Theoretical and Numerical Predictions of Burst

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2(b); and the equivalent average shear stress TA and the equivalent average strain sA are obtained from Eqs. ( 1) and (5) and shown in Fig. 2(c). These figures ...
Theoretical and Numerical Predictions of Burst Pressure of Pipelines Xian-Kui Zhu

Mem.ASME e-mail: [email protected]

Brian N. Leis

Mem.ASME

Battelle Memorial Institute, 505 King Avenue, Columbus, OH 43221

To accurately characterize plastic yield behavior of metals in multiaxia/ stress states, a new yield theory, i.e., the average shear stress yield (ASSY) theory, is proposed in reference to the classical Tresca and von Mises yield theories for isotropic hardening materials. Based on the ASSY theory, a theoretical solution for predicting the burst pressure of pipelines is obtained as a function of pipe diameter, wall thickness, material hardening exponent, and ultimate tensile strength. This solution is then validated by experimental data for various pipeline steels. According to the ASSY yield theory, four failure criteria are developed for predicting the burst pressure of pipes by the use of commercial finite element softwares such as ABAQUS and ANSYS, where the von Mises yield theory and the associated flow rule are adopted as the classical metal plasticity model for isotropic hardening materials. These failure criteria include the von Mises equivalent stress criterion, the maximum principal stress criterion, the von Mises equivalent strain criterion, and the maximum tensile strain criterion. Applications demonstrate that the proposed failure criteria in conjunction with the ABAQUS or ANSYS numerical analysis can effectively predict the burst pressure of end-capped line pipes. [DOl: 10.1115/1.2767352] Keywords: von Mises theory, Tresca theory, ASSY theory, pipeline

Introduction An accurate prediction of the burst pressure of pipelines is very important in the engineering design and integrity assessment of oil and gas transmission pipelines. Experimental results showed that analytical, numerical, and empirical predictions available are generally inconsistent and inaccurate, and have limited applications. Stewart and Klever [1] pointed out that the theoretical solutions of burst pressure depend on the yield theory adopted. The Tresca yield theory generally predicts a lower bound of experimental data of burst pressure, whereas the von Mises yield theory predicts an upper bound of burst pressure for end-capped pipes or cylindrical pressure vessels, as reviewed in Refs. [2,3]. This stimulates the present work to develop a better solution to predict the burst pressure of line pipes. The application of the finite element analysis (FEA) to the burst failure prediction of pipelines with or without corrosion defects potentially offers greater accuracy, but it requires an appropriate failure criterion. Such failure criteria available are often related to the ultimate tensile stress (UTS), but inconsistent with each other. For instance, a pipeline can be considered as a burst failure when the von Mises equivalent stress on the defect ligament in FEA simulations reaches the true UTS. This failure criterion was utilized by Fu and Kirkwood [4] for X46 and X60 pipeline steels and by Karstensen et al. [5] for an X52 pipeline steel in their FEA predictions of burst pressure. However, Choi et al. [6] predicted the burst pressure for X65 corroded pipelines using a different failure criterion in the FEA calculations. They assumed that burst failure occurs when the von Mises equivalent stress in a defect reaches 90% of the true UTS for a rectangular defect and 80% of the true UTS for an elliptical defect. Therefore, further investigaContributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OP PREssURE VESSEL Tl!cHNOLOOY. Manuscript received October 6, 2006; final manuscript received February 22. 2007. Review conducted by G. E. Otto Widera. Paper presented at the 2006 ASME Pressure Vessels and Piping Conference (PVP2006), Vancouver, BC, Canada, July 23-27,2006.

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tions are needed to develop a consistent and valid failure criterion for the burst pressure prediction of line pipes using the FEA calculations. As is well known, the commercial finite element software ABAQUS [22] provides a modified RIKS method for a plastic instability analysis, which can be used to determine global failure loads for engineering structures. Most commercial FEA packages, including ABAQUS and ANSYS [23], adopt the classical metal plasticity model, i.e., the von Mises yield theory and the associated flow rule as its default plasticity model for isotropic hardening materials. Accordingly, the RIKS method that was built in ABAQUS can only determine a von Mises-based upper bound solution for the burst pressure of pipelines, as demonstrated in our recent work [7] for a defect-free pipe. Likewise, Lam et al. [8] showed that the FEA results of burst pressure determined using ABAQUS and the RIKS method overestimate the experimental data for thin-wall cylindrical pressure vessels. This indicates that ABAQUS with the RIKS method may determine an unreliable FEA result of burst pressure for end-capped pipes or cylindrical shells. To effectively predict the actual burst pressure, theoretical and numerical investigations are carried out for defect-free endcapped pipes in this paper. Based on the proposed average shear stress yield (ASSY) theory, a new theoretical solution for the burst pressure of pipes is obtained and validated by extensive experimental data. Four failure criteria are proposed for the burst pressure prediction in the von Mises-based FEA simulations. These failure criteria are the von Mises equivalent stress criterion, the maximum principal stress criterion, the von Mises equivalent strain criterion, and the maximum tensile strain criterion. Applications of these failure criteria to a numerical analysis for endcapped pipes using ABAQUS and ANSYS are discussed.

A New Multiaxial Yield Theory In the plasticity analysis of metallic materials, the classical Tresca and von Mises theories are commonly used. Many experimental investigations have indicated that the test data for initial yielding and postyielding lie between the Tresca and von Mises predictions. To more effectively describe the plastic yield behav-

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ior, the present authors [9] recently developed a new multiaxial yield theory, i.e., the ASSY theory for isotropic hardening materials. In the following sections, the ASSY theory is first introduced, and its validations for both initial and subsequent yieldings are then demonstrated with extensive experimental results. Average Shear Stress Yield Theory. An average shear stress ( TA) is defined as the average of the maximum shear stress ( Tmax> and the von Mises equivalent shear stress ( TM). Since the von

Mises shear stress is related to the octahedral shear stress by TM = .J3i2Tocto the average shear stress is a weighted average of the maximum shear stress and the octahedral shear stress. It is assumed that plastic yielding will occur if the average shear stress of a material reaches a critical value, namely,

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For convenience, this yield theory is referred to as the ASSY theory hereafter. In reference to the uniaxial tension test, the maximum shear stress and the octahedral shear stress at initial yielding are u0 12 and .f2u0 /3, respectively. Th~efore, ~e critical average shear stress at yielding is ( TA)c=(2+ \'3)u0 /4\'3, where u 0 is the yield strength of the material in tension. In the principal stress space (uttu2 ,u3 ) with an assumption of u 1 ;;:;J: u 2 ;;:;J: u 3, from Eq. (1), the ASSY equivalent stress uA can be expressed as UA

1 = --,-( ~3UT+ 2uM) 2+ ~3

Experimental Validation for Initial Yielding. Under the plane stress condition (u3 =0) and in the principal stress plane (c· From Eqs. (16) and (23), the critical von Mises equivalent strain is obtained as

P!'=J1

1 {e~)c=r ln(l +0.882ni.SSS)

(31)

\!3

Through curve fitting, the above equation can be further approximated as (32)

Maximum Tensile Strain Criterion. For a pressurized pipeline, the maximum tensile strain is the hoop strain. Similarly, one can assume that if the von Mises hoop strain reaches its critical value, a pipeline burst will be initiated and the corresponding critical load will be equal to the ASSY burst pressure, i.e., at s~=(s~)c. From {s~)c=( J312He!'>c and Eq. (32), the critical von Mises hoop strain can be approximated as

P!'

=11

(e~)c = 0.384n 1•524

(33)

Theoretically, the four failure criteria proposed above can be equally used to determine the burst pressure of pipes using the FEA calculations and the von Mises yield theory. Actually, our experience indicated that the two strain criteria are not as efficient as the two stress criteria because a small load increase can cause a large plastic strain increase near the pipe burst Therefore, only applications of the two stress failure criteria to the numerical burst prediction are demonstrated in the next sections.

Numerical Prediction of Pipe Burst Pressure Finite Element Calculations and Results by ABAQUS. Detailed FEA calculations were performed using the commercial package ABAQUS Standard [22] for a defect-free pipeline with an outside diameter of 762 mm and a thickness of 17.53 mm, i.e., Dlt=43.5. Due to the negligible axial strain, the long pipe was simplified as a plane strain problem (note that this plane strain model is equivalent to the axisymmetric model for the pipeline). Only one quarter of the circular section was modeled because of symmetry. The uniform FEA mesh has four elements in thickness and 90 elements in circumferential direction, which lead to 360 elements and 1269 nodes in total. The eight-node quadratic parametric element with reduced integration was used in the numerical simulation. The applied load was internal pressure only, and the symmetric displacement boundary conditions were employed in the FEA model. The elastic-plastic finite strain formulation and the modified RIKS method built in ABAQUS have been adopted in the FEA simulation. Since ABAQUS adopts the classical metal plasticity model, i.e., the von Mises yield theory and the associated flow rule, as its default plasticity model, all calculated results are the von Mises-based numerical solutions. The material considered is an X65 pipeline steel. Experimental data of true stress-true plastic strain curve for the X65 is shown in Fig. 9, where the input data of material properties used in the FEA calculations are also marked. The yield stress defined at the 0.5% total strain is 508 MPa, the UTS is 645 MPa, and thus YIT =0.788. From Eq. (lOa), the strain hardening exponent is estimated as n=0.113, and the measured value is n=0.112. In the FEA calculation, the elastic modulus E=207 GPa and the Poisson's ratio v=0.3. From Eq. (8), the von Mises solution and the ASSY solution for the burst pressure of this pipe are detennined as pl{=32.96 MPa and 11=30.51 MPa, respectively. Figure 10 shows the variation of the von Mises equivalent stress with internal pressure obtained from the FEA calculations

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Since the commercial finite element codes, ABAQUS and ANSYS, adopt the von Mises yield criterion and the associated flow rule as the default plasticity model for isotropic hardening metals, only the von Mises-based burst pressure of pipes can be determined using these FEA codes, as shown in the examples. To effectively predict the actual or ASSY burst pressure using these FEA codes, four burst failure criteria: the von Mises equivalent stress criterion, the maximum principal stress criterion, the von Mises equivalent strain criterion, and the maximum tensile strain crite-----~n.... ·oUJn..__wu...a.ere__de_vekm.ed in reference_to the_UTS_anQthe_strain_hard-

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and the theoretical formula in Eq. (24), respectively. It is found that an excellent agreement exists between the numerical results and theoretical solutions for a full-range loading from elastic deformation to plastic instability. At plastic instability, the numerical result of the von Mises pressure is 32.84 MPa, which is almost identical to its theoretical value of 32.96 MPa. This indicates that ABAQUS can well predict the von Mises-based stress-load relation and the burst pressure for defect-free pipelines. From Eq. (26), one has the critical von Mises equivalent stress (~)c =0.895ufrr5 =577.13 MPa for this X65 pipeline steel. When this critical von Mises equivalent stress is reached in the FEA simulation, a corresponding critical pressure is obtained as =30.85 MPa, which is nearly equal to the ASSY burst pressure of 30.51 MPa. Similarly, the same critical pressure can be determined by the use of the maximum principal stress failure criterion. Therefore, it is concluded that the proposed stress failure criteria can be effectively used to determine the burst pressure of pipes in the FEA simulations by ABAQUS, and the burst pressure for the X65 pipeline is -30.85 MPa.

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Finite Element Calculations and Results by ANSYS. Recently, Xue et al. [24] performed detailed FEA calculations for static and dynamic burst analyses of a cylindrical shell using the commercial

of Failure Pressures in Line Pipes," Int. J. Offshore Polar Eng., 14, pp. 125131. [8] Lam, P. S., Morgan, M. J., Imrich, K. J., and Chapman, G. K., 2006. "Predicting Tritium and Decay Helium Effects on Burst Properties of Pressure Vessels," Proceedings of the ASME Pressure Vessel Piping Conference, Vancouver, BC, Canada, July 23-27, Paper No. PVP2006-ICPVTll-93274. [9] Zhu, X. K., and Leis, B. N., 2006, "Average Shear Stress Yield Criterion and Its Application to Plastic Collapse Analysis of Pipelines," Int. J. Pressure Vessels Piping, 83, pp. 663-671. [10] Lode, W., 1926, "Versuche ueber den Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen, Kupfer, und Nickel," Z. Pbys., 36, pp. ~~~- ~--91_3_-9_3_9. - - - - - - - -

FEA codes ANSYS [23] and LS-DYNA, respectively. Note that these two FEA codes adopted the von Mises yield theory and the associated flow rule as the classical metal plasticity model. The cylindrical sheii is a thin-wall end-capped pipe with a mean diameter of 606 nun, a wall thickness of 6 nun, and a length of 2400 mm. The material of the cylinder is a low carbon steel, Q235-A. The yield stress defined at the 0.2% offset strain is 339.4 MPa, the UTS is 472 MPa, and YIT=0.119. From Eq. (lOb), the strain hardening exponent for this material is estimated as n=0.127. From Eq. (8), the theoretical result of the von Mises burst pressure for this cylinder is determined as 10.06 MPa, and the ASSY burst pressure for this cylinder is 9.31 MPa. The FEA result of the burst pressure for this pipe determined in Ref. [24] is 10.0 MPa in the static analysis using ANSYS and 9.6 MPa in the dynamic analysis using LS-DYNA. Apparently, these two numerical predictions coincide with the theoretical result of the von Mises burst pressure. Accordingly, both ANSYS and LS-DYNA can numerically determine the von Mises burst pressure. Similar to ABAQUS, when the von Mises equivalent stress criterion or the maximum principal stress criterion developed previously is used in the FEA simulations, the actual burst pressure for this cylinder can be similar to the ASSY burst pressure of 9.31 MPa.

Concluding Remarks This paper investigated the theoretical and numerical predictions of the accurate burst pressure of pipes or pipelines. Since the Tresca yield theory predicts a lower bound of burst pressure and the von Mises yield theory provides an upper bound of burst pressure for pipelines, a new multiaxial yield theory, i.e., the ASSY theory, was developed for isotropic hardening materials so as to improve the prediction of burst pressure. The comparisons with classical experimental data showed that the ASSY yield theory can much better correlate the stress-strain relations for both initial yielding and subsequent yielding than the Tresca or von Mises yield theory can. A theoretical solution of burst pressure based on the ASSY yield theory was obtained as a function of pipe diameter, wall thickness, material hardening exponent, and ultimate tensile strength for defect-free pipelines. This solution was validated by extensive experimental data of burst pressure for different pipeline sizes and grades. Therefore, the proposed ASSYbased solution for the burst pressure can be considered as an effective prediction of burst pressure for pressurized pipes and pipelines.

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von Mises hoop stress. It is assumed that if the von Mises hoop stress reaches its critical value during loading, a pipeline burst will occur and the corresponding critical load will be the ASSY burst pressure, i.e., J>!'=J1 at a-Wo=(c~>c· From Eqs. (22), (23), and (28), the critical von Mises hoop stress is determined and has an equation similar to Eq. (25). Moreover, (~)c=(2/,/J)(~)c holds. From Eq. (26), an approximate explicit equation for the critical von Mises hoop stress is obtained as (~)c=(0.920n 2 -0.48ln+ 1.076)u~

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Using this equation, the critical von Mises hoop stress is obtained as {~)c= 1.025u:rrs for n=0.15. This critical hoop stress is about 83% of the von Mises burst hoop stress. The big difference is clearly shown in Fig. 8. If the true UTS is used, Eq. (29) becomes (u%,)c=(l.454n 2 -1.510n+ 1.076)uUTS

(30)

Due to the similar reason for the use of Eq. (27), Eq. (30) can be used to define a critical nominal von Mises hoop stress in reference to the engineering UTS.

von Mises Equivalent Strain Criterion. Similar to the von Mises equivalent stress criterion, one can assume that if the von Mises equivalent strain reaches its critical value, a pipeline burst will occur and the corresponding critical load will be the ASSY burst pressure, i.e., ~=J1 at e~=(e~>c· From Eqs. (16) and (23), the critical von Mises equivalent strain is obtained as 1 (e~)c=-,;: ln(l +0.882nl. 585 )

-..3

(31)

Through curve fitting, the above equation can be further approximated as (32) Maximum Tensile Strain Criterion. For a pressurized pipeline, the maximum tensile strain is the hoop strain. Similarly, one can assume that if the von Mises hoop strain reaches its critical value, a pipeline burst will be initiated and the corresponding critical load will be equal to the A~SY burst pressure, i.e., =J1 at e~=(e:O>c· From {e~)c=( ~312){e!')c and Eq. (32), the critical von Mises hoop strain can be approximated as

J>!'

(e~)c = 0.384n 1.524

(33)

Theoretically, the four failure criteria proposed above can be equally used to determine the burst pressure of pipes using the FEA calculations and the von Mises yield theory. Actually, our experience indicated that the two strain criteria are not as efficient as the two stress criteria because a small load increase can cause a large plastic strain increase near the pipe burst. Therefore, only applications of the two stress failure criteria to the numerical burst prediction are demonstrated in the next sections.

Numerical Prediction of Pipe Burst Pressure Finite Element Calculations and Results by ABAQUS. Detailed FEA calculations were performed using the commercial package ABAQUS Standard [22] for a defect-free pipeline with an outside diameter of 762 mm and a thickness of 17.53 mm, i.e., Dlt=43.5. Due to the negligible axial strain, the long pipe was simplified as a plane strain problem (note that this plane strain model is equivalent to the axisymmetric model for the pipeline). Only one quarter of the circular section was modeled because of symmetry. The uniform FEA mesh has four elements in thickness and 90 elements in circumferential direction, which lead to 360 elements and 1269 nodes in total. The eight-node quadratic parametric element with reduced integration was used in the numerical simulation. The applied load was internal pressure only, and the symmetric displacement boundary conditions were employed in the FEA model. The elastic-plastic finite strain formulation and the modified RIKS method built in ABAQUS have been adopted in the FEA simulation. Since ABAQUS adopts the classical metal plasticity model, i.e., the von Mises yield theory and the associated flow rule, as its default plasticity model, all calculated results are the von Mises-based numerical solutions. The material considered is an X65 pipeline steel. Experimental data of true stress-true plastic strain curve for the X65 is shown in Fig. 9, where the input data of material properties used in the FEA calculations are also marked. The yield stress defined at the 0.5% total strain is 508 MPa, the UTS is 645 MPa, and thus Y IT =0.788. From Eq. (lOa), the strain hardening exponent is estimated as n=0.113, and the measured value is n=0.112. In the FEA calculation, the elastic modulus E=207 GPa and the Poisson's ratio v=0.3. From Eq. (8), the von Mises solution and the ASSY solution for the burst pressure of this pipe are determined as ~=32.96 MPa and J1=30.51 MPa, respectively. Figure 10 shows the variation of the von Mises equivalent stress with internal pressure obtained from the FEA calculations

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and the theoretical formula in Eq. (24), respectively. It is found that an excellent agreement exists between the numerical results and theoretical solutions for a full-range loading from elastic deformation to plastic instability. At plastic instability, the numerical result of the von Mises pressure is 32.84 MPa, which is almost identical to its theoretical value of 32.96 MPa. This indicates that ABAQUS can well predict the von Mises-based stress-load relation and the burst pressure for defect-free pipelines. From Eq. (26), one has the critical von Mises equivalent stress (~)c =0.895u{rr5 =577.13 MPa for this X65 pipeline steel. When this critical von Mises equivalent stress is reached in the FEA simulation, a corresponding critical pressure is obtained as =30.85 MPa, which is nearly equal to the ASSY burst pressure of 30.51 MPa. Similarly, the same critical pressure can be determined by the use of the maximum principal stress failure criterion. Therefore, it is concluded that the proposed stress failure criteria can be effectively used to determine the burst pressure of pipes in the FEA simulations by ABAQUS, and the burst pressure for the X65 pipeline is -30.85 MPa.

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Finite Element Calculations and Results by ANSYS. Recently, Xue et al. [24] performed detailed FEA calculations for static and dynamic burst analyses of a cylindrical shell using the commercial

FEA codes ANSYS [23] and LS-DYNA, respectively. Note that these two FEA codes adopted the von Mises yield theory and the associated flow rule as the classical metal plasticity model. The cylindrical shell is a thin-wall end-capped pipe with a mean diameter of 606 mm, a wall thickness of 6 mm, and a length of 2400 mm. The material of the cylinder is a low carbon steel, Q235-A. The yield stress defined at the 0.2% offset strain is 339.4 MPa, the UTS is 472 MPa, and YIT=0.119. From Eq. (lOb), the strain hardening exponent for this material is estimated as n=0.127. From Eq. (8), the theoretical result of the von Mises burst pressure for this cylinder is determined as 10.06 MPa, and the ASSY burst pressure for this cylinder is 9.31 MPa The FEA result of the burst pressure for this pipe determined in Ref. [24] is 10.0 MPa in the static analysis using ANSYS and 9.6 MPa in the dynamic analysis using LS-DYNA. Apparently, these two numerical predictions coincide with the theoretical result of the von Mises burst pressure. Accordingly, both ANSYS and LS-DYNA can numerically determine the von Mises burst pressure. Similar to ABAQUS, when the von Mises equivalent stress criterion or the maximum principal stress criterion developed previously is used in the FEA simulations, the actual burst pressure for this cylinder can be similar to the ASSY burst pressure of 9.31 MPa.

Concluding Remarks This paper investigated the theoretical and numerical predictions of the accurate burst pressure of pipes or pipelines. Since the Tresca yield theory predicts a lower bound of burst pressure and the von Mises yield theory provides an upper bound of burst pressure for pipelines, a new multiaxial yield theory, i.e., the ASSY theory, was developed for isotropic hardening materials so as to improve the prediction of burst pressure. The comparisons with classical experimental data showed that the ASSY yield theory can much better correlate the stress-strain relations for both initial yielding and subsequent yielding than the Tresca or von Mises yield theory can. A theoretical solution of burst pressure based on the ASSY yield theory was obtained as a function of pipe diameter, wall thickness, material hardening exponent, and ultimate tensile strength for defect-free pipelines. This solution was validated by extensive experimental data of burst pressure for different pipeline sizes and grades. Therefore, the proposed ASSYbased solution for the burst pressure can be considered as an effective prediction of burst pressure for pressurized pipes and pipelines.

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