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Fatigue life prediction of shielded metal arc welded cruciform joints containing LOP defects by a mathematical model
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International Journal of Pressure Vessels and Piping 76 (1999) 283–290
Fatigue life prediction of shielded metal arc welded cruciform joints containing LOP defects by a mathematical model V. Balasubramanian*, B. Guha Mechanical Testing Laboratory, Department of Metallurgical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India Received 22 November 1998; accepted 3 December 1998
Abstract A new mathematical model is developed to predict the fatigue life of Shielded Metal Arc Welded (SMAW) cruciform joints containing Lack of Penetration (LOP) defect. High strength, quenched and tempered steel (ASTM 517 ‘F’ Grade) is used as the base material throughout the investigation. Four factors, five level, central composite, rotatable design matrix is used to optimise the required number of experiments. Fatigue experiments have been conducted in a mechanical resonance pulsator, under constant amplitude loading. The model is developed by Response Surface Method (RSM). Analysis of Variance (ANOVA) technique is applied to check the validity of the model. Student’s t-test is utilised to find out the significant factors. The effect of joint dimensions on fatigue life have been analysed in detail. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Shielded metal arc welding; Lack of penetration; Cruciform joint; Fatigue life; ANOVA
1. Introduction Fillet welded cruciform joints are widely used in many structures, including offshore and nuclear installations. In such joints, a commonly encountered defect is the Lack of Penetration (LOP) which occurs in the joint as a result of lack of access to the root during welding [1]. The fatigue crack will initiate at the LOP defect and propagate in the weldment and lead to the failure of the joint, if these structures are subjected to fatigue type of loading [2]. Fatigue life prediction of welded joints is complex, costly and time consuming. This is as a result of the multiplicity of stress concentration locations and heterogeneity of the weld metal properties. The traditional approach is to apply the S–N curve covered by BS 5400 or IIW Documents [3]. However, the fatigue life estimation of a welded joint with defects can be made by performing a crack growth experiment and subsequently the fatigue life is evaluated in terms of crack growth parameters such as da/dN versus DK. Such data merely indicate the fatigue crack growth behaviour of the component and do not predict the actual fatigue life [4]. Recent investigation [5] of the influence of flux cored arc welded cruciform joint dimensions on fatigue life of ASTM * Corresponding author. Tel.: ⫹91-44-235-1365 X3820; Fax: ⫹91-44235-2545. E-mail address: [email protected] (V. Balasubramanian)
517 ‘F’ Grade steel, was carried out using four factors, two level, full factorial design of experiments concept. This method utilised only two levels of a factor and has not taken care of intermediate levels and hence the accuracy of the developed model is limited to 95% confidence level. Hence, in this paper an attempt is made to develop a new mathematical model to predict the fatigue life of Shielded Metal Arc Welded (SMAW) cruciform joints of ASTM 517 ‘F’ grade steels containing LOP defects with more accuracy.
2. Plans of investigation In order to achieve the desired aim, the investigations were planned in the following sequence: 1. Identifying the predominant factors (joint dimensions) which are having influence on fatigue life of cruciform joints. 2. Fabricating the cruciform joints. 3. Finding the upper and lower limits of chosen factors. 4. Developing the experimental design matrix. 5. Conducting the experiments as per the design matrix. 6. Developing the mathematical model. 7. Calculating the coefficients of the model. 8. Checking the adequacy of the developed model by ANOVA method.
0308-0161/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(98)00137-9
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Fig. 1. Dimensions of cruciform specimen.
9. Testing the significance of the coefficients by Student’s t-test. 10. Analysing the effect of joint dimensions on fatigue life. 2.1. Identification of predominant factors (joint dimensions) Many of the fatigue failure modes that can occur in the welded joints involve fatigue cracking from severe imperfections which are actually an inherent part of the joint. Root failures cannot be prevented unless the weld dimensions are appropriate to the plate thickness [6]. The lack of penetration discontinuity will affect the fatigue behaviour of the fillet welds when it exceeds a critical value of half of the plate thickness to be welded [7]. For partial penetration joints, the situation is more complex as the load carrying area of the weld will depend upon the depth of the penetration [8]. From the literature [9,10] and the previous work done in our lab [4,5] the predominant factors which are having influence on fatigue life of cruciform joints have been identified. They are: 1. the ratio between initial LOP (refer Fig. 1) size (2a) to fillet width (2W), 2. the ratio between leg length (L) to plate thickness (Tp), 3. fillet angle (u ) or weld profile and 4. stress range (Ds ). 2.2. Fabrication of the joints A high strength quenched and tempered steel (ASTM 517 ‘F’ Grade) of weldable quality in the form of rolled plates of
8 mm thickness is used as the base material throughout the investigation. This material is widely used for welded constructions of all kinds such as pressure vessels, penstocks, bridges and structures as well as transport vehicles, hoisting and earthmoving equipments utilised in many different types of climatic conditions. Cruciform joints were fabricated using SMAW process with matching weld metal consumable (AWS E11018-M). The chemical composition and mechanical properties of the base metal and weld metal are given in Tables 1 and 2. All the four fillets, forming the joint, were made identical leaving an unfused gap, i.e., LOP, was controlled by prior machining process (bevelling). The various root faces enabled the joints to have different LOP lengths after welding. The fillet leg length (L) was varied by controlling number of weld passes. Weld profile (or fillet angle) was varied by controlling the arc movement. Further details of fabrication and specimen sectioning were given elsewhere [5]. The dimensions of the test specimen are shown in Fig. 1. 2.3. Finding the limits of the factors Trial experiments were conducted on large number of specimens to find out the feasible limits of the aforementioned chosen factors in such a way that the failure should occur from the LOP defect. From the experimental results, the following conditions were existed: 1. If a/W ratio is less than 0.25, then toe failure is predominant. 2. If a/W ratio is in between 0.25 and 0.45, then root (LOP) failures are more common. 3. If a/W ratio is greater than 0.45, then failure occurs quickly (⬍10 4 cycles) i.e., the failure is considered to be Low Cycle Fatigue (LCF) failure. 4. If L/Tp ratio is less than 0.4, then failure occurs much faster, like in the previous condition. 5. If L/Tp ratio is in between 0.4 and 1.2, then failure occurs from root(LOP) region. 6. If L/Tp ratio is greater than 1.2, then the failure is from toe region only. 7. If the weld profile is either concave (u ⬍ 45⬚) or straight (u 45⬚), then failure occurs from the root (LOP) region. 8. If the weld profile is convex (u ⬎ 45⬚), then toe cracking is predominant. 9. If the stress range is less than 120 MPa, then most of the specimen endured upto 10 7 cycles. 10. If the stress range is greater than 280 MPa, then majority of the specimen failed within 10 4 cycles (LCF region).
Table 1 Chemical composition (wt.%) of the base and weld metals Material
C
Si
Mn
P
S
Cr
Mo
Ni
Cu
Co
V
Base metal Weld metal
0.19 0.08
0.72 0.5
0.95 1.6
0.01 0.01
0.002 0.03
0.8 0.35
0.35 0.27
0.07 1.73
0.03 0.1
0.004 0.01
0.002 0.001
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Table 2 Mechanical properties of the base and weld metals Material
Yield strength (MPa)
Tensile strength(MPa)
Vicker’s hardness (v30 kg)
Impact value (J)
Percentage of elongation (%)
Base metal Weld metal
690 740
790 845
210 320
110 175
19 24.5
By considering all the aforesaid conditions, the feasible limits of the factors were chosen in such a way that the failure should occur from LOP defect and in High Cycle Fatigue (⬎10 4 cycles) region and they are presented in the Table 3. For the convenience of recording and processing the experimental data, the upper and lower levels of the factors are coded as ⫹2 and ⫺2, respectively and the coded values of any intermediate levels can be calculated by using the expression in Ref. [11]: Xi
2X ⫺ Xmax ⫹ Xmin Xmax ⫺ Xmin =2
machine (SCHENCK 200 kN capacity) with a frequency of 30 Hz under constant amplitude loading (R 0). Care was taken to see that the load was axial and no bending component was present in the joint. The number of cycles to complete failure of each specimen was recorded and presented in the Table 4. Even though the experiments were conducted in a random order, the Table 4 shows the standard order to avoid systematic errors creeping the results. 2.6. Developing the model
where Xi is the required coded value of a factor of any value X from Xmin to Xmax; Xmin is the lower level of the factor and Xmax is the upper level of the factor.
Representing the fatigue life of welded cruciform joints containing LOP defects by Nf, then the response function (cf. Ref. [13]) can be expressed as
2.4. Developing the experimental design matrix
Nf f a=W; L=Tp ; u; Ds f A; L; P; S:
Owing to slightly wider ranges of the factors, it was decided to use a five level, central composite, rotatable design matrix to optimise the experimental conditions. Table 4 shows the 31 set of coded conditions used to form the design matrix. The first 16 experimental conditions (rows) have been formed for the main effects by using the formula 2 nc⫺1 for the low (⫺1) and high (⫹1) values; where ‘nc’ refers to the column number. For example, in Table 4, the first four rows are coded as ⫺1 and the next four rows are coded as ⫹1, alternatively, in the third column (⬗ nc 3 and 2 3⫺1 4). All chosen variables at the intermediate level (0) constitute the centre points and the combinations of each of the variables at either its lowest (⫺2) or highest (⫹2) with the other three variables of the intermediate levels constitute the star points. The method of designing such a matrix is dealt with elsewhere [11,12].
The model selected was a second degree response surface expressed as follows:
2.5. Conducting the experiments The fatigue experiments were conducted as per the conditions dictated by the design matrix (Table 4), by using a mechanical resonance controlled universal fatigue testing Table 3 Important factors and their levels S. no.
1 2 3 4
(0)
(⫹1)
(⫹2)
a/W L/Tp u Ds
0.35 0.8 37.5 200
0.4 1.0 45 240
0.45 1.2 52.5 280
— — deg MPa
⫹ N22 L2 ⫹ N33 P2 ⫹ N44 S2 ⫹ N12 AL ⫹ N13 AP ⫹ N14 AS ⫹ N23 LP ⫹ N24 LS ⫹ N34 PS: The values of the coefficients were calculated by regression with the help of the following equations (cf. Ref. [11]): X X N0 0:142857 Y ⫺ 0:035714 Xii Y; Ni 0:041667
0.25 0.4 22.5 120
0.3 0.6 30 160
X Xi Y;
X X Nii 0:03125 Xii Y ⫹ 0:003720 Xii Y ⫺ 0:035714 X Y; Nij 0:0625
Factor Notation Unit Levels (⫺2) (⫺1) A L P S
Nf N0 ⫹ N1 A ⫹ N2 L ⫹ N3 P ⫹ N4 S ⫹ N11 A2
X
Xij Y;
Student’s t-test [14] was applied to eliminate the significant coefficients without sacrificing much of the accuracy to avoid cumbersome mathematical labour. After determining the significant coefficients, the final model was developed including only those coefficients and is given in the following text:
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Fatigue life,
Table 4 Design matrix and experimental results Expt. no.
A (X1)
L (X2)
P (X3)
S (X4)
Nf (life) × 10 5 cycles
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 ⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 0 0 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 0 0 0 0 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0
8.6 5.2 17.3 10.8 11.4 7.9 19.2 12.8 1.2 0.7 2.5 1.6 1.7 1.0 3.3 2.2 9.4 4.2 3.7 6.3 2.8 7.8 24.2 0.8 7.0 7.2 6.4 5.8 6.8 6.2 5.6
Table 5 Analysis of variance (ANOVA) test results First order terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Second order terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Error terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Lack of fit Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Fratio (MS of lack of fit)/ (MS of error terms) F(10,6,0.99) from table Whether the model is adequate or not?
782.72 4 195.7 129.64 10 12.96 2.2 6 0.37 18.35 10 1.84 4.96 7.87 Yes, adequate
Nf {6:44 ⫺ 1:39 A ⫹ 1:55 L ⫹ 0:9 P ⫺ 5:24 S ⫺ 0:29 A2 ⫺ 0:47 L2 ⫺ 0:4 P2 ⫹ 1:4 S2 ⫺ 0:43 AL ⫹ 1:04 AS ⫺ 1:38 LS ⫺ 0:45 PS} × 105 cycles: 2.7. Checking the adequacy of the developed model The adequacy of the model was then checked by using the Analysis of Variance (ANOVA) technique [11–14]. As per this technique, if the calculated value of the Fratio of the developed model does not exceed the standard tabulated value of Fratio for a desired level of confidence (say 99%), then the model is considered to be adequate within the confidence limit. ANOVA test results are presented in the Table 5. Further, the experimental data and the predicted data by the using the aforesaid model are plotted as shown in Fig. 2, which indicates a good correlation. 3. Discussions Using the developed model, the fatigue lives have been predicted for different combinations of joint dimensions and they are presented in the graphical form (Figs. 3–6). The effects of joint dimensions on fatigue life have been discussed in the following sections. 3.1. Effect of L/Tp ratio on fatigue life The L/Tp ratio, i.e., the ratio between leg length and plate thickness, decides the final weld size of the cruciform joints. The effect of L/Tp ratio on fatigue life, for different values of LOP size and fillet angle, have been shown in Figs. 3–6. From the graphs, it is evident that the higher the L/Tp ratio, the larger will be the fatigue lives and viceversa. The reason can be easily understood from the following Stress Intensity Factor (SIF) range expression, for a load carrying cruciform joint at the apex of a root (LOP) defect [7]. DK
DsA1 ⫹ A2 a*pa sec pa*=21=2 1 ⫹ 2 L=Tp
where Ds a a* ( a/W) L/Tp A1, A2
nominal stress range; half crack (LOP) length; normalised crack length; weldsize; are constants which depend on weldsize.
From the earlier expression, it is clear that the SIF range is inversely proportional to the L/Tp ratio, i.e., if the L/Tp ratio is more (for larger welds), the SIF range value will become lower and hence the crack initiation, crack propagation and failure will be delayed. More importantly, the variations in the fatigue crack growth behaviour and fatigue
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Fig. 2. Correlation graph.
Fig. 3. S–N curves.
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Fig. 4. S–N curves.
Fig. 5. S–N curves.
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Fig. 6. S–N curves.
life is mainly attributed to the difference in the number of weld passes involved during the fabrication of the joints. In multipass deposits (larger L/Tp), the weld metal consists partly of a tertiary transformation micro structure because in multipass welding, each successive bead tempers previous ones, consequently the secondary micro structures, such as pro-eutectoid (Grain boundary) ferrite, side plate ferrite and acicular ferrite, will partly be heated into the region. This extra transformation results in the formation of tertiary micro structure. This structure normally has higher toughness than the non-transformed weld metal (the secondary micro structure) [15,16]. Further, the larger welds are having more weld metals and hence the fatigue crack has to propagate longer distance before final failure to occur. 3.2. Effect of a/W ratio on fatigue life The a/W ratio, i.e., the ratio between initial LOP size and the fillet width, decides the defect size in cruciform joints. The effect of LOP size on fatigue life, for different values of L/Tp and u have been depicted in Figs. 3–6. From the graphs, it is clear that the lower the LOP size, higher will be the fatigue life and vice versa, for any weld size and fillet angle. This is caused by the fact that the joints which contains smaller defects will give longer life than its counterpart. It is also evident from the SIF range expression, given earlier, that the SIF range value is directly proportional to the a/W (a*) and hence the joints with smaller LOP
defect will endure more number of cycles than its counterpart. If the LOP defect size is small, then crack initiation will be delayed as a result of the lower values of SIF range but the reverse will be the result, if the LOP defect size is large. Moreover, the crack has to propagate a smaller distance, in comparison, when a/W is larger, as a result of the reduced effective fillet width. 3.3. Effect of weld profile on fatigue life The fillet angle will decide the weld profile in cruciform joints. Figs. 3–6, reveals the effect of weld profile on fatigue life of cruciform joints, for various L/Tp and a/W values. From the graphs, it is inferred that the straight fillets (u 45⬚) are superior compared to concave fillets but the difference is very small. This can be easily understood from the following SIF range expression for the cracks emanating from toe region [17] DK
Ms × Mt × Mk × Ds pa1=2 f0
where Ms Mt Mk
f0
correction factor for the effect of a free surface; correction factor for the effect of plate thickness; correction factor for the effect of stress concentration owing to the weld toe angle; correction factor for the effect of crack front shape.
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From the aforesaid expression it is obvious that the fillet toe angle is directly proportional to the SIF range values at weld toe. In load carrying cruciform joint, if the fillet angle is more than 45⬚, then the SIF range is sufficient to initiate and propagate a fatigue crack and cause failure in the joint prematurely as a result of its high level of stress concentration effect near the toe region [18]. If the fillet angle is less than 45⬚, then the SIF range is not sufficient to initiate a fatigue crack from toe region but the SIF range at the tip of LOP defect is increased as a result of the reduction in effective weld size. Further, in the concave fillets less weld metal is available to resist the propagation of fatigue crack and hence the failure occurs somewhat earlier than in the straight fillet welds. 4. Conclusions
1. By using this new model, the fatigue life of shielded metal arc welded cruciform joints containing LOP defects can be predicted at 99% confidence level. The accuracy of the predicted values is improved by considering more number of experimental conditions than the earlier investigation. 2. Though the number of experimental conditions are more, factorial experimentation technique is still economical to predict the effects of various factors on fatigue life by conducting optimum number of experiments. 3. Moreover, the ANOVA technique is convenient to check the validity of the developed model and the Student’s ttest is very simple to identify the significance of main effects and interaction effects of various combinations of factors. 4. The effect of cruciform joint dimensions on fatigue life have been analysed in detail. Larger weld sizes, smaller LOP sizes and straight profile fillet welds are showing better fatigue lives compared to other combinations.
Acknowledgements The authors are grateful to the Indian Institute of Technology, Madras, Chennai-36, for the support rendered and for making available the facilities of Metal Joining
Laboratory and Mechanical Testing Laboratory of Metallurgical Engineering Department. The authors are indebted to M/s. Bharat Heavy Electricals Ltd., Ranipet, Tamil Nadu for the materials supplied to carryout the investigation.
References [1] Nadkarni SV. Modern arc welding technology. New Delhi: Oxford and IBH Publishing Co. Pvt. Ltd, 1996 pp. 650–651. [2] Jakubczak H, Glinka G. Fatigue anaysis of manufacturing defects in weldments. Int. J. Fatigue 1986;8:51–57. [3] Hobbacher A. Recommendations for assessment of weld imperfections in respect to fatigue, IIW Doc. IIW-XII-1266-88/XV-659-88, 1988. [4] Guha B. A new fracture mechanics method to predict the fatigue life of welded cruciform joints. Engineering Fracture Mechanics 1995;52:215–229. [5] Balasubramanian V, Guha B. Influence of flux cored arc welded cruciform joint dimensions on fatigue life of ASTM 517 ‘F’ grade steels. Int. J. Pres. Ves. and Piping 1998;10:765–772. [6] Maddox SJ. Assessing the significance of flaws in welds subject to fatigue. Weld Res. Suppl. 1974;9:401s–409s. [7] Frank KH, Fisher JW. Fatigue strength of fillet welded cruciform joints. Journal of Struc. Div. 1979;105:1727–1739. [8] Blodgett OW. Designing weldments for fatigue loading. Welding Journal 1992;7:39–42. [9] Usami S, Kusumoto S. Fatigue strengths at roots of cruciform, tee and LOP joints. Trans. Japan Welding Society 1978;9:1–10. [10] Ferreira JM, Branco CM. Influence of weld and plate geometry on the fatigue strength of cruciform joints. Theo. and Appl. Frac. Mech. 1988;9:23–32. [11] Box GEP, Hunter WH, Hunter JS. Statistics for experimenters. New York: John Wiley and Sons, 1978. [12] Montgomery DC. Design and analysis of experiments. New York: John Wiley and Sons, 1991. [13] Ravindra J, Parmar RS. Mathematical model to predict weld bead geometry for flux cored arc welding. Metal Construction 1987;19:45–52. [14] Miller I, Freund JE, Johnson R. Probability and Statistics for Engineers. New Delhi: Prentice of Hall of India Pvt. Ltd, 1996. [15] Tweed H, Knott H. Effect of reheating on microstructure and toughness of C–Mn weld metal. Journal of Material Science 1983;17:45– 54. [16] Balasubramanian V, Guha B. Effect of L/Tp ratio on fatigue life prediction of SMAW cruciform joints of ASTM 517 ‘F’ grade steels. Int. J. Pres. Ves. and Piping 1998;13:907–918. [17] Maddox SJ. An analysis of fatigue cracks in fillet welded joints. Int. J. of Fract. 1975;11:221–243. [18] Ferreira JM, Branco CM. Fatigue analysis and prediction in fillet welded joints in the low thickness range. Fat. Fract. Engng. Mater. Struct. 1990;13:201–212.
International Journal of Pressure Vessels and Piping 76 (1999) 283–290
Fatigue life prediction of shielded metal arc welded cruciform joints containing LOP defects by a mathematical model V. Balasubramanian*, B. Guha Mechanical Testing Laboratory, Department of Metallurgical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India Received 22 November 1998; accepted 3 December 1998
Abstract A new mathematical model is developed to predict the fatigue life of Shielded Metal Arc Welded (SMAW) cruciform joints containing Lack of Penetration (LOP) defect. High strength, quenched and tempered steel (ASTM 517 ‘F’ Grade) is used as the base material throughout the investigation. Four factors, five level, central composite, rotatable design matrix is used to optimise the required number of experiments. Fatigue experiments have been conducted in a mechanical resonance pulsator, under constant amplitude loading. The model is developed by Response Surface Method (RSM). Analysis of Variance (ANOVA) technique is applied to check the validity of the model. Student’s t-test is utilised to find out the significant factors. The effect of joint dimensions on fatigue life have been analysed in detail. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Shielded metal arc welding; Lack of penetration; Cruciform joint; Fatigue life; ANOVA
1. Introduction Fillet welded cruciform joints are widely used in many structures, including offshore and nuclear installations. In such joints, a commonly encountered defect is the Lack of Penetration (LOP) which occurs in the joint as a result of lack of access to the root during welding [1]. The fatigue crack will initiate at the LOP defect and propagate in the weldment and lead to the failure of the joint, if these structures are subjected to fatigue type of loading [2]. Fatigue life prediction of welded joints is complex, costly and time consuming. This is as a result of the multiplicity of stress concentration locations and heterogeneity of the weld metal properties. The traditional approach is to apply the S–N curve covered by BS 5400 or IIW Documents [3]. However, the fatigue life estimation of a welded joint with defects can be made by performing a crack growth experiment and subsequently the fatigue life is evaluated in terms of crack growth parameters such as da/dN versus DK. Such data merely indicate the fatigue crack growth behaviour of the component and do not predict the actual fatigue life [4]. Recent investigation [5] of the influence of flux cored arc welded cruciform joint dimensions on fatigue life of ASTM * Corresponding author. Tel.: ⫹91-44-235-1365 X3820; Fax: ⫹91-44235-2545. E-mail address: [email protected] (V. Balasubramanian)
517 ‘F’ Grade steel, was carried out using four factors, two level, full factorial design of experiments concept. This method utilised only two levels of a factor and has not taken care of intermediate levels and hence the accuracy of the developed model is limited to 95% confidence level. Hence, in this paper an attempt is made to develop a new mathematical model to predict the fatigue life of Shielded Metal Arc Welded (SMAW) cruciform joints of ASTM 517 ‘F’ grade steels containing LOP defects with more accuracy.
2. Plans of investigation In order to achieve the desired aim, the investigations were planned in the following sequence: 1. Identifying the predominant factors (joint dimensions) which are having influence on fatigue life of cruciform joints. 2. Fabricating the cruciform joints. 3. Finding the upper and lower limits of chosen factors. 4. Developing the experimental design matrix. 5. Conducting the experiments as per the design matrix. 6. Developing the mathematical model. 7. Calculating the coefficients of the model. 8. Checking the adequacy of the developed model by ANOVA method.
0308-0161/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0308-016 1(98)00137-9
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V. Balasubramanian, B. Guha / International Journal of Pressure Vessels and Piping 76 (1999) 283–290
Fig. 1. Dimensions of cruciform specimen.
9. Testing the significance of the coefficients by Student’s t-test. 10. Analysing the effect of joint dimensions on fatigue life. 2.1. Identification of predominant factors (joint dimensions) Many of the fatigue failure modes that can occur in the welded joints involve fatigue cracking from severe imperfections which are actually an inherent part of the joint. Root failures cannot be prevented unless the weld dimensions are appropriate to the plate thickness [6]. The lack of penetration discontinuity will affect the fatigue behaviour of the fillet welds when it exceeds a critical value of half of the plate thickness to be welded [7]. For partial penetration joints, the situation is more complex as the load carrying area of the weld will depend upon the depth of the penetration [8]. From the literature [9,10] and the previous work done in our lab [4,5] the predominant factors which are having influence on fatigue life of cruciform joints have been identified. They are: 1. the ratio between initial LOP (refer Fig. 1) size (2a) to fillet width (2W), 2. the ratio between leg length (L) to plate thickness (Tp), 3. fillet angle (u ) or weld profile and 4. stress range (Ds ). 2.2. Fabrication of the joints A high strength quenched and tempered steel (ASTM 517 ‘F’ Grade) of weldable quality in the form of rolled plates of
8 mm thickness is used as the base material throughout the investigation. This material is widely used for welded constructions of all kinds such as pressure vessels, penstocks, bridges and structures as well as transport vehicles, hoisting and earthmoving equipments utilised in many different types of climatic conditions. Cruciform joints were fabricated using SMAW process with matching weld metal consumable (AWS E11018-M). The chemical composition and mechanical properties of the base metal and weld metal are given in Tables 1 and 2. All the four fillets, forming the joint, were made identical leaving an unfused gap, i.e., LOP, was controlled by prior machining process (bevelling). The various root faces enabled the joints to have different LOP lengths after welding. The fillet leg length (L) was varied by controlling number of weld passes. Weld profile (or fillet angle) was varied by controlling the arc movement. Further details of fabrication and specimen sectioning were given elsewhere [5]. The dimensions of the test specimen are shown in Fig. 1. 2.3. Finding the limits of the factors Trial experiments were conducted on large number of specimens to find out the feasible limits of the aforementioned chosen factors in such a way that the failure should occur from the LOP defect. From the experimental results, the following conditions were existed: 1. If a/W ratio is less than 0.25, then toe failure is predominant. 2. If a/W ratio is in between 0.25 and 0.45, then root (LOP) failures are more common. 3. If a/W ratio is greater than 0.45, then failure occurs quickly (⬍10 4 cycles) i.e., the failure is considered to be Low Cycle Fatigue (LCF) failure. 4. If L/Tp ratio is less than 0.4, then failure occurs much faster, like in the previous condition. 5. If L/Tp ratio is in between 0.4 and 1.2, then failure occurs from root(LOP) region. 6. If L/Tp ratio is greater than 1.2, then the failure is from toe region only. 7. If the weld profile is either concave (u ⬍ 45⬚) or straight (u 45⬚), then failure occurs from the root (LOP) region. 8. If the weld profile is convex (u ⬎ 45⬚), then toe cracking is predominant. 9. If the stress range is less than 120 MPa, then most of the specimen endured upto 10 7 cycles. 10. If the stress range is greater than 280 MPa, then majority of the specimen failed within 10 4 cycles (LCF region).
Table 1 Chemical composition (wt.%) of the base and weld metals Material
C
Si
Mn
P
S
Cr
Mo
Ni
Cu
Co
V
Base metal Weld metal
0.19 0.08
0.72 0.5
0.95 1.6
0.01 0.01
0.002 0.03
0.8 0.35
0.35 0.27
0.07 1.73
0.03 0.1
0.004 0.01
0.002 0.001
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Table 2 Mechanical properties of the base and weld metals Material
Yield strength (MPa)
Tensile strength(MPa)
Vicker’s hardness (v30 kg)
Impact value (J)
Percentage of elongation (%)
Base metal Weld metal
690 740
790 845
210 320
110 175
19 24.5
By considering all the aforesaid conditions, the feasible limits of the factors were chosen in such a way that the failure should occur from LOP defect and in High Cycle Fatigue (⬎10 4 cycles) region and they are presented in the Table 3. For the convenience of recording and processing the experimental data, the upper and lower levels of the factors are coded as ⫹2 and ⫺2, respectively and the coded values of any intermediate levels can be calculated by using the expression in Ref. [11]: Xi
2X ⫺ Xmax ⫹ Xmin Xmax ⫺ Xmin =2
machine (SCHENCK 200 kN capacity) with a frequency of 30 Hz under constant amplitude loading (R 0). Care was taken to see that the load was axial and no bending component was present in the joint. The number of cycles to complete failure of each specimen was recorded and presented in the Table 4. Even though the experiments were conducted in a random order, the Table 4 shows the standard order to avoid systematic errors creeping the results. 2.6. Developing the model
where Xi is the required coded value of a factor of any value X from Xmin to Xmax; Xmin is the lower level of the factor and Xmax is the upper level of the factor.
Representing the fatigue life of welded cruciform joints containing LOP defects by Nf, then the response function (cf. Ref. [13]) can be expressed as
2.4. Developing the experimental design matrix
Nf f a=W; L=Tp ; u; Ds f A; L; P; S:
Owing to slightly wider ranges of the factors, it was decided to use a five level, central composite, rotatable design matrix to optimise the experimental conditions. Table 4 shows the 31 set of coded conditions used to form the design matrix. The first 16 experimental conditions (rows) have been formed for the main effects by using the formula 2 nc⫺1 for the low (⫺1) and high (⫹1) values; where ‘nc’ refers to the column number. For example, in Table 4, the first four rows are coded as ⫺1 and the next four rows are coded as ⫹1, alternatively, in the third column (⬗ nc 3 and 2 3⫺1 4). All chosen variables at the intermediate level (0) constitute the centre points and the combinations of each of the variables at either its lowest (⫺2) or highest (⫹2) with the other three variables of the intermediate levels constitute the star points. The method of designing such a matrix is dealt with elsewhere [11,12].
The model selected was a second degree response surface expressed as follows:
2.5. Conducting the experiments The fatigue experiments were conducted as per the conditions dictated by the design matrix (Table 4), by using a mechanical resonance controlled universal fatigue testing Table 3 Important factors and their levels S. no.
1 2 3 4
(0)
(⫹1)
(⫹2)
a/W L/Tp u Ds
0.35 0.8 37.5 200
0.4 1.0 45 240
0.45 1.2 52.5 280
— — deg MPa
⫹ N22 L2 ⫹ N33 P2 ⫹ N44 S2 ⫹ N12 AL ⫹ N13 AP ⫹ N14 AS ⫹ N23 LP ⫹ N24 LS ⫹ N34 PS: The values of the coefficients were calculated by regression with the help of the following equations (cf. Ref. [11]): X X N0 0:142857 Y ⫺ 0:035714 Xii Y; Ni 0:041667
0.25 0.4 22.5 120
0.3 0.6 30 160
X Xi Y;
X X Nii 0:03125 Xii Y ⫹ 0:003720 Xii Y ⫺ 0:035714 X Y; Nij 0:0625
Factor Notation Unit Levels (⫺2) (⫺1) A L P S
Nf N0 ⫹ N1 A ⫹ N2 L ⫹ N3 P ⫹ N4 S ⫹ N11 A2
X
Xij Y;
Student’s t-test [14] was applied to eliminate the significant coefficients without sacrificing much of the accuracy to avoid cumbersome mathematical labour. After determining the significant coefficients, the final model was developed including only those coefficients and is given in the following text:
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Fatigue life,
Table 4 Design matrix and experimental results Expt. no.
A (X1)
L (X2)
P (X3)
S (X4)
Nf (life) × 10 5 cycles
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺1 ⫹1 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 ⫺1 ⫺1 ⫹1 ⫹1 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 ⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 0 0 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0 0 0
⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 ⫹1 0 0 0 0 0 0 ⫺2 ⫹2 0 0 0 0 0 0 0
8.6 5.2 17.3 10.8 11.4 7.9 19.2 12.8 1.2 0.7 2.5 1.6 1.7 1.0 3.3 2.2 9.4 4.2 3.7 6.3 2.8 7.8 24.2 0.8 7.0 7.2 6.4 5.8 6.8 6.2 5.6
Table 5 Analysis of variance (ANOVA) test results First order terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Second order terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Error terms Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Lack of fit Sum of squares (SS) Degrees of freedom (d.o.f) Mean square (MS) Fratio (MS of lack of fit)/ (MS of error terms) F(10,6,0.99) from table Whether the model is adequate or not?
782.72 4 195.7 129.64 10 12.96 2.2 6 0.37 18.35 10 1.84 4.96 7.87 Yes, adequate
Nf {6:44 ⫺ 1:39 A ⫹ 1:55 L ⫹ 0:9 P ⫺ 5:24 S ⫺ 0:29 A2 ⫺ 0:47 L2 ⫺ 0:4 P2 ⫹ 1:4 S2 ⫺ 0:43 AL ⫹ 1:04 AS ⫺ 1:38 LS ⫺ 0:45 PS} × 105 cycles: 2.7. Checking the adequacy of the developed model The adequacy of the model was then checked by using the Analysis of Variance (ANOVA) technique [11–14]. As per this technique, if the calculated value of the Fratio of the developed model does not exceed the standard tabulated value of Fratio for a desired level of confidence (say 99%), then the model is considered to be adequate within the confidence limit. ANOVA test results are presented in the Table 5. Further, the experimental data and the predicted data by the using the aforesaid model are plotted as shown in Fig. 2, which indicates a good correlation. 3. Discussions Using the developed model, the fatigue lives have been predicted for different combinations of joint dimensions and they are presented in the graphical form (Figs. 3–6). The effects of joint dimensions on fatigue life have been discussed in the following sections. 3.1. Effect of L/Tp ratio on fatigue life The L/Tp ratio, i.e., the ratio between leg length and plate thickness, decides the final weld size of the cruciform joints. The effect of L/Tp ratio on fatigue life, for different values of LOP size and fillet angle, have been shown in Figs. 3–6. From the graphs, it is evident that the higher the L/Tp ratio, the larger will be the fatigue lives and viceversa. The reason can be easily understood from the following Stress Intensity Factor (SIF) range expression, for a load carrying cruciform joint at the apex of a root (LOP) defect [7]. DK
DsA1 ⫹ A2 a*pa sec pa*=21=2 1 ⫹ 2 L=Tp
where Ds a a* ( a/W) L/Tp A1, A2
nominal stress range; half crack (LOP) length; normalised crack length; weldsize; are constants which depend on weldsize.
From the earlier expression, it is clear that the SIF range is inversely proportional to the L/Tp ratio, i.e., if the L/Tp ratio is more (for larger welds), the SIF range value will become lower and hence the crack initiation, crack propagation and failure will be delayed. More importantly, the variations in the fatigue crack growth behaviour and fatigue
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Fig. 2. Correlation graph.
Fig. 3. S–N curves.
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Fig. 4. S–N curves.
Fig. 5. S–N curves.
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Fig. 6. S–N curves.
life is mainly attributed to the difference in the number of weld passes involved during the fabrication of the joints. In multipass deposits (larger L/Tp), the weld metal consists partly of a tertiary transformation micro structure because in multipass welding, each successive bead tempers previous ones, consequently the secondary micro structures, such as pro-eutectoid (Grain boundary) ferrite, side plate ferrite and acicular ferrite, will partly be heated into the region. This extra transformation results in the formation of tertiary micro structure. This structure normally has higher toughness than the non-transformed weld metal (the secondary micro structure) [15,16]. Further, the larger welds are having more weld metals and hence the fatigue crack has to propagate longer distance before final failure to occur. 3.2. Effect of a/W ratio on fatigue life The a/W ratio, i.e., the ratio between initial LOP size and the fillet width, decides the defect size in cruciform joints. The effect of LOP size on fatigue life, for different values of L/Tp and u have been depicted in Figs. 3–6. From the graphs, it is clear that the lower the LOP size, higher will be the fatigue life and vice versa, for any weld size and fillet angle. This is caused by the fact that the joints which contains smaller defects will give longer life than its counterpart. It is also evident from the SIF range expression, given earlier, that the SIF range value is directly proportional to the a/W (a*) and hence the joints with smaller LOP
defect will endure more number of cycles than its counterpart. If the LOP defect size is small, then crack initiation will be delayed as a result of the lower values of SIF range but the reverse will be the result, if the LOP defect size is large. Moreover, the crack has to propagate a smaller distance, in comparison, when a/W is larger, as a result of the reduced effective fillet width. 3.3. Effect of weld profile on fatigue life The fillet angle will decide the weld profile in cruciform joints. Figs. 3–6, reveals the effect of weld profile on fatigue life of cruciform joints, for various L/Tp and a/W values. From the graphs, it is inferred that the straight fillets (u 45⬚) are superior compared to concave fillets but the difference is very small. This can be easily understood from the following SIF range expression for the cracks emanating from toe region [17] DK
Ms × Mt × Mk × Ds pa1=2 f0
where Ms Mt Mk
f0
correction factor for the effect of a free surface; correction factor for the effect of plate thickness; correction factor for the effect of stress concentration owing to the weld toe angle; correction factor for the effect of crack front shape.
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From the aforesaid expression it is obvious that the fillet toe angle is directly proportional to the SIF range values at weld toe. In load carrying cruciform joint, if the fillet angle is more than 45⬚, then the SIF range is sufficient to initiate and propagate a fatigue crack and cause failure in the joint prematurely as a result of its high level of stress concentration effect near the toe region [18]. If the fillet angle is less than 45⬚, then the SIF range is not sufficient to initiate a fatigue crack from toe region but the SIF range at the tip of LOP defect is increased as a result of the reduction in effective weld size. Further, in the concave fillets less weld metal is available to resist the propagation of fatigue crack and hence the failure occurs somewhat earlier than in the straight fillet welds. 4. Conclusions
1. By using this new model, the fatigue life of shielded metal arc welded cruciform joints containing LOP defects can be predicted at 99% confidence level. The accuracy of the predicted values is improved by considering more number of experimental conditions than the earlier investigation. 2. Though the number of experimental conditions are more, factorial experimentation technique is still economical to predict the effects of various factors on fatigue life by conducting optimum number of experiments. 3. Moreover, the ANOVA technique is convenient to check the validity of the developed model and the Student’s ttest is very simple to identify the significance of main effects and interaction effects of various combinations of factors. 4. The effect of cruciform joint dimensions on fatigue life have been analysed in detail. Larger weld sizes, smaller LOP sizes and straight profile fillet welds are showing better fatigue lives compared to other combinations.
Acknowledgements The authors are grateful to the Indian Institute of Technology, Madras, Chennai-36, for the support rendered and for making available the facilities of Metal Joining
Laboratory and Mechanical Testing Laboratory of Metallurgical Engineering Department. The authors are indebted to M/s. Bharat Heavy Electricals Ltd., Ranipet, Tamil Nadu for the materials supplied to carryout the investigation.
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