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Plastic limit loads of defective pipes under combined internal pressure and axial tension
International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
Plastic limit loads of defective pipes under combined internal pressure and axial tension M. Heitzer Forschungszentrum Julich GmbH, Central Institute for Applied Mathematics (ZAM), D-52425 Julich, Germany Received 28 November 2000; received in revised form 5 January 2002
Abstract In this paper, a +nite element mathematical programming formulation is presented for the statical limit analysis of 3-D perfectly plastic structures. A direct iterative algorithm is employed in solving the above optimization formulation. The numerical procedure has been applied to carry out the plastic collapse analysis of defective pipes under combined internal pressure and axial tension. The engineering situation considered has a practical importance in the pipeline industry. The e2ects of four kinds of typical part-through slots on the collapse loads of pipes are investigated and evaluated. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The plastic limit load, which determines the load-carrying capacity of structures, is an important parameter in performing the two-criteria assessment of structural integrity [1]. Therefore, the knowledge of limit loads of mechanical components and structures is useful for designers to address the modes of failure associated with the loading. The determination of limit loads is by no means an easy task, especially for complex con+gurations and loading systems. Therefore, how to determine the limit load e9ciently and accurately has attracted the attention of many researchers. The lower and upper bound analyses of the limit load can be approached by mathematical programming processes based on the static and kinematic theorems of limit analysis. With the progress in the +nite element technique and mathematical optimization theory, the mathematical programming methods can determine the load-carrying capacity of a plastic body. Thus, the time-consuming step-by-step elastic–plastic +nite element analyses can be replaced. Pipelines are widely used in the various +elds such as the petrochemical industry, energy and electric power engineering, etc. During their operation, many local defects such as part-through slots can be produced by corrosion, mechanical damage or surface cracks. These defects may in
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M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
integrity of pipes, such that an investigation of the possible decrease of the load-carrying capacity due to part-through slots is of bene+t. In the present paper, a static approach to limit analysis of 3-D structures is proposed by means of a direct iterative algorithm [2]. The limit analysis of defective pipes is performed under multi-loading systems. The defects considered here include part-through spherical and ellipsoidal slots. The effects of various shapes and sizes of typical part-through slots on the collapse loads of pipes are investigated. 2. Limit analysis The structure V is loaded monotonously by load P = (q; p). The engineer is interested in the load factor ¿ 1 by which P can be increased up to the collapse at P. The limit load analysis investigates the collapse state, in which the structure fails with unrestricted
in V;
div = −q n = p
in V;
on @V :
(1)
For each stress +eld , which ful+ls the conditions of the static theorem, is a safety factor, so that the load-carrying capacity of the structure is not yet exhausted. One calculates a lower bound of the limit load factor as the largest safety factor which ful+ls the conditions of Eq. (1). This is in the case of the von Mises yield function F a nonlinear optimization problem with the unknowns and . The constraints of the optimization problem are validated only in the Gaussian points, de+ned by a FEM discretization. The number of Gaussian points becomes huge for industrial structures and no e2ective solution algorithms for the nonlinear optimization problem are available. The basis reduction technique for handling such large-scale optimization problems was used in Ref. [2]. The basis of the subspaces are generated by the general purpose +nite element code PERMAS [2,3]. 3. Analytical limit load solution The plastic behaviour of an in+nite tube subjected to internal pressure and axial tension is analysed +rstly. The analytical limit loads P0 and N0 for the tube with Ra outer and Ri inner radius subjected to pure pressure and pure tension are 2y Ra and N0 = y ; (2) P0 = √ ln 3 Ri respectively [4].
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
1221
In the following, a lower bound for the limit load solution is given if the tube is loaded by tension N , combined with an internal pressure P. The loading vector is de+ned by: (N; (1 − )P) where ∈ [0; 1]. The limit load factor for di2erent values of is calculated to determine the admissible (N; P) domain. By choosing the following stress +eld a lower bound of the limit load factor is generated by using the static approach T = ((1 − )r (r)P; (1 − ) (r)P; (1 − )z P + N )
(3)
using the equilibrium equation r(@r =@r) − + r = 0 an admissible stress +eld is de+ned by 2y Ra Ra 1 Ra : (4) −ln ; 1 − ln ; − ln (r ; ; z )(r) = √ r r 2 r 3 The von Mises stress for (P = 1; N = 1) is given by F() = 2 + (1 − )2 , such that the limit load factor with reference to P0 and N0 is given by y y = ; (5) lim = min 2 r F() + (1 − )2 which de+nes a quarter of the unit-circle in the tension-pressure space normalized to N0 and P0 . 4. Limit analysis of defective pipe under internal pressure and axial tension The geometry of the defective closed pipe subjected to internal pressure and axial tension is shown in Fig. 1. The defects considered are part-through slots of various geometrical con+gurations. The axial tension N includes the independent axial tension N1 and the additional axial tension N2 induced by an internal pressure P. The limit load interaction curves of a cylindrical pipe with four di2erent sizes and shapes of part-through slots are computed by the proposed numerical algorithm. The geometric parameters of the defective pipes are presented in Table 1 (see Ref. [5]). Considering the symmetry of the structure, a quarter of the pipe is taken with the above four kinds of defects discretized by 3-D 20-node isoparametric +nite elements as shown in Fig. 2. The limit load curves of the pipes under the combined loads are plotted in Fig. 3 together with the analytical and numerical solution for the in+nite undisturbed tube.
Fig. 1. The geometry of the pipe with a part-through slot subjected to internal pressure and axial tension.
1222
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224 Table 1 The geometric parameters and dimensions of the defective pipes (mm) Defect type
Ri
Ra
L
A1
A
B
C
Case Case Case Case
17 17 17 17
21 21 21 21
250 250 250 250
0 0 =4 =4
4 4 4 4
4 20 4 20
4 4 4 4
2 2 2 2
(a) (b) (c) (d)
(a)
(b)
(c)
(d)
Fig. 2. The +nite element mesh of pipes with defects. (a) small slot; (b) axial slot; (c) circumferential slot; and (d) large slot.
From this +gure, it is clear that the small area slot has little e2ect on the limit load curve of the pipe. For a large area slot, the corresponding failure mode is a local collapse in the weakening. The analytical limit pressure for a pipe with R˜ a = 19 mm and Ri = 17 mm according to the large slot is given by P˜ = 0:1284y = 0:5264P0 . This corresponds very well with the numerical result of 0:5228P0 . i.e. in this case the failure mode of the weakened pipe dominates the structural behaviour of the defective pipe. An analytical lower bound for the limit tension of a circumferential external cracked pipe with crack depth a mm and thickness t is given for thin-walled pipes (conservative solution for thick-walled pipes) by Miller [4] and Al-Laham [6] 2y (Ra + Ri )(t − a) N˜ = √ = 0:5774y : (6) R2a − R2i 3 The limit load of the pipe with a circumferential slot is calculated as N = 0:5611y . This corresponds well with the solution of the cracked pipe, such that global failure occurs in the slot and the length of the slot in circumferential direction has no signi+cant in
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224 1
1223
no defect no defect (analytical)
0.9
small slot axial slot circumferential slot
0.8
large slot 0.7
P/P 0
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 N/N
0.8
1
0
Fig. 3. The limit load curves of pipeline under an internal pressure and an independent axial tension.
In general, the results correspond well with the upper bound obtained in Ref. [5]. For the circumferential slot the lower bound is up to 30% lower than the upper bound, such that the e2ect of the circumferential slot on the limit load seems to be larger than expected from the kinematic approach. Nevertheless, the e2ect of the slot on the limit axial load is higher than on the limit pressure. Because of the di2erent lower and upper bound approaches the exact limit loads are limited by these bounds. Therefore, the results of the lower bound approach can support the conclusions drawn in Ref. [5]. On the basis of these solutions of limit load interaction curves, the integrity assessment of pipeline with various kinds of slots can be performed by the widely used integrity assessment procedures such as Nuclear Electric’s (the former CEGB in the UK) R5 and R6 (the standards and codes of assessment of defective structures). 5. Conclusions By using a direct algorithm for lower limit analysis of 3-D structures, a plastic collapse analysis of defective pipes under multi-loading systems has been performed here. The main conclusions can be drawn as follows: (1) The proposed iterative algorithm is e9cient and reliable for performing the plastic collapse analysis of 3-D problems with complicated geometric forms and loading conditions. (2) The failure modes for defective pipes include a global collapse and a local leakage. Which failure mode occurs actually at the limit state for a defective pipeline should be determined by the size, position and orientation of the slot.
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M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
(3) The small area slot has little e2ect on the plastic load-carrying capacity of a pipeline, which collapses globally. For a pipe with a large area slot, the corresponding load-carrying capacity is decreased greatly. A local leakage generally occurs within the slot in this case. (4) The axial slot has a great e2ect on the limit load of a pipe under an internal pressure and has little e2ect under an axial force. The conclusion is opposite to the above for the circumferential slot. Acknowledgements Parts of this research have been funded by the Brite-EuRam III project LISA [7]: FEM-Based Limit and Shakedown Analysis for Design and Integrity Assessment in European Industry (Project No: BE 97-4547, Contract No: BRPR-CT97-0595). References [1] Staat M, Heitzer M, Yan A, Khoi V, Nguyen DH, Voldoire F, Lahousse A. Limit analysis of defects. Berichte des Forschungszentrums, JRulich 3746, 2000. [2] Heitzer M, Staat M. FEM–computation of load carrying capacity of highly loaded passive components by direct methods. Nuclear Engineering and Design 1999;193:349–58. [3] PERMAS user’s reference manuals. Stuttgart: INTES Publications No. 202, 207, 208, 302, UM 404, UM 405, 1988. [4] Miller AG. Review of limit loads of structures containing defects. International Journal of Pressure Vessels and Piping 1988;32:197–327. [5] Liu YH, Cen ZZ, Chen HF, Xu BY. Plastic collapse analysis of defective pipelines under multi-loading systems. International Journal of Mechanical Sciences 2000;42:1607–22. [6] Al-Laham A. Stress intensity factor and limit load handbook. British Energy, Report EPD=GEN=REP=0316=98, Issue 2, 1999. [7] Staat M, Heitzer M. LISA a European project for FEM-based limit and shakedown analysis. Nuclear Engineering and Design 2001;206:151–66.
Plastic limit loads of defective pipes under combined internal pressure and axial tension M. Heitzer Forschungszentrum Julich GmbH, Central Institute for Applied Mathematics (ZAM), D-52425 Julich, Germany Received 28 November 2000; received in revised form 5 January 2002
Abstract In this paper, a +nite element mathematical programming formulation is presented for the statical limit analysis of 3-D perfectly plastic structures. A direct iterative algorithm is employed in solving the above optimization formulation. The numerical procedure has been applied to carry out the plastic collapse analysis of defective pipes under combined internal pressure and axial tension. The engineering situation considered has a practical importance in the pipeline industry. The e2ects of four kinds of typical part-through slots on the collapse loads of pipes are investigated and evaluated. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The plastic limit load, which determines the load-carrying capacity of structures, is an important parameter in performing the two-criteria assessment of structural integrity [1]. Therefore, the knowledge of limit loads of mechanical components and structures is useful for designers to address the modes of failure associated with the loading. The determination of limit loads is by no means an easy task, especially for complex con+gurations and loading systems. Therefore, how to determine the limit load e9ciently and accurately has attracted the attention of many researchers. The lower and upper bound analyses of the limit load can be approached by mathematical programming processes based on the static and kinematic theorems of limit analysis. With the progress in the +nite element technique and mathematical optimization theory, the mathematical programming methods can determine the load-carrying capacity of a plastic body. Thus, the time-consuming step-by-step elastic–plastic +nite element analyses can be replaced. Pipelines are widely used in the various +elds such as the petrochemical industry, energy and electric power engineering, etc. During their operation, many local defects such as part-through slots can be produced by corrosion, mechanical damage or surface cracks. These defects may in
1220
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
integrity of pipes, such that an investigation of the possible decrease of the load-carrying capacity due to part-through slots is of bene+t. In the present paper, a static approach to limit analysis of 3-D structures is proposed by means of a direct iterative algorithm [2]. The limit analysis of defective pipes is performed under multi-loading systems. The defects considered here include part-through spherical and ellipsoidal slots. The effects of various shapes and sizes of typical part-through slots on the collapse loads of pipes are investigated. 2. Limit analysis The structure V is loaded monotonously by load P = (q; p). The engineer is interested in the load factor ¿ 1 by which P can be increased up to the collapse at P. The limit load analysis investigates the collapse state, in which the structure fails with unrestricted
in V;
div = −q n = p
in V;
on @V :
(1)
For each stress +eld , which ful+ls the conditions of the static theorem, is a safety factor, so that the load-carrying capacity of the structure is not yet exhausted. One calculates a lower bound of the limit load factor as the largest safety factor which ful+ls the conditions of Eq. (1). This is in the case of the von Mises yield function F a nonlinear optimization problem with the unknowns and . The constraints of the optimization problem are validated only in the Gaussian points, de+ned by a FEM discretization. The number of Gaussian points becomes huge for industrial structures and no e2ective solution algorithms for the nonlinear optimization problem are available. The basis reduction technique for handling such large-scale optimization problems was used in Ref. [2]. The basis of the subspaces are generated by the general purpose +nite element code PERMAS [2,3]. 3. Analytical limit load solution The plastic behaviour of an in+nite tube subjected to internal pressure and axial tension is analysed +rstly. The analytical limit loads P0 and N0 for the tube with Ra outer and Ri inner radius subjected to pure pressure and pure tension are 2y Ra and N0 = y ; (2) P0 = √ ln 3 Ri respectively [4].
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
1221
In the following, a lower bound for the limit load solution is given if the tube is loaded by tension N , combined with an internal pressure P. The loading vector is de+ned by: (N; (1 − )P) where ∈ [0; 1]. The limit load factor for di2erent values of is calculated to determine the admissible (N; P) domain. By choosing the following stress +eld a lower bound of the limit load factor is generated by using the static approach T = ((1 − )r (r)P; (1 − ) (r)P; (1 − )z P + N )
(3)
using the equilibrium equation r(@r =@r) − + r = 0 an admissible stress +eld is de+ned by 2y Ra Ra 1 Ra : (4) −ln ; 1 − ln ; − ln (r ; ; z )(r) = √ r r 2 r 3 The von Mises stress for (P = 1; N = 1) is given by F() = 2 + (1 − )2 , such that the limit load factor with reference to P0 and N0 is given by y y = ; (5) lim = min 2 r F() + (1 − )2 which de+nes a quarter of the unit-circle in the tension-pressure space normalized to N0 and P0 . 4. Limit analysis of defective pipe under internal pressure and axial tension The geometry of the defective closed pipe subjected to internal pressure and axial tension is shown in Fig. 1. The defects considered are part-through slots of various geometrical con+gurations. The axial tension N includes the independent axial tension N1 and the additional axial tension N2 induced by an internal pressure P. The limit load interaction curves of a cylindrical pipe with four di2erent sizes and shapes of part-through slots are computed by the proposed numerical algorithm. The geometric parameters of the defective pipes are presented in Table 1 (see Ref. [5]). Considering the symmetry of the structure, a quarter of the pipe is taken with the above four kinds of defects discretized by 3-D 20-node isoparametric +nite elements as shown in Fig. 2. The limit load curves of the pipes under the combined loads are plotted in Fig. 3 together with the analytical and numerical solution for the in+nite undisturbed tube.
Fig. 1. The geometry of the pipe with a part-through slot subjected to internal pressure and axial tension.
1222
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224 Table 1 The geometric parameters and dimensions of the defective pipes (mm) Defect type
Ri
Ra
L
A1
A
B
C
Case Case Case Case
17 17 17 17
21 21 21 21
250 250 250 250
0 0 =4 =4
4 4 4 4
4 20 4 20
4 4 4 4
2 2 2 2
(a) (b) (c) (d)
(a)
(b)
(c)
(d)
Fig. 2. The +nite element mesh of pipes with defects. (a) small slot; (b) axial slot; (c) circumferential slot; and (d) large slot.
From this +gure, it is clear that the small area slot has little e2ect on the limit load curve of the pipe. For a large area slot, the corresponding failure mode is a local collapse in the weakening. The analytical limit pressure for a pipe with R˜ a = 19 mm and Ri = 17 mm according to the large slot is given by P˜ = 0:1284y = 0:5264P0 . This corresponds very well with the numerical result of 0:5228P0 . i.e. in this case the failure mode of the weakened pipe dominates the structural behaviour of the defective pipe. An analytical lower bound for the limit tension of a circumferential external cracked pipe with crack depth a mm and thickness t is given for thin-walled pipes (conservative solution for thick-walled pipes) by Miller [4] and Al-Laham [6] 2y (Ra + Ri )(t − a) N˜ = √ = 0:5774y : (6) R2a − R2i 3 The limit load of the pipe with a circumferential slot is calculated as N = 0:5611y . This corresponds well with the solution of the cracked pipe, such that global failure occurs in the slot and the length of the slot in circumferential direction has no signi+cant in
M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224 1
1223
no defect no defect (analytical)
0.9
small slot axial slot circumferential slot
0.8
large slot 0.7
P/P 0
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 N/N
0.8
1
0
Fig. 3. The limit load curves of pipeline under an internal pressure and an independent axial tension.
In general, the results correspond well with the upper bound obtained in Ref. [5]. For the circumferential slot the lower bound is up to 30% lower than the upper bound, such that the e2ect of the circumferential slot on the limit load seems to be larger than expected from the kinematic approach. Nevertheless, the e2ect of the slot on the limit axial load is higher than on the limit pressure. Because of the di2erent lower and upper bound approaches the exact limit loads are limited by these bounds. Therefore, the results of the lower bound approach can support the conclusions drawn in Ref. [5]. On the basis of these solutions of limit load interaction curves, the integrity assessment of pipeline with various kinds of slots can be performed by the widely used integrity assessment procedures such as Nuclear Electric’s (the former CEGB in the UK) R5 and R6 (the standards and codes of assessment of defective structures). 5. Conclusions By using a direct algorithm for lower limit analysis of 3-D structures, a plastic collapse analysis of defective pipes under multi-loading systems has been performed here. The main conclusions can be drawn as follows: (1) The proposed iterative algorithm is e9cient and reliable for performing the plastic collapse analysis of 3-D problems with complicated geometric forms and loading conditions. (2) The failure modes for defective pipes include a global collapse and a local leakage. Which failure mode occurs actually at the limit state for a defective pipeline should be determined by the size, position and orientation of the slot.
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M. Heitzer / International Journal of Mechanical Sciences 44 (2002) 1219 – 1224
(3) The small area slot has little e2ect on the plastic load-carrying capacity of a pipeline, which collapses globally. For a pipe with a large area slot, the corresponding load-carrying capacity is decreased greatly. A local leakage generally occurs within the slot in this case. (4) The axial slot has a great e2ect on the limit load of a pipe under an internal pressure and has little e2ect under an axial force. The conclusion is opposite to the above for the circumferential slot. Acknowledgements Parts of this research have been funded by the Brite-EuRam III project LISA [7]: FEM-Based Limit and Shakedown Analysis for Design and Integrity Assessment in European Industry (Project No: BE 97-4547, Contract No: BRPR-CT97-0595). References [1] Staat M, Heitzer M, Yan A, Khoi V, Nguyen DH, Voldoire F, Lahousse A. Limit analysis of defects. Berichte des Forschungszentrums, JRulich 3746, 2000. [2] Heitzer M, Staat M. FEM–computation of load carrying capacity of highly loaded passive components by direct methods. Nuclear Engineering and Design 1999;193:349–58. [3] PERMAS user’s reference manuals. Stuttgart: INTES Publications No. 202, 207, 208, 302, UM 404, UM 405, 1988. [4] Miller AG. Review of limit loads of structures containing defects. International Journal of Pressure Vessels and Piping 1988;32:197–327. [5] Liu YH, Cen ZZ, Chen HF, Xu BY. Plastic collapse analysis of defective pipelines under multi-loading systems. International Journal of Mechanical Sciences 2000;42:1607–22. [6] Al-Laham A. Stress intensity factor and limit load handbook. British Energy, Report EPD=GEN=REP=0316=98, Issue 2, 1999. [7] Staat M, Heitzer M. LISA a European project for FEM-based limit and shakedown analysis. Nuclear Engineering and Design 2001;206:151–66.