Plastic Collapse Behavior and Remaining Life Assessment of Statically Indeterminate Cracked Beam Using the Limit Analysis Technique

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1 Journal of Applied Science and Engineering, Vol. 15, No. 4, pp (01) 333 Plastic Collapse Behavior and Remaining ife Assessment of Statically Indeterminate Cracked Beam Using the imit Analysis Technique Sung-Po iu Department of Mechanical Engineering, Ching Yun University, Jung-i, Taiwan 30, R.O.C. Abstract This paper develops an analytical model for the plastic collapse of a statically indeterminate rectangular beam containing a crack. imit analysis, elastic-plastic fracture mechanics, compliance, and J-integral concepts are used to study J IC and dj/da, which influence the crack propagation in this study. The relations among the plastic hinge, applied load, linear displacement, rotational angle, and crack growth lead to a better understanding of the problem. The BB (eak-before-break) characteristic of the statically indeterminate rectangular beam is valid if the crack propagates before plastic collapse. Unstable ductile fractures occur when the crack propagates before plastic collapse or when dj/da is smaller than the minimum critical value. The life span of the crack extension to collapse can be computed by the 4 th order of the Runge-Kutta method. The information provided in this study can be applied to safe and reliable design structures. These analyses and design strategies developed in this paper are useful for the safety performance of a structural beam under crack deformation. Key Words: Elastic-Plastic Fracture Mechanics, Crack Propagation Resistance, Crack Growth, Statically Indeterminate Beam, eak-before-break 1. Introduction *Corresponding author. spliu@cyu.edu.tw Most research in the past twenty years has dealt with forces, displacements, buckling, torsion, etc. in structural mechanics; the deformation analysis was considered a boundary value problem [1 3]. The principles of virtual work, energy, and limit analysis of plastic structures helped solve many other problems in structural engineering. The study of this field became exciting, particularly since Van den Broeck proposed the limit-design theory [4]. The limit-design theory is valuable for its framework structure. However, the traditional research of limitstrength analysis had not been applied to the cracked system, which means there was no consideration of fracture mechanics. This type of research is urgent for the statically indeterminate framework structure. Based on the reports of previously published literature, the relationship between plastic-collapse load and plastic-collapse displacement by combines the limit analysis theory and the elastic-plastic fracture mechanics that can be applied to the constrained condition of optimization by minimizing the crack extension resistance dj/da [5]. As a result, this paper proposed an innovative method to analyze and evaluate the loading status and plastic collapse behavior of statically indeterminate framework structures containing a crack. The present model and analysis deals with three considerations that must be noticed and solved: (1) The structure and support can generate some degree of deformation in machines and structures relating to energy conversion. These structures naturally become statically indeterminate framework. It is necessary to understand the mechanism of plastic behavior, of collapse, and the interactive influence of each member in order to evaluate the true safety margin and create a design based on the eak-before-

2 334 Sung-Po iu Break (BB) phenomenon. () The fracture assessment diagram [6] is a common means to evaluate the integrity of a cracked machine element. Although the extremely tough material satisfies the condition of plastic hinge formation, the evaluation of the safety margin under the same condition is still unclear. (3) A statically indeterminate structure becomes a plastic collapse mechanism when it is overloaded. Information such as fracture toughness and crack size requires a clear knowledge in order to prevent a non-ductile fracture. In this paper, the compliance and J-integral techniques are applied to analyze the collapse load, plastic collapse displacement, and the rotational angle of a statically indeterminate rectangular beam containing a crack. The critical value of dj/da for crack extension resistance and the sustaining life of crack extension until plastic collapse can therefore be estimated through the proposed method. The relations among the physical parameters, characteristics, and plastic behavior of a general statically indeterminate cracked structure can be sufficiently understood. Several examples are proposed to illustrate the existing model, analytical approach, and the plastic behaviors of the cracked structure.. Analysis of Plastic oad and Displacement Utilizing the simple compliance and J-integral, the plastic collapse behaviors of a rectangular beam were analyzed in this paper by combining the limit-analysis theory and elastic-plastic fracture mechanics. Figure 1 shows a statically indeterminate rectangular beam that contains a crack, with a uniformly distributed load q. The locations are at the middle point or at one of the fixed ends shown on the side view, where b, w, and a, indicates the width, height, length, and crack depth of the part through wall of the beam. Some useful parameters that appear in the formula are flow stress of the material fs, Young s modulus E, and the moment of inertia of the beam I. The basic assumptions are described as follows [7]: 1. The total plastic moment of the beam without or with a crack is represented as M cp and M p, respectively. (). The ideal moment-curvature relationship of the beam with a crack is indicated in Figure (a). 3. The beam will rotate freely around the plastic hinge when the moment is beyond the plastic moment M p or M cp. 4. The area outside of the plastic hinge is the elastic area. 5. The influence of secondary induced stress can be neglected when the deflection is large. Axial force, shear force, and residual stresses do not influence the moment-curvature relationship. 6. The propagation analysis of a cracked beam meets the following J-integral where the crack propagates according to the J-R curve when J-integral reaches J IC. (3) The crack resistance curve is called the J resistance or J-R curve, which is equivalent to the R curve. The J-R curve is a straight line in which the crack blunting phase and the crack extension resistance dj/da are neglected, as shown in Figures (b) and (c). The rotational angle in a cracked beam can be computed by the increment of the bending compliance [8] under the bending moment M. In order to simplify the calculation, we define a non-dimensional parameter which is equal to 3EI /. The relationship between the rotational angle D of the crack at the middle point D and the statically indeterminate moment M D can then be written as: (4).1 Deformation Behavior of Elastic imit The moment analysis of point A, B and D for an elas- (1) Figure 1. A statically indeterminate rectangular beam with a crack.

3 Plastic Collapse & Remaining ife of Statically Indeterminate Cracked Beam 335 Figure. Plastic behavior and J-R relation. (a) Ideal plastic behavior. (b) Realistic model. (c) Ideal model. tic and statically indeterminate beam (shown in Figure 1) can be stated as: (5) (6) If the beam has a crack in the middle section, then either point A or D reaches its complete plastic moment depending on whether value of M D /M CP or M A /M P is larger. When the plastic hinge is built at the middle point D, the condition of forming the first plastic hinge is when M D equals M CP. The deflection D1 at the middle point can be obtained by the superposition method as follows: (7) The rotational angle D1 at point D by superposition method can be obtained as follows: (8) Applying Eq. (3), the J-integral at point D can be obtained as follows: (9). Deformation Behavior of the Second Plastic Hinge After the first plastic hinge occurred at the middle point, the reaction forces at A (and B)orD is represented as R A (and R B )orr D, respectively. et q and R A (and R B, R A = R B ) indicate the increment of the load and reaction force at point A (and B), respectively. The formation of the second plastic hinge, from the relation of M P = M A + M A1, M A1 is represented increment of moment at points A from the first plastic hinge to the second plastic hinge formation, q can be consequently computed through the following: (10) Along with the q and M cp, the changing increment of D and D are as follows: (11) (1) The increment of J-integral from Eq. (3) at point D can be obtained as follows: (13) The J-integral value from J D = J D1 + J D after the first plastic hinge can be represented as: (14).3 Deformation Behavior of Collapse Mechanism The final total displacement and total rotationalangle, as a collapse occurs at the point D, are given as follows: (15) (16)

4 336 Sung-Po iu The collapse load can be written as: (17) at point A. When the crack propagates following the J-R relation (shown in Figure (c)) and the J-integral reaches J IC, we can obtain D from Eq. (14) as below: After the collapse occurs, the load does not increase. The relationship between the increment of displacement D and rotational angle D of point D, taking into account the force equilibrium and moment equilibrium of the beam on both sides, is written as: (18).4 Analytical Results A statically indeterminate rectangular beam is shown in Figure 1, and the geometrical configurations are: w = 100 mm, b=50 mm, and = 3000 mm. The crack depth of the part-through wall a equals 40 mm, the yield stress ys equals 94 MPa, the Young s modulus E equals 05,947 MPa (which is computed from ) and moment of inertia I is /1 mm 4. The nondimensional bending compliance is The crack extension resistance dj/da is 0 MN/m, which is smaller than piping or pressure vessel materials with 110 MN/m for STS4 and 178 MN/m for SUS304. The elastic-plastic fracture toughness J IC is 50 kn/m. The issue is to find the displacement and rotational angle at the middle section of the beam when the first plastic hinge is formed at point D and the second plastic hinge is formed at point A. What is the displacement and rotational angle of point D with the crack growth? From the computation of Eq. (1), (), (5) and (6), we can get M D /M CP > M A /M P so that the first plastic hinge occurs at point D. Consequently, the load, displacement M P and rotational angle at the point D are q , MP MP D and D , in that 100 EI 10EI order. The result is shown in Figure 3 at point D. When the second plastic hinge is created at point A, according to Eq. (15) to (17), the collapse load, total displacement and total rotational angle of point D are calculated as q CR M P total MP total, D 4 and D = EI MP, correspondingly. The result is shown in Figure 3 10EI (19) As a result, we can obtain q =4( D ) from Eq. (11) 3 and (13), and D ( D) can be obtained from 16 Eq. (1) and (13). Therefore, when the crack begins to propagate at point D,theloadq IC, displacement IC and rotational angle IC can be computed: (0) (1) () The J 1C of the next crack propagation can be computed ( i 1) ( i) by JIC JIC J and a (i+1) = a (i) + a. In this paper, J and a are MN/m and 0.1 mm, respectively. Figure 3 shows the relationship between uniformly distributed load q and the displacement of the middle point, as well as the relationship between the rotational Figure 3. q- - diagram of middle section with a crack at middle point.

5 Plastic Collapse & Remaining ife of Statically Indeterminate Cracked Beam 337 angle of the cracked section and the displacement of the middle point. The solid line indicates the relation of q and without crack growth. The other lines marked with, and represent the relations of q- and - of crack growth with a different J IC. Using the different elastic-plastic fracture toughness J IC after the first plastic hinge, the increasing load is proportional to the displacement until plastic collapse, so eak-before-break (BB) is valid. After the plastic collapse, the more the load applies, the smaller the displacement becomes. Moreover, the expanding rate of rotational angle decreases after the plastic collapse. The BB criterion is based on the assumption that a surface flaw is approximately semi-circular when it pops through, implying that it develops a through crack of total length equal to twice the thickness. On the other hand, the results of cracks at the fixed ends are indicated in Figure 4. The issue is to find the displacement and rotational angle at the fixed section of the beam when the first plastic hinge is formed at point A, the second plastic hinge is formed at point B and the third plastic hinge is formed at point D. The initiation point of a crack and the sequence of formation of a plastic hinge are different from that of the middle crack in Figure 3, using the different elastic-plastic fracture toughness J IC (50, 150, 350 kn/m) after the first plastic hinge, the increasing load is proportional to the displacement until plastic collapse, so eak-before-break (BB) is valid. After the plastic collapse, the more the load applies, the smaller the displacement becomes. Moreover, the expanding rate of rotational angle decreases after the plastic collapse. the energy required for crack growth so the crack will extend gradually even if the load does not increase. The increasing of crack growth not only accelerates the fracture of the structure, but also decreases the endurance of the loading capacity occurring in the unstable fracture. In this case, the increment of load q is smaller than zero. To prevent the unstable ductile fracture, the value of dj/da has to be greater than the smallest critical value. In other words, the increment of load q must be greater than or equal to zero to avoid the unstable ductile fracture. For example, forming a plastic hinge from cracked point D to point A (or B), the J-integral of crack propagation from Hutchinson and Paris [10] is stated as follows: (3) The gradient of J-R curve with crack propagation is written as: (4) Substitute Eq. (3) with Eq. (4) and obtain the following result: (5) Substitute Eq. (6) with Eq. () and (4) and get the result in the following: 3. Analysis of Crack Extension Resistance 3.1 Analytical Method and Process After the load increases and the crack gradually propagates, the final fracture takes place through cleavage or ductility due to fatigue or stress corrosion as the crack develops to a certain size. A cleavage fracture is usually linked with a small plastic deformation, called brittle fracture. The growing ductile crack contains stable and unstable ductile fractures. The relationship between the displacement of structural element and the rigidity of load system in light of the J-R curve of ductile fracture mechanics [9], evaluates whether the fracture is a stable or unstable extension. The energy released after crack growth is larger than Figure 4. q- - diagram of middle section with a crack at fixed end.

6 338 Sung-Po iu (6) Then the derivative of Eq. (6) combining Eq. (5) can be written as: (7) According to dq 0, the condition of crack extension resistance can be stated below: (8) The left-hand side of the above equation represents the dj/da of the structural materials. The right-hand side of the above equation equals dj critical /da, which indicates the minimum critical value of dj/da for the stable ductile fracture. As Eq. (8) shows, the bigger the length and yielding stress y of the statically indeterminate beam is, the smaller the bending rigidity EI is, which will be great for the dj mat /da limit-value so the limited load will not reduce. In other words, the behavior of the load obtained versus the displacement after the crack growth, is not merely influenced by the dj mat /da and J 1C values, but, y and EI values will also influence it. Similarly, the initial crack at the fixed end on the left produces by plastic hinges formed of the statically indeterminate beam before plastic collapse is formed in two stages. The first plastic hinge formed, derives from Eq. (9). The second plastic hinge formed, derives from Eq. (30). (9) (30) comparing the two above-mentioned equations with Eq. (8), we see based on the crack growth situation and not allowing the dj mat /da condition loading to reduce, the initial crack at the middle point is more serious (where the dj mat /da value is much big). But, the initial crack at the fixed point is slower, because there are more numbers of statically indeterminate (its dj mat /da value is smaller). 3. Analytical Results The problem description and the parameters are the same as above section.4. Determine the critical value dj/da of stable ductile fracture with crack propagation. What is the behavior of a crack extension with J = 0.01 MN/m and a = 0.1 mm? The minimum critical value of a stable ductile fracture dj critical /da can be obtained from Eq. (8) as MN/m. The load q IC is 9.97 M P / at the middle point when a crack begins to propagate. The displacement 1C Mp and the rotational angle 1C are 389. and 100 EI Mp 0554., respectively. The unstable ductile fracture 10 EI happens because dj/da (0.1 MN/m ) is less than the critical value. From Eq. (7), the value dq is 053. p which M is smaller than zero and coincides with the abovementioned unstable ductile fracture. The second load of crack propagation consequently follows the formulation of q = q IC + dq. The other propagating points can also be computed by means of this process. The increment of rotational angle and displacement of crack propagation can be obtained by applying a similar method. Figure 5 shows the relation between the load and displacement when the crack occurred at the middle point with different extension resistance dj/da. The line marked with represents the minimum critical resistance dj/da (1.761 MN/m ). The load increment accompanying the crack propagation in this situation is very small. When dj/da equals 0.1 MN/m, which is smaller than the critical value, the load decreases at the beginning of crack propagation. This phenomenon of load-drop represents the unstable ductile fracture characteristic. When dj/da equals 50 MN/m, which is larger than the critical value, the crack always grows stably. On the other hand, the analysis of a crack at the fixed end is indicated in Figure 6. The sequence of plastic hinge formation and the critical value dj/da (.565 MN/m ) are different from that in Figure 5. When dj/da equals 0.1 MN/m, which is smaller than the critical value, the load decreases at the beginning of crack propagation. This phenomenon of load-drop represents the unstable ductile fracture characteristic. When dj/da

7 Plastic Collapse & Remaining ife of Statically Indeterminate Cracked Beam equals 50 MN/m, which is larger than the critical value, the crack always grows stably. The analytical analysis shown in Figures 5 and 6 has the same tendency to follow the experimental results of Figure 3, written in the reference [11]. Both indicate the crack propagation along circumferential direction as well as decreasing loading that occurs at the peak loading as crack happen. The unstable ductile fracture phenomenon and BB characteristic, which appeared in Figures 5 and 6, are the same in the experimental results of SI-1 and SI- curves, as shown in Figure 7 in the reference [1]. When the crack extension resistance dj/da maintains Figure 5. q-d diagram of middle section with a crack at middle point. Figure 6. q-d diagram of middle section with a crack at fixed end. 339 a high uniform value (such as in Figure 6, dj/da = 50 MN/m), the crack grows, and the difference between the increase-rate of the bending moment at point B and the decline-rate of the bending moment at the cracked section is large. Under the circumstance, as the crack grows, the total plastic moment at the cracked section will decrease, due to the sufficiency of redistribution for the bending moment. Thus, we realize that after crack growth, the limiting load will not decrease. On the other hand, while the crack grows and the crack extension resistance dj/da is under a critical value (such as in Figure 6, dj/da = 0.1 MN/m), the declinerate of the bending moment at the cracked section is large. Because of the insufficiency of redistribution for the bending moment, the reduction of dj/da caused a small rotational angle; therefore, the crack growth would become large [3]. To observe the singular behavior of a statically indeterminate beam, it is required to analyze a statically determinate beam with a crack under a uniformly distributed load q. Figure 8 shows the diagram of load q and displacement d of the middle point with a crack on a statically determinate beam. The elastic-plastic fracture toughness value JIC in this paper is 150 and 300 kn/m with a different crack extension resistance dj/da. The decline-rate of the load becomes slow when dj/da becomes smaller, as indicated in this diagram. Moreover, the rate of loading always decreases with crack propa- Figure 7. oad-displacement curve (SD-1, SD-, SI-1 NAD SI-) [14].

8 340 Sung-Po iu Sec. XI, Appendix C respectively [14]. From the above analysis, one requires transforming the J-integral value to a K value in order to apply Eq. (31). The relationship between J-integral value and K value is written as below: (3) Figure 8. q- diagram of middle section with a crack at middle point on a statically determinate beam. gation so that the BB phenomenon does not appear. Similarly, the BB characteristic shown in Figure 8 is not the same as the SD-1 and SD- curves in Figure 7 in the reference [1]. Figure 7 shows the arrow mark is represent the crack penetration, equal to the elastic-plastic fracture toughness value J IC, the rate of loading always decreases with crack propagation so that the BB phenomenon does not appear. 4. Remaining ife Assessment of Crack Extension Until Plastic Collapse 4.1 Analytical Method and Process A pre-existing defect or flaw can cause a small crack under a regular service loading. Such a small crack may not be a critical fracture, but it may gradually grow through the mechanisms of fatigue and corrosion. It will eventually reach a range in which the fracture occurs at the regular service stress [13], called a sub-critical flaw growth. The stress corrosion crack mutually influences the material property and its surrounding environment. The rate of the stress corrosion crack and the enduring time to failure are governed by the stress intensity factor K. As a result, the crack growth rate da/dt [14] increases by: (31) where the unit of Eq. (31) is inch/hour, and the value of parameters c and m are and.161,whichis obtained from ASME Boiler and Pressure Vessel Code. where the unit of Eq. (3) is ksi (inch) 0.5 and the Poisson s ratio is 0.3. The stress intensity increases gradually while the crack continuously extends. The enduring time of crack propagation until plastic collapse can be consequently computed by the 4 th order Runge- Kutta method and the analysis of the crack growth rate dt/da as follows: (33) (34) where Eq. (34) is the fundamental formula in numerical computation of the 4 th order Runge-Kutta method. One can substitute Eq. (3) into Eq. (33) and (34) to compute the enduring time from the crack initiation, through crack propagation, until structural collapse. The next section is a numerical example for life span of crack extension. 4. Analytical Results The problem description and the parameters are also the same as section.4. Compute the enduring time of crack propagation until plastic collapse. The load of q 1C (9.97 M P / ) had been obtained from section.4 at the beginning of the crack at the middle point D. In this problem, J and a are N/m and 0.1 mm, respectively, which means that dj/da = 0 MN/m M and a = 40.1 mm. The value dq (1) is p from Eq. (7), which is the increment load when the crack begins to propagate. When the crack extends, the second continuous point can apply q () = q IC + dq (1) and J () = J IC + J. One can substitute J () into Eq. (3) and (31) to compute the increment time t as hours. The same method and process can consequently be applied to compute each increment time and the accumulate time until q (i) goes to q cr, as shown in Eq. (17).

9 Plastic Collapse & Remaining ife of Statically Indeterminate Cracked Beam 341 Figure 9 shows the relation among the enduring-time of crack propagation, J-integral of propagation process, and the loading q. The line marked with,, and represents the relationship between time and loading q with different dj/da. The line marked with (1), () and (3) represents the relationship of J-integral and load q of crack growth. It is noted that the line marked with represents the minimum critical dj/da (1.761 MN/m ). The loading capacity of the crack propagation of a small dj/da is not bigger than that of a large dj/da. A relatively small dj/da results in a large time increment and long time propagation yielding slow crack propagation. When dj/da is relatively large, the structure may be collapsed in a short time due to the fast crack propagation. Moreover, a relatively small dj/da results in a large increment rate of J-integral. On the other hand, Figure 10 shows the diagram of a time-load-j of a crack at fixed ends where the minimum critical value dj/da (.565 MN/m ) is different from that in Figure 9. The other behaviors regarding crack growth are also similar to that in Figure 9. We shall do some experimental data included in the above figures for the comparison. 5. Conclusion In this paper, the variation among load, rotational angle, displacement, enduring time of crack extension to plastic collapse, and behaviors of a statically indeterminate rectangular beam containing a crack have been analyzed quantitatively. This is done by combining limit-analysis theory, elastic-plastic fracture mechanics, and by utilizing the simple compliance and J-integral. The conditions of load-drop can be derived as a function of J IC, dj/da and flow stress, span of beam, compliance, and a flexural rigidity of the structure when a crack grows before plastic collapse. The primary phenomenon obtained from this work is stated in the following: 1. The crack begins to propagate at the cracked section after the plastic hinge has been constructed.. The unstable ductile fracture appears when the crack begins to increase before the plastic collapse, because dj/da is smaller than a critical value. 3. The eak-before-break (BB) characteristic of the statically indeterminate rectangular beam is valid when the crack propagates before plastic collapse. 4. The eak-before-break (BB) of the statically determinate beam is undesirable when the crack spreads before plastic collapse. 5. The loading capacity obviously decreases right after the collapse mechanism arises. Due to the unstable ductile fracture yielding load decreases from the crack propagation when dj/da is smaller than the minimum critical value, the value of dj/da can be minimized in order to improve the safety of a structural design with a possibility of crack existence. In addition, the current life span of crack extension until collapse for a statically indeterminate structure is useful to improve the safety margin and the structural reliability. Figure 9. T-q-J diagram of middle section with a crack at middle point. Figure 10. T-q-J diagram of middle section with a crack at fixed end.

10 34 Sung-Po iu Acknowledgment The support received from the National Science Council, Taiwan, R.O.C. under the grant No. NSC 93-1-E , is gratefully acknowledged. Notation a crack depth of the partial-through wall b crack width of the partial-through wall c parameters, equal to da/dt stress corrosion cracking growth rate dj/da crack extension resistance E Young s modulus I moment of inertia of area J IC J-integral value of beginning propagation J mat material J, is the J consumed in crack extension resistance of material, or provides all the energy K stress intensity factor length of the beam M A bending moment at fixed end A M B bending moment of the rotational spring M cp bending moment of collapsing pipe is the total plastic moment under cracked condition M p total plastic moment of the non-cracked pipe Q uniformly distributed load q c collapse load R A supporting reaction force at fixed end A R B reaction force at support B t cracks propagation time t n step-time in the crack propagation w crack height of the partial-through wall a increment of crack depth q load increment under the crack propagation q B increment of load at points B from the first plastic hinge to the second plastic hinge formation q D increment of load at points D from the first plastic hinge to the second plastic hinge formation fs flow stress ys yielding stress p plastic rotation angle of cracked pipe rotational angle bending compliance non-dimensional bending compliance Poisson s ratio displacement from the elastic behaviors of plastic cr hinge formation, and plastic collapse collapse displacement References [1] Gerstle, K. H., Basic Structure Analysis, New York, Prentice-Hall (1974). [] Tauchert, T. R., Energy Principles in Structural Mechanics, New York, McGraw-Hill (1974). [3] iu, S.-P. and Ando, K., eak-before-break and Plastic Collapse Behavior of Statically Indeterminate Pipe System with Circumferential Crack, Nuclear Engng & Design, Vol. 195, pp (000). [4] Broeck, V. D., Theory of imit Design, New York, John Wily and Sons Inc (1948). [5] iu, S.-P., Remaining ife Assessment and Optimal Design of Statically Indeterminate Pipe System with Circumferential Crack, Int Communications in Heat and Mass Transfer, Vol. 37, pp (010). [6] Milne, I., Ainsworch, R. A., Dowling, A. R. and Stewart, A. T., CEGB Report, R/H/R6-Rev. 3 (1986). [7] Kihara, H., Plastic Design Method, Morikita Issue (1960). [8] Brown, W. F. J. and Srawly, J., ASTM STP, Vol. 410, pp (1966). [9] Machida, S., Ductile Fracture Mechanics, Tokyo Nikkan Industry News Inc (1984). [10] Hutchinson, J. W. and Paris, P. C., ASTM STP, Vol. 668, pp (1979). [11] Shibata, K., Kaneko, T., Yokoyama, N., Ohba, T., Kawamura, T. and Miyazono, S., Ductile Fracture Behavior and BB Evaluation of Circumferentially Cracked Type 304 Stainless Steel Piping under Bending oad, JHPI, Vol. 4, pp (1986). [1] Yoo, Y. S. and Ando, K., Plastic Collapse and BB Behavior of Statically Indeterminate Piping System Subjected to a Static oad, Nuclear Engng & Design, Vol. 07, pp (001). [13] Broek, D., Elementary Engineering Fracture Mechanics, New York, Martinus Nijhoff Publishers (1986). [14] ASME Boiler and Pressure Vessel Code. Sec. XI, Appendix C, NUREG-0313 (1995). Manuscript Received: Jan., 01 Accepted: Mar., 01