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Chapter 13 Reliability-based strength design of pipelines
219
Chapter 13 Reliability-Based Strength Design of Pipelines 13.1
General
A technical revolution in the design process is taking place in the pipeline industry as a result of new codes, e.g. ISO DIS 13623 Code (ISO 1997), and other codes. Advanced methods and analysis tools allow a more sophisticated approach to design that takes advantage of modern materials and the revised design codes. A "Design Through Analysis" (DTA) approach has been developed by Bai and Damsleth (1998) where the finite element method (FEM) is used to analyze global behavior as well as local structural strength of pipelines. The structural reliability method is used to determine the partial safety factors used in the finite element analyses. Reliability-based limit-state design principles are described in will be issued as an ISO guideline. Advanced engineering reliability are increasingly demanded by design projects to deepwater, High Pressure/High Temperature (HP/HT), new and reassessment of existing pipelines.
NORSOK Standard Y-002 and based on FEM and structural meet new challenges such as materials, harsh environments
A pipeline design typically involves the following technical aspects: 9
route optimization
9
wall-thickness design
9
on-bottom stability analysis
9
installation analysis
9
upheaval and lateral buckling design
9
free-spans design for vortex-induced vibrations (VIV) and fishing gear impacts
9
seabed intervention design
9
constructions such as tie-in and pipeline crossings
Bai and Damsleth (1998) demonstrated that reliability-based limit-state design may be applied to the above technical aspects of pipeline design.
220
Chapter 13
The purpose of this chapter is to present the experience gained from reliability-based limitstate design applied in practical design and re-qualification of pipelines to achieve costeffective solutions (e.g. Bai et al. (1997), Bai and Song (1998)). This chapter summarizes methods for reliability analysis, uncertainty measures, target reliability levels and calibration of safety factors. A limit-state design case study is then presented and discussed.
13.2 Reliability-based Design 13.2.1 General In principle, reliability-based design of offshore pipelines involves the following aspects: 9
Identification of failure modes for specified design cases;
9
Definition of design formats and Limit State Functions (LSF);
9
Uncertainty measurements of all random variables;
9
Calculation of failure probability;
9
Determination of target reliability levels;
9
Calibration of safety factors for design;
9
Evaluation of design results.
13.2.2 Deterministic vs. Probabilistic Design Structural design codes commonly specify loads and strength and appropriate safety factors for design use. Generally, two design approaches are being adopted namely deterministic and probabilistic designs. In traditional (deterministic) design, the relevant loads, load effects and material properties are defined as deterministic quantities. Two basic design equations are explicitly specified for the yielding check: the hoop stress criterion and the equivalent stress criterion. In reality, most of the design variables, such as wall-thickness and material properties, contain uncertainty. In addition, the idealized analytical model is also a source of uncertainties. Hence, a probabilistic approach is required to provide an appropriate method to deal with these uncertainties and to achieve a consistently safe level of design. Besides, different failures may occur in different design scenarios and, hence, lead to different failure consequences. Reliability methods may be applied to achieve a cost-effective design that balances both safety and costs. 13.2.3 Load Effects and Combinations In general, the following loads and load combinations in pipeline structural design should be considered:
Reliability-Based Strength Design of Pipelines
9
221
Functional loads, e.g. internal and external pressure load effects, thermal forces, pipe
weight and residual lay forces. 9
Environmental loads, e.g. wave (in shallow water) and current loads.
9
Accidental load effects, e.g. fishing gear impact, dropped objects impact, anchor impact,
etc. 9
Combinations of the above.
The functional load and environmental load effects are related to the pipeline system. While accidental load effects and load combinations may be critical to the local components. Two design phases are defined: temporary and operational.
13.2.4 LRFD Design Format To achieve a uniform safety level for a range of parameter variation, an appropriate design format, which should be simple to use in design, will be selected. The design format is usually based on LRFD (Load Resistance Factored Design). The selected design format should be a simplified representation of the actual limit state condition under consideration. The most significant variables should be included in the design format. A representative LRFD design format is expressed as: YEScE+ YFScF < R',c/~,,
(13.1)
/IR
where Sc and Rc are characteristic load effect and resistance of the considered failure mode, y is the partial safety factors to be calibrated, Subscripts E and F denote environmental loads and functional loads respectively. The design values of load effects and capacity are estimated as the product of characteristic values and partial safety factors. Four kinds of limit states and related failure modes for pipelines are generally identified namely serviceability limit state (SLS), ultimate limit state (ULS), fatigue limit state (FLS) and accidental limit state (ALS).
13.2.5 Calculation of Failure Probability Generally, limit-state function (LSF) is introduced and denoted by g(Z) where Z is the vector of all uncertainty variables. Failure occurs when g(Z)<0. For a given LSF g(Z), the probability of failure is defined as: e~(t)= Pig(z)_
(13.2)
The results can also be expressed in terms of a reliability index 13, which is uniquely related to the failure probability by: 13(t)= -~,-' (Pf(t))= ~-'(- Pf(t))
(13.3)
Chapter 13
222 where ~(.) is standard normal distribution function.
Two general approaches are available to solve Equation (13.2) namely analytical and simulation methods respectively.
Analytical Methods: Analytical methods consist of first- and second-order reliability methods (FORM and SORM). The advantage of these methods is that they do usually not require excessively large computing cost. The drawback is that they do not give exact results, but only approximations that may not always be sufficiently accurate. Details of FORM and SORM are available from standard textbooks, e.g. Thoft-Christensen and Baker (1982).
Simulation Methods: A Monte Carlo simulation technique is an alternative or complementary tool for estimation of failure probability. The advantage of this technique is that the methods are very simple and give solutions, which converge towards exact results when a sufficient number of simulations are performed. The disadvantage of the simulation methods is that their computing efficiency is low. Many refined simulation methods have been developed to improve the efficiency of simulations.
13.3 Uncertainty Measures 13.3.1 General Failure probability is evaluated based on uncertainties associated with the considered LSF, which is composed of a set of basic random variables and analysis models. Uncertainty measures are a critical and fundamental step in reliability analysis. The major steps involved in the measurement of uncertainty include the following: 9
Classification of uncertainties,
9
Selection of distribution functions,
9
Determination of statistical values of those random variables in the LSF.
13.3,2 Classification of Uncertainties Uncertainty of a random variable can be measured using a probability distribution function and statistical values. The major uncertainties considered in this study include the following (Thoft-Christensen and Baker (1982)):
Physical uncertainty: Caused by random nature of the actual variability of physical quantities, such as pipe geometry (wall-thickness), etc.
Statistical uncertainty: This is uncertainty due to incomplete information of the variable. It is a function of the type of distribution function fitted, type of estimation technique applied, value of the distribution parameters and amount of underlying data. Statistical uncertainty may further occur due to negligence of systematic variations of the observed variables. This
Reliability-BasedStrength Design of Pipelines
223
uncertainty can be reduced by additional information of the variable in terms of its statistical significance.
Model uncertainty: This is uncertainty due to simplifications and assumptions made in establishing the analytical model and reflects a general confidence in the applied model to describe the real situation. It results in the difference between actual and predicted results. Model uncertainty in a physical model for presentation of the load or resistance quantities may be represented by a stochastic factor defined as the ratio between the true quantity and the quantity described by the model. Guedes Soares (1997) discussed the common methods of representing model uncertainties and illustrated principles of assessing model uncertainties. Considering uncertainties involved in the design format, each random variable Xi can be specified as: xi = Bx" Xc
(13.4)
where Xc is the characteristic value of Xi, and Bx is a normalized variable reflecting the uncertainty in Xi.
13.3.3 Selection of Distribution Functions Usually, the determination of the distribution function is strongly influenced by the physical nature of the random variables. Also, its determination may be related to a well-known description and stochastic experiment. Experience from similar problems is also very useful. If several distributions are available, it is necessary to identify by plotting of data on probability paper, by comparisons of moments, statistical tests, etc. Normal or lognormal distributions are normally applied when no detailed information is available. For instance, resistance variables are usually modeled by normal distribution, and lognormal distribution is used for load variables. The occurrence frequency of a damage (e.g. an initial crack), is described by Poisson distribution. Exponential distribution is used to model the capacity of detecting a certain damage.
13.3.4 Determination of Statistical Values Statistical values used to describe a random variable are mean value and coefficient of variation (COV). These statistical values shall normally be obtained from recognized data sources. Regression analysis may be applied based on methods of moment, least-square fit methods, maximum likelihood estimation technique, etc.
13.4 Calibration of Safety Factors 13.4.1 General One of the important applications of structural reliability methods is to calibrate safety factors in design format in order to achieve a consistent safety level. The safety factors are determined so that the calibrated failure probability, Pf, i for various conditions is as close to the target reliability level Pf as possible:
Chapter 13
224
(13.5)
Zfi(Pc~(y)- P~)2 = minimum
where fi is the relative frequency of the design case number i.
13.4.2 Target Reliability Levels When conducting structural reliability analysis, target reliability levels in a given reference time period and reference length of pipeline should be selected. The selection is based on consequence of failure, location and contents of pipelines, relevant rules, access to inspection and repair, etc. The target reliability levels have to be met in design to ensure that certain safety levels are maintained. The following safety classes are proposed:
Low safety class: where failure implies no risk of human injury, minor environmental damage and economic consequences.
Normal safety class: classification for temporary conditions where failure implies risk of human injury, significant environmental and economical consequences.
High safety class: classification for operation conditions where failure implies risk of human injury, significant environmental and economical consequences. Target reliability levels may be specified by the operator guided by authority requirements, design philosophy and risk attitude in terms of economics. The target reliability level for damaged pipelines should be defined in the same level as intact pipeline. The target reliability level needs to be evaluated considering the implied safety level in the existing rules and codes. Sotberg et al. (1997) proposed target reliability levels as below: Table 13.1 Target reliability levels (Sotberg, et al. (1997)).
Limit States
Safety Classes Low
Normal
High
SLS
101-- 10-2
10-2-- 10.3
10-2-- 10.3
ULS
10"2-- 10.3
10.3-- 10-4
10-4-- 10.5
FLS
10.3
104
10.5
ALS
10.4
10.5
10 -6
13.5 Buckling/Collapse of Corroded Pipes 13.5.1 Buckling/Collapse In the DNV'96 rules for submarine pipelines two alternative criteria are defined for local buckling/collapse for internal over pressure cases (and load-controlled situations):
Reliability-Based Strength Design of Pipelines
225
LRFD moment format (DNV'96 Section 5, C300) ASD (Allowable Stress Design) Format - checks of equivalent- and longitudinal stress, as an alternative to the moment check. (DNV'96 Section 5, C400). The ASD format is originally from ISO DIS 13623. The ASD criteria define a maximum allowable longitudinal and equivalent stress as a percentage of SMYS. A pipe may be stressed well beyond the yield stress before it loses its capability to fulfil its function and the ASD criteria should therefore be rather conservative for this limit state. The interaction equation between moment, internal pressure and axial loads for local buckling/collapse in DNV'96 is a yield criterion from Mohareb et al. (1994). The maximum capacity of the pipe is here defined as the bending moment at which the entire cross-section yields. The moment criteria in DNV'96 (Section5, C305) is a LRFD format of the Mohareb's equation, where partial safety factors are defined based on structural reliability analysis (MCrk et al. (1997)). However, Hauch and Bai (1998) found that the allowable moment given by DNV'96 for some load conditions is at least 20% lower than that corresponding to the ASD criteria. In the following the moment criteria shall be re-visited and the causes of the conservatism will be identified, together with a newly suggested calibration of safety factors.
13.5.2 Analytical Capacity Equation Bai and Hauch (1998) modified the yield criterion from Mohareb et al. (1994), to account for the effect of corrosion defects. The ultimate moment capacity is also defined as the moment at which the entire cross section yields. The cases considered: the defect section is in compression (case 1), in compression and some in tension (case 2), in tension (case 3), in tension and some in compression (case 4). The four cases are shown in Figure 3.4.
13.5.3 Design Format For internal-over pressure cases, the design format based on DNV'96 (Section 5 C305) may be written as: YFYcMF,c+ YEME.c< Mc/YR
(13.6)
where MF,c is the characteristic functional bending moment; ME,c is the characteristic environmental bending moment; Me is the characteristic ultimate bending moment (as given in Chapter 3); YF is the functional load factor =1.1; YE is the environmental load factor which is 1.3; 7c is the conditional load factor; YRis the strength resistance factor.
13.5.4 Limit-State Function The limit-state function may be expressed as: g(Z) = Mc -(M F+ ME)
(13.7)
226
Chapter 13
where Me denotes the stochastic ultimate moment, MF and ME are stochastic applied load effects for the functional and environmental bending moment. Mc may be expressed as the product of the model uncertainty parameter XM and the moment capacity given by Bai and Hauch (1998) equations. A large amount of experimental and numerical tests are required to quantify mean value, COV and distribution function of the model uncertainty XM. All of the stochastic variables M~, MF and ME shall be defined for the parameter range of interests (e.g. hoop stress 0.8SMYS for operating conditions, 0.96SMYS for hydro test conditions). Uncertainty measures may be found from the SUPERB project, e.g. Jiao et al. (1997).
13.5.5 Calibration of Safety Factors In order to reduce the conservatism, the following shall be given considerations in the calibration of safety factors: 1) Use capacity equation for corroded pipes. Section 5 B205 of DNV'96 states that the wall-thickness used in buckling/collapse calculation for pipelines in operation shall exclude corrosion allowance. However, the width of corrosion defects is typically less than a quarter of the circle. Neglecting the whole circle would "likely lead to 10% less capacity predicted.
2) Use SMTS (Specified Minimum Tensile Stress) as o'v in Bai and Hauch equations. Figure 3.13 to 3.16 shows that the moment capacity equation agrees well with the finite element predictions (which have been validated against laboratory tests), if SMTS is used as cry in the equations. The reasoning of using SMTS is strain-hardening effects and the strengthening due to outward deformation for highly pressurized pipes in collapse. Unfortunately, in DNV'96, SMYS (Specified Minimum Yield Stress) is used as ~y. For X65 material, the ratio of SMTS and SMYS is 1.17. Laboratory tests by Mohareb et al. (1994) have confirmed that the mean bias for highly pressurized pipes is about 1.05 if SMTS is used as cry in the capacity equations. 3) Use strain-based design or a conditional load factor Yc (<1.0) for displacement dominant situations. For high pressure and high temperature (HP/HT) pipelines, pressure and temperature induced axial stress and moment could be large. Up-lift and lateral buckling behavior is typically displacement controlled. When a HP/HT pipeline is subject to fishing gear pullover load, its response is load dominant for small diameter pipelines and displacement dominant for large diameter pipelines. It is therefore suggested that a conditional load factor Yc is introduced to reflect the differences in the structural response to fishing gear load. In many practical situations, no adequate capacity equation is available for strength prediction due to the complexity of the problem. Instead, numerical tests (using FEM) and laboratory tests are conducted for strength design. It is then required to select partial safety factors that may be applied together with a (numerical) structural laboratory (e.g. finite element analysis). Selecting partial safety factors using reliability methods, FEM may be applied to strength design as an alternative to direct use of code equations. The advantage of such an innovative approach is that when information is lacking from the design codes for new materials,
Reliability-BasedStrength Design of Pipelines
227
deepwater and harsh environments, project specific design criteria can be established by use of finite elements and reliability methods. It is difficult (if at all possible) to define code criteria which can cover all combinations of design scenarios (e.g. installation, hydro tests, operation), pipe materials (e.g. corrosion defects), loads (high hoop stress and low hoops stress cases) and new phenomenon which appear in challenging projects. Therefore, subject to achievable cost-saving and technical qualification, design engineers should develop project-specific design criteria based on the direct use of the reliability-based calibration under the principles of the NORSOK standard Y002 (1997).
13.6 Conclusions
The chapter presents reliability-based limit-state design and re-qualifications of pipelines. Following a summarized discussions of reliability and uncertainty measures, target reliability levels and calibration of safety factors, the following two limit-state design applications are given. (1) Design factors for hoop stress criterion: it is shown how the usage factor has been derived using reliability-based calibration. (2) Buckling/collapse of Corroded Pipes: the causes of the conservatism in the existing codes were identified and the reliability-based determination of partial safety factors was discussed. Further study is required to refine uncertainty measures and to develop methodology for the determination of partial safety factors for finite element models and laboratory tests.
13.7 References
1. Bai, Y. and Damsleth, P.A., (1998) "Design Through Analysis Applying Limit-state Concepts and Reliability Methods", Proc. of ISOPE'98. A plenary presentation at ISOPE'98. 2. Bai, Y. and Hauch, S., (1998) "Analytical Collapse Capacity of Corroded Pipes", Proc. of ISOPE'98. 3. Bai, Y. and Song, R., (1997) "Fracture Assessment of Dented Pipes with Cracks and Reliability-based Calibration of Safety Factors", Intemational Joumal of Pressure Vessels and Piping, Vol. 24, pp. 221-229. 4. Bai, Y., Xu, T. and Bea, R., (1997) "Reliability-based Design & Re-qualification criteria for Longitudinally Corroded Pipelines", Proc. of ISOPE '97. 5. DNV (1996) "Rules for Submarine Pipelines", Det Norske Veritas. 6. Guedes Soares, C., (1997) "Quantification of Model Uncertainty in Structural Reliability", in Probabilistic Methods for Structural Design, edited by C. Guedes Soares. Kluwer Academic Publishers. 7. Hauch, S. and Bai, Y., (1998) "Use of Finite Element Methods for Local Buckling Design of Pipeline", Proc. of OMAE '98.
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8. ISO/DIS 13623(1997) "Petroleum and Natural Gas Industries; Pipeline Transportation Systems", International Standard Organisation. 9. Jiao, G., Sotberg T., Bruschi, R. and Igland, R.T., (1997) "The SUPERB Project: Linepipe Statistical Properties and Implications in Design of Offshore Pipelines", Proc. of OMAE'97. 10. Mohareb, M.E., Elwi, A.E., Kulak, G.L. and Murray, D.W., (1994) "Deformational Behaviour of Line Pipe", Structural Engineering Report No. 22, University of Alberta. 11. M0rk K.J., Spiten J., Torselletti, E., Ness O.B. and Verley R., (1997) "The SUPERB Project & DNV'96: Buckling and Collapse Limit State", Proc. of OMAE'97. 12. NORSOK Standard Y-002 (1997) "Reliability-based Limit-state Principles for Pipeline design". 13. Sotberg, T., Moan, T., Bruschi R., Jiao, G. and MCrk, K.J., (1997) "The SUPERB Project: Recommended Target Safety Levels for Limit State Based Design of Offshore Pipelines", Proc. of OMAE'97. 14. Thoft-Christensen, P. and Baker, M.J., (1982) "Structural Reliability, Theory and its Applications", Springer-Verlag, 1982.
Chapter 13 Reliability-Based Strength Design of Pipelines 13.1
General
A technical revolution in the design process is taking place in the pipeline industry as a result of new codes, e.g. ISO DIS 13623 Code (ISO 1997), and other codes. Advanced methods and analysis tools allow a more sophisticated approach to design that takes advantage of modern materials and the revised design codes. A "Design Through Analysis" (DTA) approach has been developed by Bai and Damsleth (1998) where the finite element method (FEM) is used to analyze global behavior as well as local structural strength of pipelines. The structural reliability method is used to determine the partial safety factors used in the finite element analyses. Reliability-based limit-state design principles are described in will be issued as an ISO guideline. Advanced engineering reliability are increasingly demanded by design projects to deepwater, High Pressure/High Temperature (HP/HT), new and reassessment of existing pipelines.
NORSOK Standard Y-002 and based on FEM and structural meet new challenges such as materials, harsh environments
A pipeline design typically involves the following technical aspects: 9
route optimization
9
wall-thickness design
9
on-bottom stability analysis
9
installation analysis
9
upheaval and lateral buckling design
9
free-spans design for vortex-induced vibrations (VIV) and fishing gear impacts
9
seabed intervention design
9
constructions such as tie-in and pipeline crossings
Bai and Damsleth (1998) demonstrated that reliability-based limit-state design may be applied to the above technical aspects of pipeline design.
220
Chapter 13
The purpose of this chapter is to present the experience gained from reliability-based limitstate design applied in practical design and re-qualification of pipelines to achieve costeffective solutions (e.g. Bai et al. (1997), Bai and Song (1998)). This chapter summarizes methods for reliability analysis, uncertainty measures, target reliability levels and calibration of safety factors. A limit-state design case study is then presented and discussed.
13.2 Reliability-based Design 13.2.1 General In principle, reliability-based design of offshore pipelines involves the following aspects: 9
Identification of failure modes for specified design cases;
9
Definition of design formats and Limit State Functions (LSF);
9
Uncertainty measurements of all random variables;
9
Calculation of failure probability;
9
Determination of target reliability levels;
9
Calibration of safety factors for design;
9
Evaluation of design results.
13.2.2 Deterministic vs. Probabilistic Design Structural design codes commonly specify loads and strength and appropriate safety factors for design use. Generally, two design approaches are being adopted namely deterministic and probabilistic designs. In traditional (deterministic) design, the relevant loads, load effects and material properties are defined as deterministic quantities. Two basic design equations are explicitly specified for the yielding check: the hoop stress criterion and the equivalent stress criterion. In reality, most of the design variables, such as wall-thickness and material properties, contain uncertainty. In addition, the idealized analytical model is also a source of uncertainties. Hence, a probabilistic approach is required to provide an appropriate method to deal with these uncertainties and to achieve a consistently safe level of design. Besides, different failures may occur in different design scenarios and, hence, lead to different failure consequences. Reliability methods may be applied to achieve a cost-effective design that balances both safety and costs. 13.2.3 Load Effects and Combinations In general, the following loads and load combinations in pipeline structural design should be considered:
Reliability-Based Strength Design of Pipelines
9
221
Functional loads, e.g. internal and external pressure load effects, thermal forces, pipe
weight and residual lay forces. 9
Environmental loads, e.g. wave (in shallow water) and current loads.
9
Accidental load effects, e.g. fishing gear impact, dropped objects impact, anchor impact,
etc. 9
Combinations of the above.
The functional load and environmental load effects are related to the pipeline system. While accidental load effects and load combinations may be critical to the local components. Two design phases are defined: temporary and operational.
13.2.4 LRFD Design Format To achieve a uniform safety level for a range of parameter variation, an appropriate design format, which should be simple to use in design, will be selected. The design format is usually based on LRFD (Load Resistance Factored Design). The selected design format should be a simplified representation of the actual limit state condition under consideration. The most significant variables should be included in the design format. A representative LRFD design format is expressed as: YEScE+ YFScF < R',c/~,,
(13.1)
/IR
where Sc and Rc are characteristic load effect and resistance of the considered failure mode, y is the partial safety factors to be calibrated, Subscripts E and F denote environmental loads and functional loads respectively. The design values of load effects and capacity are estimated as the product of characteristic values and partial safety factors. Four kinds of limit states and related failure modes for pipelines are generally identified namely serviceability limit state (SLS), ultimate limit state (ULS), fatigue limit state (FLS) and accidental limit state (ALS).
13.2.5 Calculation of Failure Probability Generally, limit-state function (LSF) is introduced and denoted by g(Z) where Z is the vector of all uncertainty variables. Failure occurs when g(Z)<0. For a given LSF g(Z), the probability of failure is defined as: e~(t)= Pig(z)_
(13.2)
The results can also be expressed in terms of a reliability index 13, which is uniquely related to the failure probability by: 13(t)= -~,-' (Pf(t))= ~-'(- Pf(t))
(13.3)
Chapter 13
222 where ~(.) is standard normal distribution function.
Two general approaches are available to solve Equation (13.2) namely analytical and simulation methods respectively.
Analytical Methods: Analytical methods consist of first- and second-order reliability methods (FORM and SORM). The advantage of these methods is that they do usually not require excessively large computing cost. The drawback is that they do not give exact results, but only approximations that may not always be sufficiently accurate. Details of FORM and SORM are available from standard textbooks, e.g. Thoft-Christensen and Baker (1982).
Simulation Methods: A Monte Carlo simulation technique is an alternative or complementary tool for estimation of failure probability. The advantage of this technique is that the methods are very simple and give solutions, which converge towards exact results when a sufficient number of simulations are performed. The disadvantage of the simulation methods is that their computing efficiency is low. Many refined simulation methods have been developed to improve the efficiency of simulations.
13.3 Uncertainty Measures 13.3.1 General Failure probability is evaluated based on uncertainties associated with the considered LSF, which is composed of a set of basic random variables and analysis models. Uncertainty measures are a critical and fundamental step in reliability analysis. The major steps involved in the measurement of uncertainty include the following: 9
Classification of uncertainties,
9
Selection of distribution functions,
9
Determination of statistical values of those random variables in the LSF.
13.3,2 Classification of Uncertainties Uncertainty of a random variable can be measured using a probability distribution function and statistical values. The major uncertainties considered in this study include the following (Thoft-Christensen and Baker (1982)):
Physical uncertainty: Caused by random nature of the actual variability of physical quantities, such as pipe geometry (wall-thickness), etc.
Statistical uncertainty: This is uncertainty due to incomplete information of the variable. It is a function of the type of distribution function fitted, type of estimation technique applied, value of the distribution parameters and amount of underlying data. Statistical uncertainty may further occur due to negligence of systematic variations of the observed variables. This
Reliability-BasedStrength Design of Pipelines
223
uncertainty can be reduced by additional information of the variable in terms of its statistical significance.
Model uncertainty: This is uncertainty due to simplifications and assumptions made in establishing the analytical model and reflects a general confidence in the applied model to describe the real situation. It results in the difference between actual and predicted results. Model uncertainty in a physical model for presentation of the load or resistance quantities may be represented by a stochastic factor defined as the ratio between the true quantity and the quantity described by the model. Guedes Soares (1997) discussed the common methods of representing model uncertainties and illustrated principles of assessing model uncertainties. Considering uncertainties involved in the design format, each random variable Xi can be specified as: xi = Bx" Xc
(13.4)
where Xc is the characteristic value of Xi, and Bx is a normalized variable reflecting the uncertainty in Xi.
13.3.3 Selection of Distribution Functions Usually, the determination of the distribution function is strongly influenced by the physical nature of the random variables. Also, its determination may be related to a well-known description and stochastic experiment. Experience from similar problems is also very useful. If several distributions are available, it is necessary to identify by plotting of data on probability paper, by comparisons of moments, statistical tests, etc. Normal or lognormal distributions are normally applied when no detailed information is available. For instance, resistance variables are usually modeled by normal distribution, and lognormal distribution is used for load variables. The occurrence frequency of a damage (e.g. an initial crack), is described by Poisson distribution. Exponential distribution is used to model the capacity of detecting a certain damage.
13.3.4 Determination of Statistical Values Statistical values used to describe a random variable are mean value and coefficient of variation (COV). These statistical values shall normally be obtained from recognized data sources. Regression analysis may be applied based on methods of moment, least-square fit methods, maximum likelihood estimation technique, etc.
13.4 Calibration of Safety Factors 13.4.1 General One of the important applications of structural reliability methods is to calibrate safety factors in design format in order to achieve a consistent safety level. The safety factors are determined so that the calibrated failure probability, Pf, i for various conditions is as close to the target reliability level Pf as possible:
Chapter 13
224
(13.5)
Zfi(Pc~(y)- P~)2 = minimum
where fi is the relative frequency of the design case number i.
13.4.2 Target Reliability Levels When conducting structural reliability analysis, target reliability levels in a given reference time period and reference length of pipeline should be selected. The selection is based on consequence of failure, location and contents of pipelines, relevant rules, access to inspection and repair, etc. The target reliability levels have to be met in design to ensure that certain safety levels are maintained. The following safety classes are proposed:
Low safety class: where failure implies no risk of human injury, minor environmental damage and economic consequences.
Normal safety class: classification for temporary conditions where failure implies risk of human injury, significant environmental and economical consequences.
High safety class: classification for operation conditions where failure implies risk of human injury, significant environmental and economical consequences. Target reliability levels may be specified by the operator guided by authority requirements, design philosophy and risk attitude in terms of economics. The target reliability level for damaged pipelines should be defined in the same level as intact pipeline. The target reliability level needs to be evaluated considering the implied safety level in the existing rules and codes. Sotberg et al. (1997) proposed target reliability levels as below: Table 13.1 Target reliability levels (Sotberg, et al. (1997)).
Limit States
Safety Classes Low
Normal
High
SLS
101-- 10-2
10-2-- 10.3
10-2-- 10.3
ULS
10"2-- 10.3
10.3-- 10-4
10-4-- 10.5
FLS
10.3
104
10.5
ALS
10.4
10.5
10 -6
13.5 Buckling/Collapse of Corroded Pipes 13.5.1 Buckling/Collapse In the DNV'96 rules for submarine pipelines two alternative criteria are defined for local buckling/collapse for internal over pressure cases (and load-controlled situations):
Reliability-Based Strength Design of Pipelines
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LRFD moment format (DNV'96 Section 5, C300) ASD (Allowable Stress Design) Format - checks of equivalent- and longitudinal stress, as an alternative to the moment check. (DNV'96 Section 5, C400). The ASD format is originally from ISO DIS 13623. The ASD criteria define a maximum allowable longitudinal and equivalent stress as a percentage of SMYS. A pipe may be stressed well beyond the yield stress before it loses its capability to fulfil its function and the ASD criteria should therefore be rather conservative for this limit state. The interaction equation between moment, internal pressure and axial loads for local buckling/collapse in DNV'96 is a yield criterion from Mohareb et al. (1994). The maximum capacity of the pipe is here defined as the bending moment at which the entire cross-section yields. The moment criteria in DNV'96 (Section5, C305) is a LRFD format of the Mohareb's equation, where partial safety factors are defined based on structural reliability analysis (MCrk et al. (1997)). However, Hauch and Bai (1998) found that the allowable moment given by DNV'96 for some load conditions is at least 20% lower than that corresponding to the ASD criteria. In the following the moment criteria shall be re-visited and the causes of the conservatism will be identified, together with a newly suggested calibration of safety factors.
13.5.2 Analytical Capacity Equation Bai and Hauch (1998) modified the yield criterion from Mohareb et al. (1994), to account for the effect of corrosion defects. The ultimate moment capacity is also defined as the moment at which the entire cross section yields. The cases considered: the defect section is in compression (case 1), in compression and some in tension (case 2), in tension (case 3), in tension and some in compression (case 4). The four cases are shown in Figure 3.4.
13.5.3 Design Format For internal-over pressure cases, the design format based on DNV'96 (Section 5 C305) may be written as: YFYcMF,c+ YEME.c< Mc/YR
(13.6)
where MF,c is the characteristic functional bending moment; ME,c is the characteristic environmental bending moment; Me is the characteristic ultimate bending moment (as given in Chapter 3); YF is the functional load factor =1.1; YE is the environmental load factor which is 1.3; 7c is the conditional load factor; YRis the strength resistance factor.
13.5.4 Limit-State Function The limit-state function may be expressed as: g(Z) = Mc -(M F+ ME)
(13.7)
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where Me denotes the stochastic ultimate moment, MF and ME are stochastic applied load effects for the functional and environmental bending moment. Mc may be expressed as the product of the model uncertainty parameter XM and the moment capacity given by Bai and Hauch (1998) equations. A large amount of experimental and numerical tests are required to quantify mean value, COV and distribution function of the model uncertainty XM. All of the stochastic variables M~, MF and ME shall be defined for the parameter range of interests (e.g. hoop stress 0.8SMYS for operating conditions, 0.96SMYS for hydro test conditions). Uncertainty measures may be found from the SUPERB project, e.g. Jiao et al. (1997).
13.5.5 Calibration of Safety Factors In order to reduce the conservatism, the following shall be given considerations in the calibration of safety factors: 1) Use capacity equation for corroded pipes. Section 5 B205 of DNV'96 states that the wall-thickness used in buckling/collapse calculation for pipelines in operation shall exclude corrosion allowance. However, the width of corrosion defects is typically less than a quarter of the circle. Neglecting the whole circle would "likely lead to 10% less capacity predicted.
2) Use SMTS (Specified Minimum Tensile Stress) as o'v in Bai and Hauch equations. Figure 3.13 to 3.16 shows that the moment capacity equation agrees well with the finite element predictions (which have been validated against laboratory tests), if SMTS is used as cry in the equations. The reasoning of using SMTS is strain-hardening effects and the strengthening due to outward deformation for highly pressurized pipes in collapse. Unfortunately, in DNV'96, SMYS (Specified Minimum Yield Stress) is used as ~y. For X65 material, the ratio of SMTS and SMYS is 1.17. Laboratory tests by Mohareb et al. (1994) have confirmed that the mean bias for highly pressurized pipes is about 1.05 if SMTS is used as cry in the capacity equations. 3) Use strain-based design or a conditional load factor Yc (<1.0) for displacement dominant situations. For high pressure and high temperature (HP/HT) pipelines, pressure and temperature induced axial stress and moment could be large. Up-lift and lateral buckling behavior is typically displacement controlled. When a HP/HT pipeline is subject to fishing gear pullover load, its response is load dominant for small diameter pipelines and displacement dominant for large diameter pipelines. It is therefore suggested that a conditional load factor Yc is introduced to reflect the differences in the structural response to fishing gear load. In many practical situations, no adequate capacity equation is available for strength prediction due to the complexity of the problem. Instead, numerical tests (using FEM) and laboratory tests are conducted for strength design. It is then required to select partial safety factors that may be applied together with a (numerical) structural laboratory (e.g. finite element analysis). Selecting partial safety factors using reliability methods, FEM may be applied to strength design as an alternative to direct use of code equations. The advantage of such an innovative approach is that when information is lacking from the design codes for new materials,
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deepwater and harsh environments, project specific design criteria can be established by use of finite elements and reliability methods. It is difficult (if at all possible) to define code criteria which can cover all combinations of design scenarios (e.g. installation, hydro tests, operation), pipe materials (e.g. corrosion defects), loads (high hoop stress and low hoops stress cases) and new phenomenon which appear in challenging projects. Therefore, subject to achievable cost-saving and technical qualification, design engineers should develop project-specific design criteria based on the direct use of the reliability-based calibration under the principles of the NORSOK standard Y002 (1997).
13.6 Conclusions
The chapter presents reliability-based limit-state design and re-qualifications of pipelines. Following a summarized discussions of reliability and uncertainty measures, target reliability levels and calibration of safety factors, the following two limit-state design applications are given. (1) Design factors for hoop stress criterion: it is shown how the usage factor has been derived using reliability-based calibration. (2) Buckling/collapse of Corroded Pipes: the causes of the conservatism in the existing codes were identified and the reliability-based determination of partial safety factors was discussed. Further study is required to refine uncertainty measures and to develop methodology for the determination of partial safety factors for finite element models and laboratory tests.
13.7 References
1. Bai, Y. and Damsleth, P.A., (1998) "Design Through Analysis Applying Limit-state Concepts and Reliability Methods", Proc. of ISOPE'98. A plenary presentation at ISOPE'98. 2. Bai, Y. and Hauch, S., (1998) "Analytical Collapse Capacity of Corroded Pipes", Proc. of ISOPE'98. 3. Bai, Y. and Song, R., (1997) "Fracture Assessment of Dented Pipes with Cracks and Reliability-based Calibration of Safety Factors", Intemational Joumal of Pressure Vessels and Piping, Vol. 24, pp. 221-229. 4. Bai, Y., Xu, T. and Bea, R., (1997) "Reliability-based Design & Re-qualification criteria for Longitudinally Corroded Pipelines", Proc. of ISOPE '97. 5. DNV (1996) "Rules for Submarine Pipelines", Det Norske Veritas. 6. Guedes Soares, C., (1997) "Quantification of Model Uncertainty in Structural Reliability", in Probabilistic Methods for Structural Design, edited by C. Guedes Soares. Kluwer Academic Publishers. 7. Hauch, S. and Bai, Y., (1998) "Use of Finite Element Methods for Local Buckling Design of Pipeline", Proc. of OMAE '98.
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8. ISO/DIS 13623(1997) "Petroleum and Natural Gas Industries; Pipeline Transportation Systems", International Standard Organisation. 9. Jiao, G., Sotberg T., Bruschi, R. and Igland, R.T., (1997) "The SUPERB Project: Linepipe Statistical Properties and Implications in Design of Offshore Pipelines", Proc. of OMAE'97. 10. Mohareb, M.E., Elwi, A.E., Kulak, G.L. and Murray, D.W., (1994) "Deformational Behaviour of Line Pipe", Structural Engineering Report No. 22, University of Alberta. 11. M0rk K.J., Spiten J., Torselletti, E., Ness O.B. and Verley R., (1997) "The SUPERB Project & DNV'96: Buckling and Collapse Limit State", Proc. of OMAE'97. 12. NORSOK Standard Y-002 (1997) "Reliability-based Limit-state Principles for Pipeline design". 13. Sotberg, T., Moan, T., Bruschi R., Jiao, G. and MCrk, K.J., (1997) "The SUPERB Project: Recommended Target Safety Levels for Limit State Based Design of Offshore Pipelines", Proc. of OMAE'97. 14. Thoft-Christensen, P. and Baker, M.J., (1982) "Structural Reliability, Theory and its Applications", Springer-Verlag, 1982.