An Estimation of Critical Buckling Strain for Pipe Subjected Plastic Bending

Cent. Eur. J. Eng. 4(3) 014 36-333 DOI: 10.478/s13531-013-0168-8 Central European Journal of Engineering An Estimation of Critical Buckling Strain for Pipe Subjected Plastic Bending Research Article L. K. Ji 1, M. Zheng 3, H. Y. Chen, Y. Zhao 3, L. J. Yu 3, J. Hu 3, H. P. Teng 3 1 School of Materials Sci. & Eng., Xi an Jiaotong University, Xi an 710049, China CNPC Tubular Goods Research Institute, Xi an, 710065, China 3 Institute for Energy Transmission Technology and Application, School of Chemical Engineering, Northwest University, Xi an 710069, China Received 6 January 014; accepted 5 April 014 Abstract: An approach for estimating critical buckling strain of pipe subjected plastic bending is established in the present paper. A rigid - perfectly plastic material model and cross section ovalization of pipe during bending are employed for the approach. The energy rates of the ovalised pipe bending and the cross section ovalising are proposed firstly. Furthermore, these energy rates are combined to perform the buckling analysis of pipe bending, an estimation formula of critical buckling strain for pipe subjected plastic bending is proposed. The predicting result of the new critical buckling strain formula is compared with the available experimental data, it shows that the formula is valid. Keywords: pipe bending rigid-perfectly plastic material cross section ovalization critical buckling strain Versita sp. z o.o. 1. Introduction Pipeline is the most reasonable, economical transport mode for oil and natural gas. At present, the total length of the global natural gas and petroleum pipeline is more than 30 10 4 KM, and rapid annual speed reaches to ( 3) 10 4 KM. As to China, it has began the construction of long distance oil pipeline in Xinjiang since 1958, and the construction of long distance natural gas pipeline started in 1963 in Sichuan. It is expected that the total length of China s oil and gas transmission pipeline will reach to 5 10 4 km by 015 1]. In practice, the environment for the E-mail: mszheng@yahoo.com pipeline is usually complex and changeable, the region is even multiple seismic and geological disasters, and pipeline has to suffer from larger strain and displacement during operation. The failure of pipeline is no longer controlled by stress, but in part or in whole by strain or displacement. As to a bending load, the failure criterion of pipeline is often defined by the buckling initiation. For a bending pipe, the pipe bends and the cross section shape changes simultaneously with the bending load increasing. When the pipe bending and the cross section shape changing exceed a certain value, the bending load or moment could no longer increase or even decreases rapidly, which is called buckling of the bending pipe. The critical strain of a bending pipe buckling is considered to be an important index in the design of the pipeline 36

L. K. Ji, M. Zheng, H. Y. Chen, Y. Zhao, L. J. Yu, J. Hu, H. P. Teng in nowadays 1 3]. The critical strain formula currently used (including the classical analytical solution, and the regressive formulae of experimental data), is with low prediction accuracy though relatively simple, or short of real physical meaning, it cannot satisfy the actual application in engineering, 3]. Therefore, it is necessary to establish a more reasonable critical buckling strain formula with higher prediction accuracy, so as to meet the demand of the actual engineering application in the viewpoint of operation safety and design for pipeline.. Current status of Approaches for Estimating Pipe Bending Buckling Strain.1. Elastic and elliptical cross section model for cylindrical shell bending Long-Yuan Li, et al, considered a thin-wall circular shell under static bending as the Brazier type. The longitudinal compressive and tensile stresses make the cross section ovalization, as shown in Figure 1(a). According to the Brazier assumption, the typical elliptical shape can be expressed as 4] u = aξ cos θ (1) in which u is the radial displacement of thin shell, a is the average radius of the un-deformed original shell,θ is the angle of a radius in plane coordinates, as shown in Figure 1, ξ is a dimensionless factor describing radial displacement. Long-Yuan Li proposed that the deformation of cross section occurs in its own plane section, the total potential energy per unit length can be expressed as 4] U(ξ, C) = 1 πeta3 ( 1 3 ξ + 5 8 ξ ) C + 3 πe t 3 8 1 v a ξ MC in which E is the elastic modulus of material, v is Poisson s ratio, t is the thickness of the shell wall, C is the longitudinal curvature of the bending shell, M is the instantaneous bending moment. The 1 st term on the right side of Eq. () represents the potential energy for longitudinal bending, and the nd term on the right side of Eq. () represents elliptical potential energy due to the cross section ovalization; the 3 rd term on the right side of Eq. () is the potential energy of the external loading. () Figure 1. Elliptical cross section model for tube bending. According to the principle of minimum potential energy for balance condition, it gives U C = 0, U ξ = 0 (3) And the critical condition of instability reads U CC U ξξ U Cξ U Cξ = 0 (4) It obtains the deformation parameters of the critical state shell and critical static moment 4] M C = 0.3988 πeat 1 v (5) ξ C = 0.370 (6) C C = 0.731 t a 1 1 v Accordingly, the macroscopic critical strain at the outer - fiber - line of the bending pipe can be obtained ε C = a(1 ξ C ) C C = 0.630a 0.731 1 a = 0.461 t a (8) In Eq. (8), though the numerical data 0.461 is less than the result of the classical elastic analytical solution, ε C = 0.6 t, i.e., 0.6, it is still much higher than the experimental a values, 3, 5]. Anyhow, it is the elastic result considering cross section ovalization for the pipe bending buckling problem. While, the latter doesn t take the cross section shape changing of the bending pipe into consideration and its prediction is about twice of the experimental data, 3, 5]. (7) 37

An Estimation of Critical Buckling Strain for Pipe Subjected Plastic Bending In addition, he gave an approximate relationship 6] δ = 0.533 R K t (13) where K is the longitudinal curvature of the bending pipe. Combing Eqs (1) and (13), it obtains the critical value of longitudinal curvature of the pipe bending K C = 0.154 t R (14) Accordingly, the critical strain at the outer- fiber-line of the bending pipe can be obtained Figure. Tomasz Wierzbicki s flattening cross-section model for pipe bending 6]... Flattening cross-section model for shell pipe subjected plastic bending In order to consider the effect of plastic deformation on pipe bending behavior, Tomasz Wierzbicki, et al proposed a flattening cross-section model to approximate the shape of thin- walled pipe cross section in bending, instead of elliptical cross section 6], shown in Figure. It is easily to yield the geometric relationship in Figure (a) 6] b + πr = πr, b = π δ (9) δ = R = r, r = R δ (10) Furthermore, Tomasz Wierzbicki gave the following relationship according to the total energy minimization and bending instability condition M M 0 = 1 (1 δ)(1.43 + 1.71 δ) (11) where M is the applied moment, M 0 = 4 R t is the rigid perfectly plastic bending moment of a pipe in absolute circular cross section condition, is the flow stress of the pipe material, R is the average radius of the original pipe before bending, t is the wall thickness of pipe, δ = δ/(r) is a dimensionless parameter characterizing cross section shape changing. From Eq. (11), it obtains the critical value of dimensionless deformation parameters δ C at pipe bending instability M M 0 δ, δc = 0.0819 (1) ( ε C = R δ ) C K C = 0.918R 0.154 t R = 0.141 t a (15) Although the parameter data in Eq.(15), 0.141, is much less than the result of the classical elastic theory i.e., 0.6, it is less than the experimental value as well, 3, 5]..3. Other approaches In 006, M. Khurram Wadee et al proposed a variational model to formulate the localization of deformation due to buckling under pure bending of thin-walled elastic tubes with circular cross-sections 7]. The results are compared with a number of case studies, including a nanotube, etc., but the model is still an elastic one. In 009, Philippe Le Grognec and Anh Le van devoted to the theoretical aspects of the elastoplastic buckling and initial post-buckling of plates and cylinders under uniform compression 8]. The analysis was based on the 3D plastic bifurcation theory assuming the J flow theory of plasticity with the von Mises yield criterion and a linear isotropic hardening. The proposed method showed to be a systematic and unified way to obtain the critical loads, the buckling modes and the initial slope of the bifurcated branch for rectangular plates under uniaxial or biaxial compression (-tension) and cylinders under axial compression, with various boundary conditions. In 009, S. Poonaya, U. Teeboonma and C. Thinvongpituk analyzed the Plastic collapse of thin-walled circular tubes subjected to bending 9]. 3-D geometrical collapse mechanism was analyzed by adding the oblique hinge lines along the longitudinal tube within the length of the plastically deformed zone. The internal energy dissipation rates were calculated for each of the hinge lines. Inextensional deformation and perfect plastic material behavior were assumed in the derivation of deformation energy rate. 38

L. K. Ji, M. Zheng, H. Y. Chen, Y. Zhao, L. J. Yu, J. Hu, H. P. Teng In 011, Gianluca Ranzi and Angelo Luongo proposed a new approach to illustrate the cross-sectional analysis in the context of the Generalized Beam Theory (GBT) 10]. The novelty relies in formulating the problem in the spirit of Kantorovich s semi-variational method. The new procedure aimed to describe the linear-elastic behavior of thin-walled members as well. In 01, T. Christo Michael et al studied the effect of ovality and variable wall thickness on collapse loads in pipe bends subjected to inplane bending closing moment 11]. Finite element limit analyses based on elastic- perfectly plastic material was employed to study the effect of ovality and variable wall thickness (thickening at intrados and thinning at extrados) on the collapse loads in pipe bends subjected to in-plane bending moment that tend to close the bend. The collapse moments were obtained from load-deflection curves of the models with circular (uniform wall thickness) and irregular cross sections and compared. It indicated that ovality in the pipe bends more significantly affects the collapse loads than thinning. a mathematical equation is proposed to include the effect of ovality based on finite element collapse load results. In 014, Gayan Rathnaweera et al investigated the performance of aluminium/terocore hybrid structures in quasi- static three-point bending by experimental and finite element analysis study 1]. The performance of the hybrid structures was changed with percentage volume of Terocore foam and tube wall thickness. Two failure modes were observed in this study. Top surface failure (compression) from structures made of AA7075 T6 and bottom surface failure (tensile) from structures with higher percentage volume of foam. It was also found 55% as the optimum percentage volume of foam to prevent bottom surface failure, while maximizing the performance under current experimental conditions. Above discussion indicates that a more reasonable critical buckling strain formula with higher accuracy is in need, so that the demand of the actual engineering application could be met. 3. New approach for estimating critical buckling strain for a pipe subjected plastic bending As is well known, when the pipeline is subjected to bending load, the cross section shape changes. According to the previous discussions, Long-Yuan Li took the elliptical shape for the changing cross section of the bending pipe and the elastic approximate for energy estimation ], a critical strain formula is obtained, i.e., ε C = 0.461t/a, although the numerical data 0.461 is less than the result of the classical elastic theory, i.e., 0.6, it is still much higher than the experimental values ]. In Tomasz Wierzbicki s approach6], the plastic strain energy was used, and the cross section changing of the bending pipe was approximately as a flattening shape, and some other approximations are used as well, another critical strain formula is obtained, i.e., ε C = 0.141t/a, the numerical parameter in above equation, 0.141, is much less than the result of the classical elastic theory i.e., 0.6, it is also less than the experimental values, 3], which indicates that the approach is oversimplified the ovalization of pipe bending. In this section, the ovalization of cross section of pipe bending is described by a standard ellipse. A rigid - perfectly plastic material model is employed for the approach. The energy rates of cross section ovalising and the ovalised pipe bending are established firstly. Furthermore, these energy rates for pipe bending are combined to perform the buckling analysis. 3.1. Strain energy rate corresponding to pure bending According to Brazier effect 1], as shown in Figure 1, the cross section of a circular pipe gradually becomes elliptical one due to bending. The ovalization is affected by many factors, among which the tube size is of great significance. Local buckling appears when the bending ultimate moment reaches to a critical value, and thereafter bending moment declines and the bending instability occurs. For a rigid - perfectly plastic pipe with the original radius R, wall thickness t, and length l, when it is subjected bending load and enters into fully plastic state, the pure bending moment due to cross section ovalization could be written as 13, 14] M p = 4σ xx 3 b a b i a i ] (16) in which a i and b i are the lengths of internal longer semiaxis and shorter semi-axis. a and b are the lengths of external longer semi-axis and shorter semi-axis, a = (a i + t), b = (b i + t), respectively. σ xx is a tensile stress along the pipeline. For a thin circular pipe, t R, when it becomes a standard ellipse due to bending, a dimensionless parameter γ could be introduced, such that a = R(1 + γ) and b = R(1γ), respectively. Thus, by neglecting higher terms of t/r, Eq. (16) becomes M p = 4R tσ xx 1 3 γ 1 3 γ ] = M 0 1 3 γ 1 ] ( ) (17) σxxx 3 γ 39

An Estimation of Critical Buckling Strain for Pipe Subjected Plastic Bending In Eq. (17), M 0 = 4 R t is bending moment of a perfect circular shell in perfectly plastic pipe state, is the flow stress of the pipe material. The strain energy rate corresponding pure bending is 6] Ẇ b = M p φ = M0 ( 1 3 γ 1 3 γ )] ( σxx ) φ (18) φ is the changing rate of the bending angle of the pipe subjected bending load in Figure 1(b). Since the relationship between bending angle φ and the longitudinal curvature C of the pipe with length l is, φ = lc, thus, φ = lċ. Therefore, Eq. (18) is rewritten as, Ẇ b = M p φ = M0 ( 1 3 γ 1 3 γ )] ( σxx ) lċ (19) 3.. Strain energy rate corresponding to the cross section ovalization of the bending pipe For a pipe with the length of l and thickness t, the ovalization of its cross section can be described as follows. Referring to Tomasz Wierzbicki 6], for a elliptical pipe, in the fully plastic state, the strain energy rate of the cross-section ovalising is 6], Ẇ crush = S K θθ M θθ ds (0) in which, M θθ = σ θθ (t l/4) is the plastic moment corresponding to the elliptical arc in Figure 1(a); K θθ is the changing rate of the curvature in the elliptical arc; ds is the length of the elliptical arc. In mathematics, for a standard ellipse, each point on the elliptical arc can be described by following equation in rectangular coordinate system x = a cos θ = R(1 + γ) cos θ, y = b sin θ = R(1 γ) sin θ The curvature in the elliptical arc can be derived as (1) From Eq. (), it can be seen, for θ < θ C, K > K C, and θ > θ C, K < K C. Especially, at θ = 0 and π/, the value of the curvature K is, respectively K 0 = 1 R (1 + γ) (1 γ) ], K π = 1 (1 γ) R (1 + γ) ] (4) Figure 3 gives the variation of θ C with respect to the parameter γ. Since the direct integral of Eq. (0) is quite difficult, we have to make some simplification. The whole ellipse can be divided into 4 quadrants, for the 1 st quadrant, the integral can be approximated in regions of 0 < θ < θ C and θ C < θ < π/, respectively. In regions of 0 < θ < θ C and θ C < θ < π/, the averaged curvatures can be approximated by K 0 θc = 1 K θc π/ = 1 respectively. Furthermore { } ( ) 1 (1 + γ) K0 + K θc = R (1 γ) ] + 1 { } ( ) 1 (1 + γ) KθC + K π/ = 1 + R (1 + γ) ] (5) (6) K 0 θc = 1 ] (3 + γ) γ (7) R (1 γ) 3 K θc sin π/ = 1 ] γ (8) R (1 + γ) 3 In addition, since the expression of the variation of θ C with respect to the parameter γ, says, Eq. (3), is still complex, by fitting is needed, it yields a fitted expression K = 1 (1 γ ) R (1 γ) + 4γ sin θ] 3 () θ C = π 4 (1 0.95γ) 1 (9) There exists a critical angle θ C, at which K c = 1 R, from Eq. (), it yields θ C = arcsin 3 1 γ + (1 + γ)] 3 1 γ (1 γ)] 4γ (3) Figure 3 gives comparison of Eq. (9) and the original curvature of Eq. (3). It can be seen that the fitted expression, Eq. (9), describes the variation of θ C with respect to the parameter γ very well in most valuable range. 330

L. K. Ji, M. Zheng, H. Y. Chen, Y. Zhao, L. J. Yu, J. Hu, H. P. Teng Then, the total bending strain energy rate including cross section shape changing for a pipe with length l is Ẇ = Ẇb + Ẇcrush (33) i.e., Ẇ = 1 ( M 0 1 γ 3 γ 3 (3 + γ) (1γ) 3 (10.95γ) 1 (1+γ) 3 Furthermore, it can be rearranged as ) lċ + 1 πtl 3R M 0 (10.95γ) ] ] γ 1 (34) Figure 3. Variation of θ C with respect to the parameter γ. Thus, the integral Eq. (0) can be approximately written as Ẇ crush = S K θθ M θθ ds 4 M θθ { K (R θ 0 θc C ) + K π )]} θc π/ ( θ C γ = π (3 + γ) M θθ (1 γ) (1 0.95γ) 1 3 ] ] γ (1 + γ) (1 0.95γ) 1 3 = πσ θθt l (3 + γ) 8 (1 γ) (1 0.95γ) 1 3 ] ] γ (1 + γ) (1 0.95γ) 1 3 = πtl (3 + γ) 3R M 0 (1 γ) (1 0.95γ) 1 3 ] ] ( ) (1 + γ) (1 0.95γ) 1 σθθ 3 γ (30) 3.3. Plastic yielding condition and the total strain energy rate of the pipe bending Refer to Tomasz Wierzbicki s method 6], the plastic yielding condition for pipe bending problem can be written as follows ( ) ( ) Mθ Nxx + = 1 (31) M 0 N 0 And furthermore, Tomasz Wierzbicki assumed 6] σ θ = M θ M 0 = 1, σ xx = M xx M 0 = 1 (3) in which Ẇ = M {( 0 lċ 1 γ ) 3 γ 3 (3 + γ) +α (1 γ) (1 0.95γ) 1 3 (1 + γ) 3 α = (1 0.95γ) 1 πt γ 3R Ċ ] ]} (35) (36) In Eq. (36), the adjusted coefficient α reflects the energetic relation between the pure bending and cross section shape changing of the pipe during bending, which is the ratio of the shape changing rate γ with respect to the longitudinal bending curvature rate Ċ. From formula Eq. (35), it shows the energy change rate of pipe bending depends on the parameter Ċ and adjusted coefficient α. In the actual process, α will be adjusted automatically to make the total energy required for the pipe bending minimum, i.e., Ẇ γ (37) Therefore, from Eq. (35), it obtains the relationship between α and γ α = { (1 + γ) (10.95γ) 1 0.95(1 0.95γ) 1 (3+γ) 3 (1γ) 3 (1 γ) 3 + 3(3+γ)(10.95γ) 1 + (10.95γ) 1 (1γ) 4 (1+γ) 3 0.95(3γ) (10.95γ) 1 3(3γ) + (1+γ) 3 (10.95γ) 1 (1+γ) 4 ] (38) Figure 4 gives the variation of α(γ) with respect to γ. 1 331

An Estimation of Critical Buckling Strain for Pipe Subjected Plastic Bending Substituting Eq. (38) into Eq. (35), it obtains Ẇ = 1 M 0 f 1 (γ) lċ (39) in which the function f 1 (γ) is defined as f 1 (γ) = {(1 ) γ3 γ 3 (3+γ) + α (1γ) (10.95γ) 1 (3γ) 3 (1+γ) 3 ] ]} (10.95γ) 1 (40) 3.4. Macro bending moment and buckling condition When pipe bending is carried out under the action of bending moment M e, the external moment energy rate Ṗ on the bending process is 6], Ṗ = M e φ = M e lċ (41) According to the law of conservation of energy, the external moment rate should equal to the consumed strain energy rate within the pipe due to bending, i.e., Ṗ = Ẇ (4) Substituting Eqs.(39) and (40) into (41), it yields M e = 1 M 0 f 1 (γ) (43) Eq. (43) is the expression for macroscopic bending moment of the pipeline undergoing bending deformation and entering fully plastic state with the cross section ovalising. Besides, the material of the pipe behaves as a rigid - perfectly plastic one. Eq. (43) shows that the macroscopic bending moment M e is the function of flattening cross section parameter γ in the bending process. The pipe bending instability occurs when the curve of M e with γ reaches to the peak. Figure 4 shows the variation of function f 1 (γ) with respect to γ. It can be seen from Figure 4 that f 1 (γ) reaches to a maximum value at γ = 0.11, i.e., the critical value is γ c = 0.11 (44) Combining Eq. (36) and Eq. (38), it yields numerically C C = πt γ C 3R 1 t dγ = 0.131 (45) γ R 0 Figure 4. Variations of function f(γ) with respect to γ. Accordingly, the critical strain of the outer-fiber-line of the bending pipe at buckling can be obtained ε C = R(1 γ) C C = 0.89R 0.131 t R = 0.19 t R (46) Eq. (46) is expression of macroscopic critical strain at the outer-fiber-line of pipe at bending buckling. Additionally, rigid - perfectly plastic material model and cross section ovalising are involved. The factor 0.19 in Eq. (46) is close to the most experimental results, 3, 6]. In addition, the variation of function f = γ 0 1 α d δ with respect to γ is drawn in Figure 4 as well. Figure 5 shows the comparison of the experimental data with the predictions, in which the experimental data was cited from Ref. ], and classical elastic analytical solution, the regressive expressions proposed by Sherman and the Battelle are given as well ] ε C classic = 0.6 t R (47) ( ) t ε C Sherman = 4 (48) R ( ) 1.59 t ε C Battelle = 0.804 (49) R Figure 5 shows that the experimental data agrees with the present estimation very well, and much better than the others. 33

L. K. Ji, M. Zheng, H. Y. Chen, Y. Zhao, L. J. Yu, J. Hu, H. P. Teng Figure 5. Comparison of the experimental data with different estimations. 4. Concluding remarks The new approach proposed in this paper is based on the perfectly plastic material model and ovalization assumption. The strict geometric relationship of the cross section ovalization is employed, and it derives the reasonable strain energy rate and the cross section shape changing in pipe bending process. The new critical buckling strain prediction formula is applied to the analysis of pipe bending buckling strain. It shows that the formula is valid and reasonable. References 1] Li H. L., Development and application of strain based design and anti-large-strain pipeline steel, Petroleum Sci. & Tech. Forum (in Chinese) 7(), 008, 19-5 ] Dorey A. B., Murray D. W., Cheng J. J. R., An experimental evaluation of critical buckling strain criteria, 000 International Pipeline Conference, Vol.1, Calgary, Alberta, Canada, October 1-5, 000, 71-80 3] Dorey A. B., Murray D. W., Cheng J. J., Critical buckling strain equations for energy pipelines - A Parametric Study, Transaction of the ASME 18, 48-55, 006 4] Li L. Y., Approximate estimates of dynamic instability of long circular cylindrical shells under pure bending, Int. J. Pres. Ves. & Piping 67, 1996, 37-40 5] Yang J. L., Reid S. R., Approximate estimation of hardening - softening behavior of circular pipes subjected to pure bending, Acta Mechanica Sinica 13(3), 1997, 7-40 6] Wierzbicki T., Sinmao M. V., A simplified model of Brazier effect in plastic bending of cylindrical tubes, Int. J. Pres. Ves. & Piping, 71, 1997, 19-8 7] Khurram Wadee M., Ahmer Wadee M., Bassom A. P., Aigner A. A., Longitudinally inhomogeneous deformation patterns in isotropic tubes under pure bending, Proc. R. Soc. A 46, 006, 817-838 8] Le Grognec P., Anh Le van, Some new analytical results for plastic buckling and initial post-buckling of plates and cylinders under uniform compression, Thin-Walled Structures 47, 009, 879-889 9] Poonaya S., Teeboonma U., Thinvongpituk C., Plastic collapse analysis of thin-walled circular tubes subjected to bending, hin-walled Structures 47, 009, 637-645 10] Ranzi G., Luongo A., A new approach for thin-walled member analysis in the frame work of GBT, Thin- Walled Structures 49, 011, 1404-1414 11] Michael T. Ch., Veerappan A. R., Shanmugam S., Effect of modality and variable wall thickness on collapse loads in pipe bends subjected to inplane bending closing moment, Engineering Fracture Mechanics 79, 01, 138-148 1] Rathnaweera G., Ruan D., Hajj M., Durandet Y., Performance of aluminium/ Terocore hybrid structures in quasi-static three - point bending: Experimental and finite element analysis study, Materials and Design 54, 014, 880-89 13] Elchalakani M., Zhao X. L, Grzebieta R. H., Plastic Slenderness Limits for Cold - Formed Circular Hollow Sections, Australian J. of Structural Eng. 3, 00, 17-141 14] Guo L., Yang S., Jiao H., Behavior of Thin - walled Circular Hollow Section Tubes Subjected to Bending, Thin-Walled Structures, 73, 013, 81-89 333