Limit load analysis for the piping branch junctions under in-plane moment

Transcription

1 International Journal of Mechanical Sciences 48 (2006) Limit load analysis for the piping branch junctions under in-plane moment Fu-Zhen Xuan, Pei-Ning Li, Shan-Tung Tu School of Mechanical Engineering, East China University of Science and Technology, 130, Meilong Street, PO Box 402 Shanghai , PR China Received 20 May 2003; received in revised form 21 July 2005; accepted 15 October 2005 Available online 22 ecember 2005 Abstract An approximate analysis approach for plastic limit load of piping branch junctions, by means of the relationship of the internal force between the main and branch pipes around intersection line, is presented in this work. The approach is built on the following process: based on the external force equilibrium condition, an equation between the limit load and internal force of the branch pipe around intersection is derived firstly. And then, taking this internal force as an external force acting on the intersection of the main pipe, the approximate solution of the internal force around the intersection on the main pipe is given as a function of the limit load. Finally, referring to the von-mises yield criterion, the limit load of component with two intersecting cylindrical shells is then obtained. In use of the proposed approach, a closed form of limit load solution for piping branch junction under in-plane moment is developed. Finite element (FE) models of the idealized piping branch junction with various diameters and wall-thickness of the main and branch pipes were analyzed by using nonlinear FE software. The limit loads from FE analysis, from the proposed solution and six experimental data of real piping branch junctions are compared. Overall good agreement between the different limit loads was observed which provides confidence in the use of the proposed formulae in practice. r 2005 Elsevier Ltd. All rights reserved. Keywords: Piping branch junction; Plastic limit load; In-plane moment 1. Introduction Corresponding author. Tel.: ; fax: address: fzxuan@ecust.edu.cn (F.-Z. Xuan). The limit load, which indicates the carrying capacity of structures, is an important parameter in performing the structural integrity assessment in use of FA-based approach and strength design [1]. In the ASME stress classification framework for pressure vessels and piping components design, stresses are classified as primary stress, secondary stress and peak stress with different admissible values which are determined from the stress state under limit load. Furthermore, it is specified in ASME code for Nuclear Vessels that the limits on local membrane stress intensity and primary membrane bending stress intensity need not be satisfied if it can be shown that the external loadings do not exceed 2/3 of the lower bound limit load. Therefore, the knowledge of limit loads of structures is very essential to engineering designer. For the ordinary structures, limit load calculation is very straightforward and some solutions are listed in most procedures [2 4]. However, for the piping branch junction, a complex configuration used in a number of industries such as power generations, chemical processing plants etc., there is very little information available for the limit load solutions qualified for practical applications. In the majority of cases, the load sustained by branch pipe is comprised of internal pressure and external loads. The magnitude and type of these external loads are normally difficult to estimate. However, a force or its effects could generally be decomposed to three forces and three couples acting at a specific location. In-plane moment is one of the important loading styles among the external loadings acting on the branch pipe, as shown in Fig. 1. On the models with a two-end supported main pipe, Schroeder [5] and Ellyin [6,7] have carried out the experimental study of in-plane plastic limit moment to provide the benchmark data for engineering treatment of the piping branch junctions in design codes. After this, a lower bound and /$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi: /j.ijmecsci

2 F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) Nomenclature mean diameter of the main pipe d mean diameter of the branch pipe k 1=½1 þðt=tþ 3 Š, modifying factor for effects of the opening weakening and branch pipe reinforcement; k ¼ 0:5, for equal-diameter piping branch junctions L length of the intersection line M L i In-plane plastic limit moment M L B s f td 2 plastic limit moment of a straight pipe with a same dimensions as that of the branch pipe M ov ð1=4þs f T 2, plastic limit moment capacity of a plate per unit length with thickness equal to the mean wall thickness of the main pipe m L i non-dimensionalized in-plane plastic limit moment with M L B M y ; M a ; M ya moments per unit length in the main pipe m y ; m a ; m ya non-dimensionalized moments per unit length in the main pipe with M ov. N ov s f T, plastic limit membrane strength of a plate per unit length with thickness equal to the mean wall thickness of the main pipe N y ; N a ; N ya membrane forces per unit length in the main pipe n y ; n a ; n ya non-dimensionalized membrane forces per unit length in the main pipe with N ov Q n ; Q m functions of membrane forces and moments contained in the yield surface t mean wall thickness of the branch pipe T mean wall thickness of the main pipe x; y; z Cartesian coordinates of the piping branch junctions Greek symbols a y b n s f s y circumferential coordinate of the main pipe (see Fig. 1) circumferential coordinate of the branch pipe (see Fig. 1) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 12ð1 n 2 Þ=ð 2 T 2 Þ, dimensional parameter of the main pipe shell Poisson s ratio of the material flow stress of the main pipe material yield stress of the main pipe material Fig. 1. Sketch for the shape of intersection line at mid-surface of the main and branch pipes. an upper bound solution of the limit load under this kind of loadings were developed by Ellyin and Turkkan [8] and Schroeder [9], respectively. Rodabaugh [10] compared Schroeder s upper bound with Ellyin s lower bound and found that, for d=40:63, the two theories have the wrong relationship to each other, i.e., the lower bound is higher than the upper bound. Furthermore, the two solutions have not yet been adopted in practice due to no explicit expressions for the limit load being available. For the limit load calculation of the complicated structures, the finite element method (FEM) is, currently, the best method because it can simulate various geometries and loadings [11,12]. To solve the time consuming problem in FE limit analysis, a static approach by Heitzer [13] and a kinematic approach by Liu et al. [14] were proposed by means of a direct iterative algorithm. By using the elastic compensation method, Nadarajah et al. [15] obtained the upper and lower bound limits interaction diagrams for a range of nozzle/cylinder intersections under in-plane moment loading. However, it is also attractive to have simplified and accurate estimation methods for engineering calculations. In the present work, a simplified solution of the plastic limit load for piping branch junction under in-plane moment is derived referring to the equilibrium equation between the internal force on main and branch pipes and the plastic failure criterion. In the end, finite element analyses of the idealized piping branch junctions as well as six experimental data of real piping branch junctions were employed to justify the proposed solution. 2. Mechanical model and failure criteria 2.1. Mechanical model simplification To stress main factors and facilitate the limit load analysis, the following simplifications are introduced for the mechanical model: (1) Taking the branch piping junction as two intersecting perfect cylindrical shells, for example, the ovality of the

3 462 ARTICLE IN PRESS F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) main and branch pipes and the machining error are all neglected. (2) The effects of weld reinforcement at two shell intersections and other reinforcing components are ignored. (3) An elastic-perfectly-plastic material is specified for both main and branch pipes. Both the elastic deformation and the section ovalization of pipe are not considered before reaching the limit state. To take the strain-hardening effectiveness into account, the flow stress is introduced as the material yield stress in the limit load analysis. Based on the proposed model, the coordinates of a point P locating on the intersecting line of mid-surface, as shown in Fig. 1, are therefore described by x ¼ 0:5d sin y ¼ 0:5 sin a; y ¼ 0:5d cos y, z ¼ 0:5 cos a. Using the linear integral method and referring to equation (1), the length L of the intersection line at midsurface is easily given by I L ¼ ¼ 2d dl L Z p=2 0 ð1þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " 1 d #," 2 sin 4 y 1 d # 2 t sin 2 y dy, ð2þ where dl denotes the increment on the intersection line and can be expressed by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dl ¼ _x 2 ðyþþ _y 2 ðyþþ_z 2 ðyþ dy. ots designate derivatives with variable y Plastic failure criterion For the in-plane moment, both the finite element (FE) analysis and the experimental study show that the piping branch junctions exhibit three kinds of failure mode [7,15]: (1) Collapse failure of the branch pipe. This usually happens on the piping branch junctions with a lower value of d= and the ratio of t=t far less than unity. (2) Global collapse of the branch junctions due to the plastic hinge forming along the intersection line. At this time, an elliptical yielding zone usually formed along the intersection line with an obvious deformation. (3) Contrast to the local instability occurred at the flank of the main pipe under the out-of-plane moment [16], local instability is also observed at the crotch of the main pipe for the piping branch junctions subjected to in-plane moment and with the large values of diameter/ thickness and t=t. In fact, most piping branch junctions utilized in the industry take on a large ratio of d= (usually d=40:5) and a small ratio of t=t (mostly t=t ¼ 1:0), and both failure modes (namely, collapse of branch pipe and local instability of the main pipe at the crotch) will not happen on them under the in-plane moment. They usually lose their load-bearing capacity due to a plastic hinge forming along the intersection line namely the second failure mode described above [5 7]. Thus it is of great engineering significance for investigations of the plastic load solution of piping branch junctions with such a failure mode. Accordingly, the plastic failure criterion of the piping branch junction with such a failure mode can be specified where the plastic limit state has been reached as soon as a plastic hinge forms along the whole intersection line at the main pipe side. The net section stress surrounding intersection needs to satisfy the yield condition suggested by Rosenblium [17] for a thin shell obeying von-mises yield criterion. It may be represented in a simple form: Q 2 n þ Q2 m ¼ 1, (3) where Q 2 n and Q2 m denote the functions of membrane forces and moments contained in the yield condition, respectively, and can be calculated using Q 2 n ¼ n2 y n yn a þ n 2 a þ 3n2 ya, Q 2 m ¼ m2 y m ym a þ m 2 a þ 3m2 ya. Here n y ¼ N y =N ov ;...; m y ¼ M y =M ov ;...; N ov ¼ s f T; M ov ¼ð1=4Þs f T 2 and s f is the flow stress. The bounds of this criterion with regards to an exact von-mises and its relation to other version of the liyushin yield criterion are discussed in Ref. [18]. 3. Limit load solution From the viewpoint of limit analysis, a lower bound to limit load of a piping branch junction will be available so long as a virtual stress field is found that satisfies the static force equilibrium condition and does not violate the plastic failure criterion. It does not demand that such a stress field satisfy the displacement condition. In another word, it is not necessarily for the components of virtual stress field to equal the real stress of the structure. Following such a principle, a virtual internal force field will be established at the intersection, which will be discussed in the following section. To perform the limit analysis, the piping branch junction is divided into branch pipe portion and main pipe portion at intersecting line, as shown in Fig. 2a. For the branch pipe portion, only taking the load equilibrium condition into account, a virtual axial internal force N z (it maybe the real force or not), assumed distributed uniformly along the intersection line of branch pipe, can

4 F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) Fig. 2. Schematic diagram for internal force decomposition of the piping branch junction under in-plane moment. be described as Z ¼ N z d cos y ds M L t L=2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p=2 u " ¼ N z d 2 cos y 1 d #," 2 sin 4 y 1 d # 2 t sin 2 y dy N z d 2. 0 Comparing to the numerical results from FE analyses [19], it is worth noting that such an approximation will lead to an increment of 7% for the average piping branch junctions utilized in petrochemical and power industries. However, such an approximation will cause a conservative limit load, which is to be expected in engineering practice. For the main pipe portion, the well-distributed internal force N z acting on the main pipe, translated from the branch pipe along the intersection line with the same magnitude but a reversed direction, can be decomposed into two components: a radial force N 0 z and a circumferential force N 00 z, as shown in Fig. 2b. N 0 z ¼ N z cos a; N 00 z ¼ N z sin a. (5) Along the opening fringe on the main pipe, the internal forces and moments are produced by the radial force N 0 z. The elastic-expressions of the internal moments and forces on a cylindrical shell due to a well-distributed radial load were presented by Timoshenko and Woinowsky-Krieger [20] thirty years ago. Based on these expressions, it is suggested that the internal force and moment on the main pipe along the intersection line could be obtained by considering a modifying factor that includes the effects of a opening weakening and the branch pipe reinforcement [19]. Therefore, the elastic internal moment and force produced by the radial component force N 0 z at the periphery of the main pipe shell opening are given by the following equations. The internal moment in the yoz plane is M y ¼ kn 0 z =ð4bþ (6) Following Timoshenko and Woinowsky-Krieger s suggestion [20], the internal moment in xoz plane can be ð4þ calculated by M a ¼ nm y. (7) The circumferential membrane force is N 0 a ¼ kn0 zb=4, (8) where b is a dimensional parameter of the cylindrical shell that is first derived from the governing differential equation of the shell by Timoshenko and Woinowsky-Krieger [20]. It should be pointed out that, according to Timoshenko and Woinowsky-Krieger s study, the components of the internal forces N y and M ay are insignificant and thus are neglected in this instance. Therefore, the elastic internal forces of the main pipe along the intersection line are summarized as: a circumferential internal force, N a ¼ N 00 z þ N0 a, an internal moment M y in the yoz plane and an internal moment M a in the xoz plane. Under the plastic limit state, the elastic-stresses will be uniform but should satisfy the external loading equilibrium condition and the plastic limit condition (Eq. (3)). Since it has been shown by FE analysis and the model test that a plastic zone will be formed along the entire intersection [7,15], it is reasonable to take the average value of these elastic-stresses on the entire intersection line as the stressexpression under the limit load. The components of stress under the limit load are therefore described approximately as follows: Circumferential internal force: N ap ¼ 4 Z ðn 00 z L þ N0 a Þ ds ¼ 2ML i L=4 L f 1 þ ML i bk f 2Ld 2. (9) Internal moment in yoz plane M yp ¼ 4 Z M y ds ¼ kml i L L=4 2Ldb f 2. (10) Internal moment in xoz plane M ap ¼ nm yp, (11) where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p=2 u " f 1 ¼ sin y 1 d #," 2 sin 4 y 1 d # 2 t sin 2 y dy, 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p=2 f 2 ¼ 1 d 2 sin 4 y dy. 0 Inserting M ov and N ov into Eqs. (9) (11) and taking the length of intersecting line L ¼ pd conservatively, the normalized internal force expressions are recast as n a ¼ N ap =N ov ¼ 2ML i pd f 1 þ ML i kb f 2pd 2 ðs f TÞ, (12) m y ¼ M yp =M ov ¼ ML i k 1 2pd 2 b f 2 4 s f T 2, (13) m a ¼ M ap =M ov ¼ nm y. (14)

5 464 ARTICLE IN PRESS F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) Introducing these normalized expressions into the plastic failure criterion (Eq. (3)) and taking Poisson s n ¼ 0:3, the estimation formula for the limit load of piping branch junctions subjected to the in-plane moment is then derived as m L i ¼ M L i =ML B ¼ p 2 jm i o, (15) where M L B is the plastic limit load of a straight cylindrical shell subjected to moment and can be easily derived according to the net section collapse method [21], often simplified as M L B ¼ s f td 2 for thin walled shells; j M i o is a weakening factor for the piping branch junction for the inplane moment qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p j M i o ¼ 1 C ðf 1 A þ 0:455k ffiffiffi B f 2 Þ 2 þ 0:2385Bf 2 2 k2, (16) where f 1 1 þ 1 3 A4 ; f 2 ðp=2þð A2 Þ, k ¼ 1=½1 þ C 3 Š; A ¼ d=; B ¼ =T; C ¼ t=t. Noting that the branch junction must be installed in a piping system in service, the straight pipe connected to the piping branch junction will collapse firstly if the normalized in-plane limit moment m L i is greater than unity, which makes it impossible to meet the plastic failure criterion (Eq. (3)) ultimately. Accordingly, an upper bound to the inplane limit moment should be introduced, i.e. m L i p1. 4. Validation and discussion 4.1. Validation with experimental results It should be pointed out that Eq. (15) is not a strictly theoretical expression for the plastic limit in-plane moment due to some assumptions and approximations being introduced during the previous analytical course. In order to validate its applicability in engineering practice, the experimental data collected from the published literatures [5 7] about piping branch junction under in-plane moment are adopted to compare them with the predicted results from Eq. (15). The testing models are constructed by welding two pipes. All pipes are carefully machined to the dimensions shown in Table 1 from thick-walled, cold-drawn round seamless carbon steel pipes and welded together perpendicularly, and annealed to remove the residual stresses. The typical fabricating process and welding technique were illuminated by Ellyin [6,7] and Schroeder [5] and not attended in this work. The mechanical properties of materials are obtained by using the standard tensile specimens sawed off from the tested junctions and shown in Table 1. Because the ultimate strength data of materials are not available in the published literatures, the flow stress is taken to be 1.25 times the yield stress s y for mild carbon steel in the limit load calculation according to the recommendation in API 579 [2]. The in-plane bending test was carried out in a specially designed test fixture where the two straight limbs were fixed at the free end. The force constituting the moment loading acting on the branch pipe end was generated by two hydraulically actuated pistons, whose orientation with respect to the base plate is adjustable using set screws in a frame supporting the attachment to the pipe holding the piston. isplacements were measured at the points of rigid caps which were free to move due to deformation or at points where plastic deformation was expected to occur close to the intersection. The experimental limit moments were determined based on the measured displacement load curves in use of the double angle of linear response method. A comparison of the experimental data and predicted results is shown in Table 1. As expected, for specimens reported in the existing articles (their failure mode is a global collapse due to the plastic hinges forming along the intersection line), there is a reasonable agreement between the experimental data and the predicted solutions with an average error of 3:56 percent. It is also worth noting that all predicted results are less than the experimental data which is acceptable in practice for the predicted value being conservative. It can therefore be concluded that, in the range of 0:52pd=p1:0 and 24p=Tp31 covered by specimens in Table 1, the new limit load solution is suitable for the engineering application. However, for the other branch junctions with structural dimensions beyond the discussed range, more experimental data are needed to justify its applicability FE numerical solutions validation To further assess the new closed form solution derived above, nonlinear FE analyses of the idealized piping Table 1 Comparison of experimental data with predicted results for a plastic limit in-plane moment Specimen d= =T t=t s y Experimental Predictions Error a References identification (mm) (MPa) data (kn m) (kn m) (%) T [5] T [5] T [5] B [6] [7] E [7] a Error expressed as (predicted experimental)/ experimental 100%.

6 F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) branch junctions over a wide range of dimensions have been carried out. The ANSYS (1999, version 5.6) finite element analysis package was used for nonlinear stress, strain and displacement analysis and post processing was used to calculate the limit load. A half model of the structure with a length of the junction limbs of 150 mm and a length of the attached straight pipe of 200 mm was taken because of its symmetry. The meshes were prepared using pre-processing program of ANSYS: 20 node 3-dimensional isoparametric solid elements (solid 95) were used for the main structure and 8-node isoparametric solid elements (solid 185) were used for the attached straight pipes. A typical mesh characteristic of the piping branch junction was depicted in Fig. 3. The moment load was applied via rotation of a master node attached to the nodes at the end of the branch pipe. The boundary condition of the FE model was designed by closing to the working condition of the piping branch junction, i.e., the nodes at the ends of the main pipe were constrained in the x and z directions (as defined in Fig. 3), but not in the y direction, and the symmetric boundary conditions were applied on the symmetric plane. A small deflection is assumed in limit analysis and the elastic/ perfectly-plastic material with a yield stress of 320 MPa and a Poisson s ratio of 0.3 is specified. Theoretical limit loads are strictly defined in terms of displacements as the load level at which continuous displacements occur without an increase in load, and correspond to that portion of a load displacement curve which is parallel to the displacement axis. In practice, however, no portion of a load displacement curve exists which is parallel to the displacement axis even after the limit state was reached due to large deformation and strain hardening. Various methods have been devised to use actual load displacement diagrams for the determination of the limit load [22,23]. In this report, the limit loads were obtained using the double angle of linear response method, i.e. by drawing a line at a slope of two-times that of the elastic portion of the rotation at the end of the branch pipe and moment curve, as shown in Fig. 4. Fig. 3. Typical FE mesh for the piping branch junction subjected to the in-plane moment. Fig. 4. In-plane moment against branch pipe rotation and 2-times linear response method for limit load. Before performing the limit load computation, the FE modeling technique was verified firstly by comparing the FE results with experimental data of the piping branch junction. The dimensions and mechanical properties of the specimen marked El in Table 1 was selected and the calculated limit load from FEM was 16:9 kn m. Comparing with the experimental limit moment of 16:13 kn m, the error is 4.8% which give us confidence in applying the FEM to evaluate the proposed limit load solution. In consideration of the equal-strength piping branch junctions with a definition of d=t ¼ =T being widely used in engineering and generally concerned by the researchers, in this work, such a type of component was thus selected and a total of 42 different FE equal-strength models was analyzed. The detailed dimensions of the FE models include d= ¼ 0:3, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, and =T ¼ 10, 20, 30, 40, 60, 80. The FE results and predicted solutions are shown in Fig. 5. It can be seen from the comparison that: (1) For FE model with a diameter ratio of d=40:5 and a moderate value of =T (e.g. 10o=To60), the failure mode of global collapse due to the plastic hinge formed along the intersection line was observed. The predicted solutions show a good agreement with the FE results with the maximum error being less than 10%, as seen in Fig. 5. (2) For FE models with high values of =T (e.g. =T480 in Fig. 5), failure of the piping branch junction is the result of the local instability of the main pipe at the crotch area, and the predicted solutions are greater than the FE results with the maximum error of 21% for the case of d= ¼ 1 and 80. Thus, it is concluded that Eq. (15) can give a rational evaluation for the in-plane limit moment of the piping branch junction with the global collapse failure, but cannot

7 466 ARTICLE IN PRESS F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) Fig. 5. FE results for the limit load of the piping branch junction subjected to in-plane moment and comparison with the predicted results from the proposed solution. be applied to those with the local instability of the failure mode Comparison with existing solutions In order to further elaborate the applicability and rationality of the proposed solution for limit load of the piping branch junctions under in-plane moment, a comparison is done with the study of other authors using either analytical or experiential approaches. Billington [24] has proposed an experiential formula for limit load of the T- joints under in-plane moment on the basis of the experimental data. Although the geometry of T-joint is different to the welded cylinder-cylinder intersection junction in pipeline investigated in this report, the experiential solution referred herein was used to perform a comparison with the proposed solution Fig. 6. Comparison of the proposed solution with other people s work ð=t ¼ 30; d= ¼ t=tþ. M L i =ML B ¼ 6:1T 2 =ðtþ. (17) For the existing solutions of Schroeder s upper bound [9] and Ellyin s lower bound [8] mentioned in Section 1, the only available direct comparison is for the dimensional parameters: =T ¼ 30, d= ¼ t=t. The two theories and the predicted limit loads from the proposed solution and Billington s work [24] are plotted in Fig. 6. The Schroeder curve is from Fig. 7(a) of Ref. [9], the Ellyin curve is from Fig. 3 of Ref. [8]. To provide a benchmark for the comparison, the corresponding experimental data listed in Table 1 are plotted in Fig. 6. From Fig. 6 it can be seen that, the proposed solution is below the Schroeder s upper bound line, as might be expected. However for the case of d=40:34, the proposed solution has the wrong relationship to Ellyin s lower bound line; i.e., the predicted limit load from Eq. (15) is lower than the lower bound. In the range of 0:2od=o1:0, the predicted limit load from Eq. (15) is above the Billington s experimental solution. This is comprehensible since apparent difference of structure exists between T-joint and piping branch junction concerned herein. It is worth noting that, compared to the test data of 1 and E 1, both Schroeder s upper bound and Ellyin s lower bound lead to a higher predicted limit load whereas Billington s equation is too conservative. The test data and the predicted results from Eq. (15) are very close to each in Fig. 6 and show a good agreement. 5. Conclusions For piping branch junctions subjected to the in-plane moment, three kinds of failure mode were observed in the test and FE analysis, i.e., collapse failure of the branch pipe, global collapse of the intersection due to a plastic hinge forming along the intersection line, as well as local instability of the pipe at the crotch area. In this work, limit load was analyzed for the piping branch junction with the second failure mode depicted before and the following conclusions were extracted: (1) An approximate analysis approach for plastic limit load of the piping branch junctions was proposed. Such a method involves the definition of plastic failure criterion and the internal forces determination traveling along the intersection line of the main and branch pipes. (2) A limit load solution (Eq. (15)) for the piping branch junctions subjected to in-plane moment was derived according to the proposed limit load analysis approach. (3) Comparison between the experimental data and the predicted results indicated that the two kinds of limit load were very close to each and thus provided us confidence in the use of the proposed limit load solution in practice. However, more experimental data were needed to validate the solution as the structural dimensions of the branch junction beyond the discussed range.

8 F.-Z. Xuan et al. / International Journal of Mechanical Sciences 48 (2006) (4) Nonlinear FE analyses for limit load of the idealized piping branch junctions under in-plane moment were performed. The comparison between the FE limit load and the predicted one from Eq. (15) indicated that, the new limit load solution is applicable for branch junctions with the global collapse failure mode at the intersection line. Acknowledgments The work described in this paper has been carried out with the financial support of the Tenth five National Key Technological Research and evelopment Program of China (contract No. 2001BA803B03) and China Natural Science Foundation (contract No ) for which due acknowledgement is given. References [1] Bousshine L, Chaaba A, e Saxce G. Plastic limit load of plane frames with frictional contact supports. International Journal of Mechanical Sciences 2002;44(11): [2] API 579. Recommended practice for fitness-for-service. American Petroleum Institute ISSUE 6, [3] R6. Assessment of the integrity of structures containing defects. Procedure R6-Revision 4, Gloucester, UK: Nuclear Electric Ltd.; [4] SINTAP. Structure integrity assessment procedures for European industry. European Commission, printed in Italy, [5] Schroeder J. Analysis of test data on branch-pipe connections exposed to internal pressure and/or external couples. WRC Bulletin 200, [6] Ellyin F. Experimental investigation of limit loads of nozzles in cylindrical vessels. WRC Bulletin 219, [7] Ellyin F. An experimental study of elastio-plastic response of branchpipe tee connections subjected to internal pressure, external couples and combined loadings. WRC Bulletin 230, [8] Ellyin F, Turkan N. Lower bounds to limit couples of branch-pipe tee connections part 2: an in-plane couple applied to branch. Second International Conference on Pres Ves Tech Part 1, New York, p [9] Schroeder J. Limit intersection of external couples and internal pressure for branch-pipe lateral and tee connections part I: upper bounds to in-plane and out-of-plane limit couples applied to the branch for branch/pipe diameter ratio smaller than 0.8. Second International Conference Pres Ves Tech Part 1, New York, p [10] Rodabaugh EC. Interpretive report on limit analysis and plastic behaviour of piping products. WRC Bulletin 254, [11] Kim YJ, Shim J, Nikbin K, et al. Finite element based plastic limit loads for cylinders with part-through surface cracks under combined loadings. International Journal of Pressure Vessels and Piping 2003;80: [12] Xuan FZ, Li PN. Finite element-based limit load of piping branch junctions under combined loadings. Nuclear Engineering and esign 2004;231(2): [13] Heitzer M. Plastic limit loads of defective pipes under combined internal pressure and axial tension. International Journal of Mechanical Sciences 2002;44: [14] Liu YH, Cen ZZ, Chen HF, Xu BY. Plastic collapse analysis of defective pipelines under multi-loading systems. International Journal of Mechanical Sciences 2000;42: [15] Nadarajah C, Mackenzie, Boyle JT. Limit and shakedown analysis of nozzle/cylinder intersections under internal pressure and in-plane moment loading. International Journal of Pressure Vessels and Piping 1996;68: [16] Xuan FZ, Li PN, Tu S-T. Evaluation of plastic limit load of piping branch junctions subjected to out-of-plane moment loadings. Journal of Strain Analysis for Engineering esign 2003;38(5): [17] Rozemblium VI. Plasticity Conditions for Thin Shells, Applied Mathematics and Mechanics. PMM 1960;24(3): [18] Ellyin F, Turkkan N. Limit pressure of Nozzles in cylindrical shells. ASME Paper No. WA-PVP-1 Winter Annual Meeting, New York, November [19] Xuan FZ. An engineering analysis for plastic limit load of pressurized piping branch junctions. Ph thesis, East China University of Science and Technology, Shanghai, China (in Chinese). [20] Timoshenko S, Woinowsky-Krieger S. Theory of Plate and Shells. 2nd ed. New York: McGraw-Hill Book Co. Inc; [21] Ansary A, El-omiaty M, Shabara M. Short circumferential through-wall cracked pipes subjected to stretch-bending load. Nuclear Engineering and esign 1998;180(2): [22] Muscat M, Mackenzie, Hamilton R. A work criterion for plastic collapse. International Journal of Pressure Vessels and Piping 2003;80: [23] Moffat G, Hsien MF, Lynch M. An assessment of ASME III and CEN TC54 methods of determining plastic and limit loads for pressure system components. Journal of Strain Analysis 2001;36(3): [24] Billington CJ. Background to new formulae for the ultimate limit state of tubular joints. OTC 4189, p