Coefficients B1 0.6298 B2 0.2043 B3 −0.0671 B4 0.0338 B5 0.2583 B6 −0.5870 B7 −0.0247 B8 0.6731 B9 −0.7144
Coefficients D1 0.4375 D2 −0.5980 D3 0.1585 D4 1.6900 D5 3.4680 D6 −2.5080 D7 −1.7230 D8 −1.2780 D9 1.4380
Coefficients F1 −0.3988 F2 1.2750 F3 −0.5387 F4 0.6945 F5 −1.6970 F6 0.6507 F7 0.3480 F8 −1.0580 F9 0.4433
6.1.2. Shear load For the shear load, we assume that the regression function takes the form Mk = f1 + f2 where f1 = A1 2
a A2 A
(8)
+ A3
(9) 2
A1 = B1 exp(a/c) + B2 (a/c) + B3 (a/c), A2 = B4 exp(a/c) + B5 (a/c) + B6 (a/c) A3 = B7 exp(a/c) + B8 (a/c)2 + B9 (a/c) a a 2 a f2 = C1 exp + C2 + C3 A A A C1 = D1 θ2 + D2 θ + D3 , C2 = D4 θ2 + D5 θ + D6 , C3 = D7 θ2 + D8 θ + D9 D1 = E1 (L/A)2 + E2 (L/A) + E3 , D2 = E4 (L/A)2 + E5 (L/A) + E6 2 D3 = E7 (L/A) + E8 (L/A) + E9 , D4 = E10 (L/A)2 + E11 (L/A) + E12 2 D5 = E13 (L/A) + E14 (L/A) + E15 , D6 = E16 (L/A)2 + E17 (L/A) + E18 D7 = E19 (L/A)2 + E20 (L/A) + E21 , D8 = E22 (L/A)2 + E23 (L/A) + E24 D9 = E25 (L/A)2 + E26 (L/A) + E27
(10)
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The aim of the regression is to determine the suitable constants B1 ∼ B9 and E1 ∼ E27 to obtain a minimum residual between the calculated data and the Mk function values. This is completed by the following two steps: (a) Determination of function Mk1 = f1 The first stage to develop the equation is to determine the coefficient of function f1 , B1 ∼ B9 . The total 750 Mk data are split into the five subsets with different values a/c = (0.2, 0.4, 0.6, 0.8, 1). Each set of Mk data with a fixed value a/c results in a set of coefficients, A1 ∼ A3 . The obtained five sets of values of A1 ∼ A3 are used to determine B1 to B9 and hence the function f1 . (b) Determination of function Mk2 = Mk1 + f2 In the second stage, it is to determine the coefficient of f2 , i.e., E1 ∼ E27 . The 750 Mk data are split into the five subsets according to the values L/A = (0.5, 0.8, 1, 1.2, 1.5). Each set data with a given value of L/A are further divided into three sub-subsets with θ = (30◦ , 45◦ , 60◦ ). Each set of data with given value θ results in a set of coefficients, C1 ∼ C3 . Based on all values of C1 ∼ C3 the constants D1 ∼ D9 are calculated. Each set of data with different values of L/A and D1 ∼ D9 data results in the constants E1 ∼ E27 , and then the function f2 is obtained. Table 2 presents the obtained coefficients of the regression function for the shear load. Table 2. The suitable coefficients of the regression Eq.(8) for the shear load
Coefficients B1 0.3213 B2 −0.4329 B3 −0.3072 B4 −0.5676 B5 0.1885 B6 0.6037 B7 0.9224 B8 −0.7856 B9 −0.3953
E1 E2 E3 E4 E5 E6 E7 E8 E9
Coefficients 0 0.0020130 −0.0007634 0.0530400 −0.2160000 0.0740100 −1.2570000 3.7500000 −1.0620000
Coefficients E10 0.001300 E11 0.007760 E12 −0.001673 E13 0.114300 E14 −0.603100 E15 0.098620 E16 −2.441000 E17 7.382000 E18 0.680900
Coefficients E19 0 E20 −0.010750 E21 0.003747 E22 −0.224500 E23 0.924500 E24 −0.273200 E25 5.08500 E26 −14.6300 E27 3.11500
6.2. Regression Evaluation Equations are generated by a number of influencing parameters and the large amount of data mean that the estimation equations are rather complex. Some selected results are shown in the following figures to illustrate that the estimation equations are a good fit to the data from which they were derived. From these figures, we can observe that the obtained regression equations provide a good agreement with the original FEA results although there exist small deviations in Fig.9(b) and Fig.9(d). The error of the regression equation is defined by Eq.(11). Based on the total 1500 calculated data of 750 for tension load and 750 for shearing load, the error curves derived from Eq.(11) are drawn in Fig.10 which shows the normal distribution form. The error of Mk equation under the shear load is larger than the one under the tension load. The error of most of the data is less than 10%, and therefore the regression equation provides a good approximation of Mk value for engineering applications. Error =
MkFE − Mkformula × 100% MkFE
(11)
VII. CONCLUSIONS The numerical investigations for T-butt joints containing semi-elliptical cracks on the weld root have been presented. The element size in the crack region is small to consider singularity effects and it is large in other regions to improve calculation efficiency. The convergence test has been undertaken to obtain the element size in the order of 0.0167 mm for the crack front region, which has been further confirmed by investigating and comparing the results on the Newman problem. Results from the parametric study, presented in the form of weld root magnification (Mk) factors, revealed the following effects of the considered parameters:
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Fig. 9. The selected cases to compare the Mk curves of regression equation with FE data.
(1) Mk decreases as the crack depth ratio increases for both tensile and shear loading conditions; (2) As the crack shape ratio increases, Mk decreases largely for tensile load, but marginally for shear load; (3) As the weld leg ratio increases, Mk increases for tensile load and its curve for shear load behaves as an upward facing parabolic form; (4) As weld angle increases, Mk increases for tensile load but decreases for shear load. The large database of weld root magnifications (Mk) factors for T-butt joints generated from the finite element study is used to develop engineering estimation equations. Two ap- Fig. 10 The error distribution of the regression proximate equations representing Mk as a function of the equations. four important parameters are obtained for the two load cases using a multiple regression method. The error distributions of the approximate equations relating to finite element data are given, which confirms that the obtained approximate equations fit very well to the database from which they are derived. The developed equations and the analytical results provide valuable reference for engineering applications in T-butt joint designs.
References [1] Balasubramanian,V. and Guha,B., Analyzing the influences of weld size on fatigue life prediction of FCAW cruciform joints by strain energy concept. Int. J. Pressure Vessel Piping, 1999, 76: 759-768. [2] Lu,Y.L., A practical procedure for evaluating SIFs along fronts of semi-elliptical surface cracks at weld toes in complex stress fields. Int. J. Fatigue, 1996, 18: 127-135. [3] Lie,S.T., Zhao,Z. and Yan,S.H., Two-dimensional and three-dimensional magnification factors, Mk, for non-load-carrying fillet welds cruciform joints. Eng. Fract. Mech., 2000, 65: 435-453.
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[4] Kirkhope,K.J., Bell,R., Caron,L., Basu,R.I. and Ma,K.T., Weld detail fatigue life improvement techniques. Mar. Struct., 1999, 12: 447-474. [5] Fricke,W., Review fatigue analysis of welded joints: state of development. Mar. Struct., 2003, 16: 185-200. [6] Poutiainen,I. and Marquis,G., A fatigue assessment method based on weld stress. Int. J. Fatigue, 2006, 28: 1037-1046. [7] Khodadad Motarjenmi,A., Kokabi,A.H., Ziaie A.A., Manteghi,S. and Burdekin,F.M., Comparison of the stress intensity factor of T and cruciform welded joints with different main and attachment plate thickness. Eng. Fract. Mech., 2000, 65: 55-66. [8] Choi,D.H., Choi,H.Y. and Lee,D., Fatigue life prediction of in-plane gusset welded joints using strain energy density factor approach. Theor Appl Fract Mech, 2006, 45: 108-116. [9] Al,Mukhtar,A.M., Henkel,S., Biermann,H. and Hubner,P., A finite element calculation of stress intensity factor of cruciform and butt welded joints for some geometrical parameters. Jordan Journal of Mechanical and Industrial Engineering, 2009, 3: 236-245. [10] Dijkstra,O.D., Snijder,H.H. and van Straalen,I.J., Fatigue crack growth calculations using stress intensity factors for weld toe geometries. In: Proc. Eighth Int Conf Offshore Mech Arctic Engng, The Hague, 1989: 137-143. [11] Thurlbeck,S.D., A Fracture Mechanics Based Methodology for the Assessment of Weld Toe Cracks in Tubular Offshore Joints. PhD thesis, UMIST, 1991. [12] Hou,C.Y., Fatigue analysis of welded joints with the aid of real three-dimensional weld toe geometry. Int J Fatigue, 2007, 29: 772-785. [13] Pommier,S., Sakae,C. and Murakami,Y. An empirical stress intensity factor set of Equations for a semielliptical crack in a semi-infinite body subjected to a polynomial stress distribution. Int J Fatigue, 1999, 21: 243-251. [14] Balasubramanian,V. and Guha,B., Effect of weld size on fatigue crack growth behaviour of cruciform joints by strain energy density factor approach. Theor Appl Fract Mech, 1999, 31: 141-148. [15] Brennan,F.P., Dover,W.D., Kare,R.F. and Hellier,A.K., Parametric equations for T-butt weld toe stress intensity factors. Int J Fatigue, 1999, 21: 1051-1062. [16] Maddox,S.J., An analysis of fatigue cracks in fillet welded joints. Int J Fract, 1975, 11(2): 221-243. [17] Newman,J.C. and Raju,I.S., An empirical stress intensity factor equation for the surface crack. Eng. Fract. Mech., 1981, 15: 185-192. [18] Bowness,D. and Lee,M.M.K., Weld toe magnification factors semi-elliptical cracks in T-butt joints comparison with existing solutions. Int. J. Fatigue, 2000, 22: 389-396. [19] Bowness,D. and Lee,M.M.K., Prediction of weld toe magnification factors for semi-elliptical cracks in T-butt joints. Int. J. Fatigue, 2000, 22: 369-387. [20] Fu,B., Haswell,J.V. and Bettess,P., Weld magnification factors for semi-elliptical surface cracks in fillet welded T-butt joint models. Int. J. Fract., 1993, 63: 155-171. [21] BS 7910: Guide to Methods for Assessing the Acceptability of Flaws in Metallic Structures, BSI 2005. [22] Gilles,P. and Franco,C., A new J-estimation scheme for cracks in mismatching welds—the ARAMIS method. In: Mismatching of Welds, Eds by Schwalbe,M. and Kocak,K.H., London: Mechanical Engineering Publications, 1994: 661-683. [23] Eripret,C. and Hornet,P., Prediction of overmatching effects on the fracture of stainless steel cracked welds. In: Mismatching of weld, Eds by Schwalbe,K.H. and Kocak,M., London: Mechanical Engineering Publications, 1994: 685-708. [24] Wang,Y.Y. and Kirk,M.T., Geometry effects on failure assessment diagrams. In: Proceedings of Second International Symposium on Mismatching of welds, Reinstorf-Luneburg, Germany, 1996. [25] Francis,M. and Rahman,S., Probabilistic analysis of weld cracks in center-cracked tension specimens. Computers and Structures, 2000, 76: 483-506. [26] Ganta,B.R. and Ayres,D.J., Analysis of Cracked Pipe Weldments, EPRI Report No. NP-5057, Electric Power Research Institute, Palo Alto, February, 1987. [27] Rahman,S. and Brust,F., An estimation method for evaluating energy release rates of circumferential through-wall cracked pipe welds. Eng. Fract. Mech., 1992, 43: 417-430. [28] O’Donoghue,P.E., Atluri,S.N. and Pipkins,D.S., Computational strategies for fatigue crack growth in three dimensions with application to aircraft components. Eng. Fract. Mech., 1995, 52: 51-64. [29] Gurney,T.R., An analysis of some fatigue crack propagation data for steels, subjected to pulsating tension loading. Weld Res. Int., 1979, 9: 45-52. [30] Fricke,W. and Kahl,A., Numerical and experimental investigation of weld root fatigue in fillet-welded structures. International Shipbuilding Progress, 2008, 55: 29-45. [31] Zhou,C.Z., Yang,X.Q. and Luan,G.H., Effect of root flaws on the fatigue property of friction stir welds in 2024-T3 aluminum alloys. Mater. Sci. Eng. A, 2006, 418: 155-160.
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[32] Kanvinde,A.M., Gomez,I.R., Roberts,M., Fell,B.V. and Grondin,G.Y., Strength and ductility of fillet welds with transverse root notch. Journal of Constructional Steel Research, 2009, 65: 948-958. [33] Chung,H.Y., Liu,S.H., Lin,R.S. and Ju,S.H., Assessment of stress intensity factors for load-carrying fillet welded cruciform joints using a digital camera. Int. J. Fatigue, 2008, 30: 1861-1872. [34] Kim,I.T., Weld root crack propagation under mixed mode and cyclic loading. Eng. Fract. Mech., 2005, 72: 523-534. [35] Bowness,D. and Lee,M.M.K., Weld toe magnification factors for semi-elliptical cracks in T-butt joints. OTO Report 99014, Health and Safety Executive, 1999. [36] Henshell,R.D. and Shaw,K.G., Crack tip finite elements are unnecessary. Int. J. Numer. Meth. Engng., 1975, 9: 495-507. [37] Barsoum,R.S., On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Numer. Meth. Engng., 1976, 10: 25-37. [38] ABAQUS User’s Manual, Version 6.3, Hibbitt, Karlsson and Sorensen, Inc. 2002. [39] Fung,Y.C., A First Course in Continuum Mechanics, 2nd ed., Prentice-Hall, 1977. [40] Cohen,J., Applied multiple regression/correlation analysis for the behavioral sciences, 3rd ed., Mahwah, NJ: L Erlbaum Associates, 2003. [41] Curve Fitting Toolbox, Functions, Data fitting for MATLAB, notes for R2009b. The Math Works Inc, 2009.
ISSN 0894-9166
WELD ROOT MAGNIFICATION FACTORS FOR SEMI-ELLIPTICAL CRACKS IN T-BUTT JOINTS⋆⋆ Zhanxun Song1,2
Yeping Xiong2⋆
Jilong Xie1
Jing Tang Xing2
(1 School of Mechanical, Electronic & Control Engineering, Beijing Jiaotong University, Beijing 100044, China) (2 Faculty of Engineering & the Environments, Fluid-Structure Interaction Research Group, University of Southampton, Highfield, Southampton SO17 1BJ, UK) Received 20 July 2011, revision received 29 December 2012
ABSTRACT Many researchers have focused their efforts on fatigue failures occurring on weld toes. In recent years, more and more fatigue failures occur on weld roots. Therefore, it is important to explore the behaviour of weld root fatigues. This paper investigates numerically the Magnification factors (Mk) for types of semi-elliptical cracks on the weld root of a T-butt joint. The geometry of the joint is determined by four important parameters: crack depth ratio, crack shape ratio, weld leg ratio and weld angle. A singular element approach is used to generate the corresponding finite element meshes. For each set of given four parameters of the semi-elliptical root crack, the corresponding T-butt joint is numerically simulated and its Mk at the deepest point of the weld root crack is obtained for the respective tension and shear loads. The variation range of the four parameters covers 750 cases for each load, totaling 1500 simulations are completed. The numerical results obtained are then represented by the curve to explore the effects of four parameters on the Mk. To obtain an approximate equation representing Mk as a function of the four parameters for each load, a multiple regression method is adopted and the related regression analysis is performed. The error distributions of the two approximate equations are compared with the finite element data. It is confirmed that the obtained approximate functions fit very well to the database from which they are derived. Therefore, these two equations present a valuable reference for engineering applications in T-butt joint designs.
KEY WORDS welded root cracks, singular element method, stress intensity factors, magnification factors, multiple regressions
I. INTRODUCTION T-butt joints are widely used in weld metallic structures, for examples, offshore platforms and highspeed heavy-load trains. Fatigue failures of this type of joints often occur on its stress concentration region. Sharp changes in the section geometry of T-butt weld structures cause fatigue cracks initiating from two important stress concentration regions: weld toe and root regions[1]. With initial cracks propagation, fatigue failures often occur in these regions. As reported in references[2, 3], for a double fillet welded T joint, fatigue failures usually occur on its weld toe region, however, for single fillet welded T joint there exist fatigue failures occurred on both the weld toe and weld root regions. Since fatigue failures are one of the extremely important issues affecting the safety operation of weld structures, Corresponding author. E-mail: [email protected] Project supported by the National Basic Research Program (973 Program) of China (No. 2011CB711102). The authors would like to acknowledge Professor M.M.K. Lee for providing information and deep discussions. ⋆
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scientists and engineers have been developing many techniques and analysis approaches to improve the fatigue life of weld structures. In a well-written review paper by Kirkhope[4] , a quite wide range of successful techniques were described in detailed. Fricke[5] discussed four approaches to analyze the fatigue life of welded joints. These four methods are based on the nominal stress, hot-spot stress, notch stress, and stress intensity factor (SIF) using fracture mechanics. It has been accepted that the method in base of fracture mechanics is more accurate than the other methods. Therefore, the method of fracture mechanics is often used to verify some new methods proposed for the fatigue analysis of weld structures[6] . A T-butt with a crack is a 3-dimensional structure, for which to find an analytical solution of SIF is very difficult. Experimental investigations could provide an accurate result but costs are too high. Therefore, numerical methods for SIF estimations of weld structures are often used[7–9] . Weld toe and root cracks normally have an approximate shape of a semi-ellipse, so that a semi-elliptical crack model of weld toe and root cracks is adopted in many publications[10–13] to calculate SIF. An accurate estimation of SIF of a weld structure affected by different geometrical parameters is necessary and important to reliably predict the fatigue life of structures operated in complex environments. As reported in reference[14] , SIF of T-butt weld joints is mainly affected by the four geometric parameters: weld leg, weld angle, crack depth and crack ratio. Brennan et al.[15] investigated the SIF of a T-butt weld toe considering different geometric parameters. They reported the obtained SIF in the forms of parametric equations. For a convenience of practical applications, Maddox[16] proposed a magnification factor, denoted by Mk, which is defined as the ratio of calculated SIF for a 3-dimensional T-butt weld structure crack and the plain plate crack. SIF of the plate crack has been investigated by Newman & Raju[17] , and the wellknown Newman’s equation provides SIF of semi-elliptical surface crack. The concept of Mk has widely accepted, and the previous works[18–20] have provided several sets of Mk results. More importantly, Mk of weld toe cracks obtained by Bowness et al.[18, 19] using a finite element method have been included in the new British Standard BS7910[21]. The evaluation of Mk for welded structure cracks could be performed by using: (a) stress strain method and (b) J-integral method. For two-dimensional weld cracks, these two methods have been confirmed to solve weld crack problems[22–24] and have been used to investigate three-dimensional weld cracks[25–27]. In these approaches, the strain field of crack tip was obtained by a singular element method using a crack virtual technology[28] . As experimentally demonstrated by Gurney[29] and further investigated by Balasubramanian and Guha[14] that weld root failures cannot be prevented unless the weld dimensions are appropriate to the plate thickness. In recent years, more and more fatigue failures on weld root region occurred and therefore much effort have been undertaken to explore the behavior of weld root fatigues by using numerical and experiment approaches[30–34]. Wolfgang[30] investigated this problem numerically and experimentally using the notch stress theory. Zhou[31] analyzed the effect of root flaws on the fatigue property of friction stir welds based on the experimental investigations using normal stress method. Kanvinde[32] presented the results on the strength and ductility of weld root notch obtained by the twenty-four cruciform weld experiments in association with the complementary finite element simulations. Chung[33] designed an experimental method to measure the crack opening displacements by using a digital camera to determine SIF. Fatigue crack propagation behavior and fatigue life of weld root cracks under mixed mode I and II were studied through many experiments[34] . However, to the best knowledge of the authors, there are no available publications that provide a set of complete integrated results on Mk for weld root cracks considering the related geometric parameters. This paper intends to address this issue to provide the required results for engineering designs. In this research, the strain field of a crack tip is obtained by a singular element method with the crack virtual technology as discussed in Ref.[28]. The software ABAQUS is used in the numerical simulations. To choose a suitable mesh structure, a convergence test of element size on the crack front region is performed, and a comparison of the result by the singular element method with Newman’s equation is carried out to confirm the element size around the crack front region. The numerical investigations for the 3-dimensional T-butt with semi-elliptical crack at weld root region take into account of the effects of the four geometrical parameters on SIF and Mk. The related curves affected by the four parameters are provided. The approximate equations of Mk are obtained by a multiple regressions method based
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on the numerical results. The results obtained for weld root cracks are similar to the ones for weld toe cracks given by Ref.[18,19], and it is expected that the results in this paper might be included in the British Standard[21] to provide a more complete data for engineering design applications.
II. PROBLEM DESCRIPTION Figure 1 shows the model of T-butt joint studied in this paper. The T-butt consists of a horizontal base plate of thickness B = 20 mm and length Wb = 160 mm and a vertical attachment plate of thickness A = 20 mm and height H = 80 mm. The two plates of same width Wa = 100 mm are single-fillet welded with a weld leg L and a weld angle θ to form the T butt joint. It is assumed that there exists a semi-elliptical crack of depth a and length c, as shown in Fig.1, on the horizontal melded interface in the weld root region of two plates. The depth and length axes of the semi-elliptical crack are parallel to the length and width directions of the base plate, respectively. The T-butt joint is fixed at the two sides of the base plate and is subject to a normal stress σyy = 100 MPa or a shear stress σyz = 100 MPa which are uniformly applied on the top section of the vertical attachment plate. The materials of both plates are constructed by the same isotropic steel of Young’s modulus E = 210 kNmm−2 and Poisson’s ratio µ = 0.3.
Fig. 1. Half-model of T-butt joint.
In the initial investigation, we will not consider the details of different weld materials but only assume that the weld material is same as the one of the two plates. Furthermore, we assume that the residual stress produced in the weld process is released. Also, for high computation efficiency in numerical analysis, the T-butt shown in Fig.1 is treated as a half-model cut out along the vertical symmetry plane of the full T-butt of double width 2Wa = 200 mm. Therefore, the condition of symmetry constraints on the symmetry plane is used in the analysis. To reveal the characteristics of weld root Mk affected by different geometrical sizes, we define the following three non-dimensional parameters: (1) the crack depth ratio a/A, which is the ratio of the crack depth and the thickness of attachment plate; (2) the weld leg ratio L/A, which defines the ratio of the weld leg and the thickness of attachment plate; (3) the crack shape ratio a/c, which is determined by the two axes of the semi-elliptical crack. For each loading case, i.e., a normal or a shear load, we choose different values of the three non-dimensional parameters and the weld angle θ, to numerically investigate 750 cases to obtain the Mk values. The considerations on how to choose the parameters and the ones used in our calculations are as follows: (a) The crack depth ratio a/A = 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5. We have noticed the fact that Mk decreases rapidly for shallow cracks as reported by Ref.[18], so that the five values of a/A are chosen below 0.1. As demonstrated in practices, weld joints fail if crack is very deep. Therefore, an investigation of cases with higher values of the crack depth ratio has no practical interests, so that the maximum crack depth 0.5 is considered in this paper.
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(b) The weld leg ratio L/A = 0.5, 0.8, 1, 1.2 and 1.5. Here, we consider that the fatigue intensity of a weld joint with a too short weld leg is very low. Therefore the minimum weld leg ratio chosen in this study is 0.5. On the other hand, the fatigue intensity of heat-affected zone would decreases if weld leg ratios are too big. This is why the maximum weld leg ratio 1.5 is used in this paper. Three ratios (0.8, 1 and 1.2) between the chosen minimum and maximum ratios are investigated to reveal the effects of the weld leg ratios on Mk. (c) The crack shape ratio a/c = 0.2, 0.4, 0.6, 0.8 and 1. The effect of crack shape ratios on SIF is very important demonstrated by Newman’s Equation[17] . However, the change rate of SIF with respect to the crack shape ratio is not high. We choose the above 5 crack shape ratios in the calculations. (d) The weld angle θ = 30◦ , 45◦ and 60◦ . As usually observed in weld products, the weld angle for joints is 45◦ , and other different weld angles are hardly found. In this research we consider the above three cases to obtain the information for calculations of Mk affected by weld angles. To provide useful results for engineering, we choose the related geometric parameters of T-butt joints investigated in the paper according to the previous works[18–20] which have been included in BS7910[21].
III. FINITE ELEMENT MODEL OF T-BUTT JOINT 3.1. Mesh Structure √ As it is well known that there exists the 1/ r singularity of the strain field in the crack front region according to linear elastic theory[35] , where r represents the distance from crack tip. To model this singularity, Henshell[36] ] and Barsoum[37] proposed a quadrilateral eight-node element and Barosum[37] proposed a triangle six-node element. For both of the two elements, the nodes located at a distance of 1/4 element side length from the crack tip are chosen to calculate the strain and the displacement. The Software ABAQUS[38] designed a 15 node three-dimensional singular element C3D20R based on the idea developed by Henshell[36] and Barsoum[37] to deal with the singularity problems around threedimensional crack tip regions. This 15 node three-dimensional singular element is used in this research. Figure 2(a) shows a global mesh structure of the T-butt joint simulated in this paper. To build a suitable mesh structure, we consider the following two characteristics of the studied problem. The first one is that the singularity locates in the region near to the crack tip curve. To address this, there are 80 3-D wedge singular elements attached to the crack tip curve. Figure 2(b) shows the details of the mesh structure in the region near to the 3-D semi-elliptical crack of depth a, length c and open distance b = 0.005 mm. In this figure, all elements contacting to the tip curve of the crack are the singular elements. The second one involves the boundary effects caused by the crack. According to the Saint-Venant’s principle[39] , at a point sufficiently distant from the crack the stress distribution is almost not affected by the boundary effects of the crack. Therefore, for our mesh structure, the points inside the internal domain with a fine finite element mesh structure are less than 3∼4c distance away from the crack tip, as shown in Fig.2(c). In the outside domain, the large size elements are adopted. All of elements except the singular elements used in the simulation are generated using the conventional element C3D20R of the software ABAQUS. 3.2. Boundary Conditions As shown in Fig.2(a), on the top surface, the uniform averaged node forces are added to model the normal and shear stresses on the top surface shown in Fig.1. On both ends, the fixed boundary conditions are used, and therefore the six degrees of freedom of each node are constrained. On the symmetrical surface, the symmetry condition is added, so that the displacement in X direction and the rotations about Y and Z axes are constrained. 3.3. Convergence Check to Determine Element Sizes Considering that the displacement element method based on conventional elements is well developed and coded, the size of conventional elements in outside domain can be determined by experience. A corresponding suitable element size for the crack domain is determined based on a convergence test. We choose a square of side length 0.2 mm of which the centre is located at the crack tip point on the symmetrical surface of the T-butt as shown in Fig.2(a). This square is divided into 16 triangles with a
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Fig. 2. The finite element mesh structure for the Half T-butt joint.
same centre angle. Drawing different number n = 1, 2, 3, 4, 5, 6 of circles, which are centered at the centre of the square and have an interval distance d = 0.1/(n + 1) between any two neighboring circles, we generate the different mesh lines of this square. Moving this square with its mesh division along the crack tip arch of the semi-elliptical crack produces a 3-D mesh structure around the crack tip arch. The convergence test is carried out by adjusting the number of the circles in the square but keeping an unchanged element number and mesh structure outside this 3-D domain around crack arch. For example, for all test cases the half arch of the semi-elliptical crack is divided into 5 parts as shown in Fig.2(b). Since the distance between the singular node and the √ crack tip is different when the element size changes, the ratio R of singular node’s displacement and r is used as a measure to choose the suitable element size. As shown in Fig.3, for both load cases: tension and shear, the ratio R tends to a constant when 5 circle mesh is adopted, which determines the suitable distance 0.0167 mm between two circles of the meshes used in this paper. With the parameters changing, the number of total elements used is different. In our calculations, the minimum element number is 5130 with 23187 nodes for the case of crack depth ratio a/A = 0.01, weld leg ratio L/A = 0.5, crack shape ratio a/c = 1 and weld angle θ = 30◦ as well as the maximum element number 44837 with 187796 nodes for the case of crack depth ratio a/A = 0.5, weld leg ratio L/A = 1.5, crack shape ratio a/c = 0.2 and weld angle θ = 60◦ .
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Fig. 3. Convergence test curves.
3.4. Comparison with Newman Equation To confirm the mesh size determined in §3.3, we study the same problem reported in Ref.[17] using the similar size mesh structure of singular element method. The obtained results are compared with Newman’s Equation as shown in Fig.4. Here, SE indicates the results of singular element method and New for the results of Newman Equation. As the crack depth ratio increasing, non-dimensional SIFs at the deepest point of the crack are increasing under tension load and decreasing under shear load. The results of singular element method show a good agreement with the ones from Newman’s Equation; therefore the chosen element size in the front region of the crack is further confirmed for the following numerical analysis.
Fig. 4. A comparison of the singular element method and Newman equation.
IV. SOLUTION OF SIF AND Mk For a 3-D weld toe crack of T-Butt joints, the Ref.[19] obtained the SIF at the deepest point of the crack using a two-dimensional formulation and the results have been included in BS7910[21]. The same two-dimensional formulation is used to obtain the SIF at the deepest point on the symmetrical plane of the weld root crack in this paper. The opening displacement vz(1/4) between the singular nodes is obtained form the finite element analysis with singular element method[37] and the SIF is calculated by using the following equation: s Evz(1/4) 2π K= (1) 4 (1 − µ2 ) r(1/4) where K is stress intensity factor (SIF), E and µ are the Young’s modulus and the Poisson’s ratio of the material, respectively. Since the symmetrical plane with the crack is approximately considered as a plain strain state, the Poisson’s ratio is chosen for plane strain cases.
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Mk[16] is defined as a ratio of the SIF obtained by Eq.(1) over the SIF of a plate crack with the same crack depth and crack shape ratio, i.e. Mk =
Kweld structure Kplate
(2)
where the SIF of the plate crack based on the Newman equation is obtained by the singular element method with the same FEA model used in Ref.[17].
V. RESULTS AND DISCUSSION In this paper, the 1500 values of Mk of the cracks are obtained, of which the 750 data for the case of tension load and another 750 data for the case of shear load. Some selected results are shown in the following figures in this section to illustrate the relationships between Mk and the four parameters: crack depth ratio a/A, crack shape ratio a/c, weld leg ratio L/A and weld angle θ. 5.1. Effect of the Crack Depth Ratio a/A As shown in Fig.5, considering the weld leg ratio L/A = 1.5 and weld angle θ = 60◦ in association with each case of crack shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the crack depth ratio a/A. At the deepest point under tension load and shear load, an increasing crack depth ratio a/A causes a decreasing in Mk. This decrease is large for shallow crack a/A < 0.1, and it becomes smaller as the crack depth ratio a/A is deeper for a/A > 0.1.
Fig. 5. The curves of Mk affected by the crack depth ratio a/A and the crack shape ratio a/c.
The notch stress of weld root region increases since the attachment plate stiffens the weld root region, so that the SIF of weld structure crack is larger than the one of plate crack and therefore Mk > 1. The notch stress effect on SIF of weld structure crack sharply decreases as crack depth ratio a/A increases for the shallow crack of a/A < 0.1. However, the SIF of the plate crack decreases not so sharply[17] , so that Mk at the deepest point of crack sharply decreases based on Eq.(2). The notch stress effect on weld structure crack is weakened as the crack depth ratio a/A increasing in the range of a/A > 0.1. The SIF of weld structure crack tends to the SIF of plate crack as a/A > 0.1 further increases, so that Mk at the deepest point of crack tends to a horizontal line. 5.2. Effect of the Crack Shape Ratio a/c As shown in Fig.6, considering the weld leg ratio L/A = 1.5 and crack depth ratio a/A = 0.1 in association with each case of weld angle θ (30◦ , 45◦ , 60◦ ), we obtain the curves of Mk affected by the crack shape ratio a/c. At the deepest point under tension load and shear load, an increase in crack shape ratio a/c causes a decrease in Mk. This decrease is large for tension load and marginal for shear load as the crack shape ratio a/c becomes larger. The slopes of the curves for both loading cases are approximately negative constants, which is not affected by the values of weld angle θ. An increase of the weld angle θ reduces the slopes of the curves.
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Fig. 6. The curves of Mk affected by the crack shape ratio a/c and the weld angle θ.
In the case of tension load, as shown in Fig.6(a), the increase of the Mk value due to the weld angle θ changed from 30◦ to 45◦ is larger than that when θ changed from 45◦ to 60◦ . However, for the shear load case as shown in Fig.6(b), the Mk increase for the weld angle changed from 30◦ to 45◦ is smaller than the one from 45◦ to 60◦ . 5.3. Effect of the Weld Leg Ratio L/A As shown in Fig.7, considering the crack depth ratio a/A = 0.1 and weld angle θ = 45◦ in association with each case of crack shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the weld leg ratio L/A. At the deepest point under tension load, an increase of weld leg ratio L/A leads to an increase in Mk (Fig.7(a)). This change becomes smaller as the weld leg ratio L/A becomes larger. At the deepest point under shear load, an increase of weld leg ratio L/A causes a reduction in Mk when L/A < 1 but an increase when L/A > 1 (Fig.7(b)). Therefore the curves behave a parabolic form. Furthermore, an increase of the crack shape ratio a/c results in a decrease in Mk. When the weld leg ratio L/A = 1, there is an interesting trend in Mk for tension load while a/c = 1. For the increase of crack shape ratio from a/c = 0.8 to 1, there is marginal decrease in Mk for shear load as shown in Fig.7(b).
Fig. 7. The curves of Mk affected by the weld leg ratio L/A and the crack shape ratio a/c.
5.4. Effect of the Weld Angle θ As shown in Fig.8, considering the crack depth ratio a/A = 0.1 and weld leg ratio L/A = 0.5 in association with each case of weld shape ratio a/c (= 0.2, 0.4, 0.6, 0.8, 1.0), we obtain the curves of Mk affected by the weld angle θ. Again, at the deepest point, an increase in weld angle θ causes an increase in Mk for tension load but a decrease for shear load. This change becomes small as weld angle θ becomes larger for both tension and shear loads. In addition, an increase in crack shape ratio a/c
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causes a decrease in Mk. For the larger crack shape ratios a/c = 0.8 and 1, there is a marginal change in Mk for shear load.
Fig. 8. The curves of Mk affected by the weld angle θ and the crack shape ratio a/c.
VI. REGRESSION EQUATIONS OF Mk 6.1. Regression Analyses The multiple regression[40] function of MATLAB[41] is used to generate the approximate analytical equation based on the data obtained in our parametric study. As discussed in §V, the Mk is a function of the four parameters: the crack depth ratio a/A, the crack shape ratio a/c, the weld leg ratio L/A and the weld angle θ, as represented in the general form: a a L Mk = f , , ,θ (3) A c A For each load case, the total calculated Mk values obtained by simulating 750 cases covers the following parameter range: a/A = 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5; a/c = 0.2, 0.4, 0.6, 0.8, 1; L/A = 0.5, 0.8, 1, 1.2, 1.5 and θ = 30◦ , 45◦ , 60◦ . According to the curves obtained in §V, we can assume the corresponding forms of the functions for the tension load and the shear load as follows. 6.1.1. Tension load For the case of tension load, we assume that the Mk function takes the following form: Mk = f1 + f2 + f3
(4)
where f1 = A1
a A2 A
A1 = B1 θB2 + B3 A2 = B4 θB5 + B6 A3 = B7 θB8 + B9
+ A3 ,
(5) C1 = D1
f2 = C1 exp
f3 = E1
a A
a E2 A
+ C2
+ E3 ,
a 2 A
+ C3
a A
,
C2 = D4
a 2 ac 2 ac 2
C3 = D7 c 2 L L E1 = F1 + F2 + F3 A A 2 L L E2 = F4 + F5 + F6 A A 2 L L E3 = F7 + F8 + F9 A A
+ D2 + D5 + D8
a
+ D3
ac
+ D9
ac c
+ D6
(6)
(7)
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The aim of the regression is to determine the suitable constants B1 ∼ B9 , D1 ∼ D9 and F1 ∼ F9 to obtain a minimum residual between the calculated data and the Mk function values. This is completed by the following three steps: (a) Determination of function Mk1 = f1 In the first development stage of the equation, we consider only the function f1 as shown in Eq.(5) to determine the related 9 constants B1 ∼ B9 . The total 750 values of calculated Mk are divided into three subsets according to different values θ = (30◦ , 45◦ , 60◦ ). For each set data with the given θ values, the corresponding constants A1 ∼ A3 can be obtained by the regression analysis considering different values of parameter a/A. As a result of this, we obtain the three sets of values of constants A1 ∼ A3 from which the 9 contents B1 ∼ B9 are calculated by the following regression process. Finally, we determine the function f1 . (b) Determination of function Mk2 = Mk1 + f2 In the second stage, it is aimed to determine the function f2 based on the obtained function f1 in the first stage. The 750 Mk data are split into the five subsets according to different values a/c = (0.2, 0.4, 0.6, 0.8, 1). Each set data with a given value a/c and different crack depth ratio a/A are used to determine a set of constants C1 ∼ C3 . Therefore, after obtained the five sets of data of constants C1 ∼ C3 , the constants D1 ∼ D9 and the function f2 are further obtained by regression method. (c) Determination of function Mk = Mk1 + Mk2 + f3 In the third stage, we aim to determine the function f3 . The 750 Mk data are split into the five subsets according to different values L/A = (0.5, 0.8, 1, 1.2, 1.5). Each set data with a given value L/A results in a set of coefficients, E1 ∼ E3 , based on which the constants F1 ∼ F9 are calculated and the function f3 is obtained. After completing these calculations, the suitable constants, as listed in Table 1, are obtained. Table 1. The suitable coefficients of the regression Eq.(4) for the tension load
Coefficients B1 0.6298 B2 0.2043 B3 −0.0671 B4 0.0338 B5 0.2583 B6 −0.5870 B7 −0.0247 B8 0.6731 B9 −0.7144
Coefficients D1 0.4375 D2 −0.5980 D3 0.1585 D4 1.6900 D5 3.4680 D6 −2.5080 D7 −1.7230 D8 −1.2780 D9 1.4380
Coefficients F1 −0.3988 F2 1.2750 F3 −0.5387 F4 0.6945 F5 −1.6970 F6 0.6507 F7 0.3480 F8 −1.0580 F9 0.4433
6.1.2. Shear load For the shear load, we assume that the regression function takes the form Mk = f1 + f2 where f1 = A1 2
a A2 A
(8)
+ A3
(9) 2
A1 = B1 exp(a/c) + B2 (a/c) + B3 (a/c), A2 = B4 exp(a/c) + B5 (a/c) + B6 (a/c) A3 = B7 exp(a/c) + B8 (a/c)2 + B9 (a/c) a a 2 a f2 = C1 exp + C2 + C3 A A A C1 = D1 θ2 + D2 θ + D3 , C2 = D4 θ2 + D5 θ + D6 , C3 = D7 θ2 + D8 θ + D9 D1 = E1 (L/A)2 + E2 (L/A) + E3 , D2 = E4 (L/A)2 + E5 (L/A) + E6 2 D3 = E7 (L/A) + E8 (L/A) + E9 , D4 = E10 (L/A)2 + E11 (L/A) + E12 2 D5 = E13 (L/A) + E14 (L/A) + E15 , D6 = E16 (L/A)2 + E17 (L/A) + E18 D7 = E19 (L/A)2 + E20 (L/A) + E21 , D8 = E22 (L/A)2 + E23 (L/A) + E24 D9 = E25 (L/A)2 + E26 (L/A) + E27
(10)
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The aim of the regression is to determine the suitable constants B1 ∼ B9 and E1 ∼ E27 to obtain a minimum residual between the calculated data and the Mk function values. This is completed by the following two steps: (a) Determination of function Mk1 = f1 The first stage to develop the equation is to determine the coefficient of function f1 , B1 ∼ B9 . The total 750 Mk data are split into the five subsets with different values a/c = (0.2, 0.4, 0.6, 0.8, 1). Each set of Mk data with a fixed value a/c results in a set of coefficients, A1 ∼ A3 . The obtained five sets of values of A1 ∼ A3 are used to determine B1 to B9 and hence the function f1 . (b) Determination of function Mk2 = Mk1 + f2 In the second stage, it is to determine the coefficient of f2 , i.e., E1 ∼ E27 . The 750 Mk data are split into the five subsets according to the values L/A = (0.5, 0.8, 1, 1.2, 1.5). Each set data with a given value of L/A are further divided into three sub-subsets with θ = (30◦ , 45◦ , 60◦ ). Each set of data with given value θ results in a set of coefficients, C1 ∼ C3 . Based on all values of C1 ∼ C3 the constants D1 ∼ D9 are calculated. Each set of data with different values of L/A and D1 ∼ D9 data results in the constants E1 ∼ E27 , and then the function f2 is obtained. Table 2 presents the obtained coefficients of the regression function for the shear load. Table 2. The suitable coefficients of the regression Eq.(8) for the shear load
Coefficients B1 0.3213 B2 −0.4329 B3 −0.3072 B4 −0.5676 B5 0.1885 B6 0.6037 B7 0.9224 B8 −0.7856 B9 −0.3953
E1 E2 E3 E4 E5 E6 E7 E8 E9
Coefficients 0 0.0020130 −0.0007634 0.0530400 −0.2160000 0.0740100 −1.2570000 3.7500000 −1.0620000
Coefficients E10 0.001300 E11 0.007760 E12 −0.001673 E13 0.114300 E14 −0.603100 E15 0.098620 E16 −2.441000 E17 7.382000 E18 0.680900
Coefficients E19 0 E20 −0.010750 E21 0.003747 E22 −0.224500 E23 0.924500 E24 −0.273200 E25 5.08500 E26 −14.6300 E27 3.11500
6.2. Regression Evaluation Equations are generated by a number of influencing parameters and the large amount of data mean that the estimation equations are rather complex. Some selected results are shown in the following figures to illustrate that the estimation equations are a good fit to the data from which they were derived. From these figures, we can observe that the obtained regression equations provide a good agreement with the original FEA results although there exist small deviations in Fig.9(b) and Fig.9(d). The error of the regression equation is defined by Eq.(11). Based on the total 1500 calculated data of 750 for tension load and 750 for shearing load, the error curves derived from Eq.(11) are drawn in Fig.10 which shows the normal distribution form. The error of Mk equation under the shear load is larger than the one under the tension load. The error of most of the data is less than 10%, and therefore the regression equation provides a good approximation of Mk value for engineering applications. Error =
MkFE − Mkformula × 100% MkFE
(11)
VII. CONCLUSIONS The numerical investigations for T-butt joints containing semi-elliptical cracks on the weld root have been presented. The element size in the crack region is small to consider singularity effects and it is large in other regions to improve calculation efficiency. The convergence test has been undertaken to obtain the element size in the order of 0.0167 mm for the crack front region, which has been further confirmed by investigating and comparing the results on the Newman problem. Results from the parametric study, presented in the form of weld root magnification (Mk) factors, revealed the following effects of the considered parameters:
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Fig. 9. The selected cases to compare the Mk curves of regression equation with FE data.
(1) Mk decreases as the crack depth ratio increases for both tensile and shear loading conditions; (2) As the crack shape ratio increases, Mk decreases largely for tensile load, but marginally for shear load; (3) As the weld leg ratio increases, Mk increases for tensile load and its curve for shear load behaves as an upward facing parabolic form; (4) As weld angle increases, Mk increases for tensile load but decreases for shear load. The large database of weld root magnifications (Mk) factors for T-butt joints generated from the finite element study is used to develop engineering estimation equations. Two ap- Fig. 10 The error distribution of the regression proximate equations representing Mk as a function of the equations. four important parameters are obtained for the two load cases using a multiple regression method. The error distributions of the approximate equations relating to finite element data are given, which confirms that the obtained approximate equations fit very well to the database from which they are derived. The developed equations and the analytical results provide valuable reference for engineering applications in T-butt joint designs.
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