A Systematic Review of Structural Reliability Methods for Deformation and Fatigue Analysis of Offshore Jacket Structures
Abstract
:1. Introduction
2. Level III Analytical Structural Reliability Analysis Methods
2.1. FirstOrder Reliability Method (FORM)
 (1)
 Define the PF for the corresponding LS, e.g., ultimate limit state (ULS), serviceability limit state (SLS), fatigue limits state (FLS), etc.
 (2)
 Let the mean value point be the initial design point, i.e., ${\mathit{x}}_{i,k}={\mu}_{{\mathit{x}}_{i}}$ $i=1,2,\dots ,n$, and evaluate the gradients $\nabla g\left({\mathit{X}}_{k}\right)$ of the LSF at this design point, where ${\mathit{x}}_{i,k}$ represents the $i\mathrm{th}$ element in the vector ${\mathit{X}}_{k}$ of the $k\mathrm{th}$ iteration and ${\mu}_{{\mathit{x}}_{i}}$ is the mean value of the $i\mathrm{th}$ element.
 (3)
 Compute the initial RI, $\beta $ adopting the meanvalue approach, i.e., $\beta =\frac{{\mu}_{\tilde{g}}}{{\sigma}_{\tilde{g}}}$ and its direction cosine $\alpha $.$$\beta =\frac{{\mu}_{\tilde{g}}}{{\sigma}_{\tilde{g}}}=\frac{g\left({\mathit{\mu}}_{\mathit{X}}\right)}{{\left[{{\displaystyle \sum}}_{i=1}^{n}{\left(\frac{\partial g\left({\mathit{\mu}}_{\mathit{X}}\right)}{\partial {\mathit{x}}_{i}}\right)}^{2}\xb7{\sigma}_{\mathit{x}i}^{2}\right]}^{\frac{1}{2}}}$$$${\alpha}_{i}=\frac{\left(\frac{\partial g\left({\mathit{X}}^{*}\right)}{\partial {\mathit{x}}_{i}}\right)\xb7{\sigma}_{{\mathit{x}}_{i}}}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left(\frac{\partial g\left({\mathit{X}}^{*}\right)}{\partial {\mathit{x}}_{i}}{\sigma}_{{\mathit{x}}_{i}}\right)}^{2}}}$$
 (4)
 Calculate a new design point ${\mathit{X}}_{k}$ and ${U}_{k}$ function value, as well as gradients at this new design point.$${\mathit{x}}_{i,k}={\mathit{\mu}}_{{\mathit{x}}_{i}}+\beta {\sigma}_{{\mathit{x}}_{i}}{\alpha}_{i}$$$${\mathit{u}}_{i,k}=\frac{{\mathit{x}}_{i,k}{\mathit{\mu}}_{{\mathit{x}}_{i}}}{{\sigma}_{{\mathit{x}}_{i}}}$$
 (5)
 Compute the RI $\beta $ and direction cosine ${\alpha}_{i}$ using Equations (4) and (5) respectively.$$\beta =\frac{g\left({U}^{*}\right){{\displaystyle \sum}}_{i=1}^{n}\frac{\partial g\left(U\right)}{\partial {\mathit{x}}_{i}}{\sigma}_{{\mathit{x}}_{i}}{\mathit{u}}_{i}^{*}}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left(\frac{\partial g\left({U}^{*}\right)}{\partial {\mathit{x}}_{i}}{\sigma}_{{\mathit{x}}_{i}}\right)}^{2}}}$$$${\alpha}_{i}=\frac{\left(\frac{\partial g\left({\mathit{X}}^{*}\right)}{\partial {\mathit{x}}_{i}}\right)\xb7{\sigma}_{{\mathit{x}}_{i}}}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left(\frac{\partial g\left({\mathit{X}}^{*}\right)}{\partial {\mathit{x}}_{i}}{\sigma}_{{\mathit{x}}_{i}}\right)}^{2}}}$$
2.2. SecondOrder Reliability Method (SORM)
3. Level III (Direct) Reliability Methods
4. Advanced Approximation Modeling Methods
4.1. Response Surface Method
4.2. Surrogate Models (SMs)
4.2.1. Kriging Incorporated with FORM
4.2.2. Kriging Incorporated with IS/SS
4.2.3. Efficient Global Reliability Analysis (EGRA)
4.2.4. Sequential Kriging Reliability Analysis (SKRA)
4.2.5. Support Vector Approach
4.2.6. Artificial Neural Network Approach
4.2.7. Radial Basis Function Approach
5. Probabilistic Fatigue and Fracture Mechanics Approaches
6. Other Methods and Applications
6.1. Stochastic Finite Element Method (SFEM)
6.2. Reliability Analysis of Systems
6.3. TimeVariant Reliability of Systems
7. Critical Discussion
8. Conclusions
 The FORM was improved by the development of the conjugate search direction, finitebased Armijo search direction method, Hybrid Relaxed HL–RF, stability transformation method (STM) with chaos feedback control, STM with chaos feedback control, and STM with chaotic conjugate search direction, among others. The combination of Maximum Entropy Fitting Method and the FORM was applied to problems of implicit LSFs. The SORM is an improvement on the FORM, to provide solutions to highly nonlinear LSFs. A new SORM for RA was developed using the SAP in order to overcome some of the issues inherent in the traditional SORM.
 The MCS method was improved by the development of interval MC method, which combines simulation process with interval analysis, new MCbased methods involving the use of brute force MCS methods for complicated structural systems, IMCIFEM, merging IS with directional simulation, etc. Improvements in variance reduction techniques were achieved, such as the development of interval importance sampling (IS) method, which applies the IS technique and imprecise probability, and the LHSbased quasirandom polar sampling technique.
 The advanced approximation modeling methods include the wellestablished Response Surface Models/Method (RSM) and the Surrogate Models (SM) as well as the Stochastic RSM (SRSM). The SRSM is a model for the RA of complex systems with low Probability of Failure (POF) for which approximate methods are inaccurate and for which Monte Carlo Simulation (MCS) is too computationally intensive. The efficiency of the RSMs developed for implicit LSFs studied herein include the Collocation Based SRSM, novel SRSM combining FEA, MPR, and FORM/SORM, incorporating the SRSM with Saddle point approximation (SPA), among others. Examples of SM include the Kriging, Adaptive Kriging, EGRA, Support vector machines, ANN, RBF, etc. These can be combined with conventional reliability methods for problems of implicit LSFs. Kriging and Adaptive Kriging interpolation models were combined with the FORM, Line sampling, IS, SS, MCS, etc.
 This study focused specifically on the probabilistic fatigue and fracture mechanics approaches because the fatigue limit state in most cases is the designdriving criterion for structural components of offshore jacket structures. Consequently, the SRA of structures considering pittingcorrosion fatigue phenomenon was identified as particularly of note and is recommended as an area open to further investigation.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AI  Analytical integration 
AKIS  active learning kriging with importance sampling 
AKMCS  Active learning kriging with Monte Carlo simulation 
ALM  Active learning methods 
ANN  Artificial neural networks 
ASCE  American Society of Civil Engineers 
ASME  American Society of Mechanical Engineers 
ASVM  Adaptive support vector machine 
BM  Bending moment 
CDF  Cumulative density function 
CGF  Cumulant generating function 
CM  Computational models 
CSRSM  Collocationbased stochastic response surface method 
DNV  Det Norske Veritas 
DoE  Design of experiment 
EGRA  Efficient global reliability analysis 
FAL  Finitebased Armijo line search direction 
FCG  Fatigue crack growth 
FEA  Finite element analysis 
FEM  Finite element method 
FLS  Fatigue limit state 
FM  Fracture mechanics 
FORM  Firstorder reliability method 
FR  Fletcher and Reeves method 
FRA  Fatigue reliability analysis 
GA  Genetic algorithm 
HL  Hasofer and Lind method 
HL–RF  Hasofer Lind–Rackwitz Fiessler method 
HRHL–RF  Hybrid relaxed Hasofer Lind–Rackwitz Fiessler method 
HSAC  Hybrid selfadaptive conjugate 
ICE  Institution of Civil Engineers 
IFEM  Interval finite element method 
IMC  Interval Monte Carlo simulation 
IS  Importance sampling 
ISKRA  Improved sequential kriging reliability analysis 
ISO  International Organisation for Standardisation 
KIM  Kriging interpolation model 
KL  Karhunen–Leove expansion 
LCoE  Levelized cost of energy 
LEFM  Linearelastic fracture mechanics 
LHS  Latin hypercube sampling 
LIF  Least improvement function 
LS  Limit state(s) 
LSF  Limit state function 
LSS  Limit state surface 
MC  Monte Carlo 
MCMC  Markov Chain Monte Carlo 
MCS  MonteCarlo simulation 
MEM  Maximum entropy fitting method 
MFEM  Multiscale finite element method 
MLS  Moving least square 
MM  Metamodel(s) 
MPP  Most probable failure point 
MPR  Multivariate (quadratic) polynomial regression 
MVFOSM  Mean value firstorder second moment method 
NF  Number of fitting points 
NI  Numerical integration 
OWT  Offshore wind turbine 
PCE  Polynomial chaos expansion 
PCKriging  Polynomial chaoskriging 
Probability density function  
PF  Performance function 
PLS  Partial least squares 
PMA  Performance measure approach 
POF  Probability of failure 
PSMFEM  Perturbationbased stochastic multiscale finite element method 
RA  Reliability analysis/assessment 
RBF  Radial basis function 
RBO  Reliabilitybased optimization 
RHL–RF  Relaxed Hasofer Lind–Rackwitz Fiessler method 
RI  Reliability index 
RS  Response surface 
RSM  Response surface method/model 
SAC  Selfadaptive conjugate 
SFEM  Stochastic finite element method 
SGFEM  Stochastic Galerkin FEM 
SHM  Structural health monitoring 
SKRA  Sequential kriging reliability Analysis 
SLS  Serviceability limit state 
SM  Surrogate modelling 
SORM  SecondOrder reliability method 
SPA  Saddle point approximation 
SR  Structural reliability 
SRA  Structural reliability assessment 
SRBDO  System reliabilitybased design optimization 
SRE  Structural reliability evaluation 
SRSM  Stochastic response surface method 
SS  Subset simulation 
SSFEM  Spectral stochastic finite element Method 
STM  Stability Transformation Method 
SVM  Support vector machines 
SVR  Support vector regression 
TD  Timedependent 
TI  Timeindependent/timeinvariant 
TV  Timevariant 
ULS  Ultimate limit state 
Nomenclature  
${h}_{\mathit{X}}\left(\mathit{x}\right)$  Density function 
${P}_{f}$  POF 
${\mathit{X}}_{i}$  Random variable 
${a}_{0},{a}_{1},\dots ,{a}_{16}$  Regression coefficients for a quadratic regression 
${f}_{\mathit{X}}\left(\mathit{x}\right)$  The joint PDF of the random variables $\mathit{X}$ 
${g}_{\mathit{X}}\left(\mathit{X}\right),{g}_{\mathit{u}}\left(\mathit{x}\right)$  Limitstate function 
${\mathit{x}}_{i}$  Realization of the random variable ${\mathit{X}}_{i}$ 
${\alpha}_{i}$  Direction cosine 
${\mathit{\mu}}_{\mathit{x}}$  Mean of random variables 
${\sigma}_{\mathit{x}}$  Standard deviation of random variables 
$\Phi (\xb7)$  Cumulative density function of the standard normal distribution 
$\mathsf{\varphi}(\xb7)$  Density function of standard normal distribution 
$I(\xb7)$  Failure domain identifier 
$N$  Number of samples 
$Pr(\xb7)$  Probability function 
$U$  Standard normal form of variables 
$V$  Vector 
$V\left({P}_{f}\right)$  Target coefficient of variation of failure probability 
$\mathit{X}$  ndimensional random vector 
$\beta $  Reliability/safety index 
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Method  Capabilities  Limitations  

First Order Reliability Method (FORM)  Mean Value First Order Second Moment Reliability Method 
 The applicability range of this method is diminished as a result of the following reasons:

Hasofer and Lind Method  FORM approximation gives adequate outcome when the function is nearly linear close to the MPP, and the LSS has only one minimal distance point. 
 
Hasofer and LindRackwitz Fiessler Method  Widely used approximate analytical method since it provides a good balance between efficiency and accuracy in realistic engineering RA. 
 
SecondOrder Reliability Method  Ideal for cases where the LSS has large or irregular curvatures (high nonlinearity), the POF estimated by FORM, using the RI $\beta $, can produce inaccurate and unreliable results. By introducing secondorder Taylor series expansions (or other polynomials), this drawback may be overcome. 

Method  Capabilities  Limitations  

Analytical Integration  Ideal for simple failure surface 
 
Numerical Integration  Standard routines are found in most computer systems  Not always feasible, owing to the growthoff errors and excessive computational times  
Crude Monte Carlo Simulation Technique (MCS)  Most versatile, clear, and well understood exact method available Requires no partial derivative of LSF; therefore, the method can be used for implicit LSF. 
 
Variance Reduction Techniques  MCS with Importance Sampling Technique 

 
Adaptive Sampling 
 Application of the directional sampling and adaptive sampling is limited to a moderate number of random variables  
Conditional Expectation Techniques  Directional Simulation 
 The technique is not very efficient for one or a few planar LS.  
Axis Orthogonal Simulation Technique  Is recommended for convex failure sets  They typically require a large number of response function evaluations, which makes them impractical if the response function is expensive to evaluate.  
Design Point Simulation  Makes use of the FORM design point which makes it less cumbersome in the search for the POF  
Subset Simulation 


Method  Capabilities  Limitations  

Parallel System  A parallel system fails when all the links (potential failure modes) fail. The most consistent function of the parallel system is for modelling the sequential failure of components in a single failure path leading to structural failure  Redundant members are introduced which introduces a computationally intensive procedure  
Series System  Ideal for pipelines  Failure of one component leads to failure of the system  
Stochastic Finite Element Method  Perturbation Method  The perturbation techniques are desirable owing to their efficiency in terms of computation times and accuracy  Too mathematically intensive  
Neumann Expansion Solution 
 Determining the covariance matrix among all elements of the fluctuation part of the stiffness matrix involves prohibitively high computational effort.  
Response Surface Method 

 
Branch and Bound Method  Are useful for the elasticplastic analysis of frame structures where effects of plasticity like the formation of plastic hinges give sharp changes in the stiffness behavior  Its application is limited  
Surrogate Models/Response Surface Model/MetaModels  Polynomial Regression Models  The most widely used due to their simple formulations and implementation 
 
Approaches Based on:  Radial Basis Function 
 Computationally efficient but at the expense of accuracy  
Local Interpolation Model (Polyhedra) 
 It is an approximate method  
Artificial Neural Network 

 
Support vector Machine  In comparison to ANNs, SVM employs the theory of minimizing the structure risk to avoid the problems of excessive study, calamity data, local minimum value etc.  Its implementation involves high computational efforts, and sufficient model sparsity cannot be guaranteed.  
Moving Least Squares 

 
Kriging Models 
 Significantly more complex compared to polynomial regression models 
Method  Capabilities  Limitations  

Stochastic Expansion  NonIntrusive  Stochastic Response Surface Method 
 It is widely used in chemical Engineering. Its application in Structural Engineering is still burgeoning. 
Intrusive  Spectral Stochastic FEM  Gives more reliable results 
 
Time variant reliability Methods 
 Practical application of TV reliability methodology appears rather limited, partially because only very few computer codes are available 
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Shittu, A.A.; Kolios, A.; Mehmanparast, A. A Systematic Review of Structural Reliability Methods for Deformation and Fatigue Analysis of Offshore Jacket Structures. Metals 2021, 11, 50. https://doi.org/10.3390/met11010050
Shittu AA, Kolios A, Mehmanparast A. A Systematic Review of Structural Reliability Methods for Deformation and Fatigue Analysis of Offshore Jacket Structures. Metals. 2021; 11(1):50. https://doi.org/10.3390/met11010050
Chicago/Turabian StyleShittu, Abdulhakim Adeoye, Athanasios Kolios, and Ali Mehmanparast. 2021. "A Systematic Review of Structural Reliability Methods for Deformation and Fatigue Analysis of Offshore Jacket Structures" Metals 11, no. 1: 50. https://doi.org/10.3390/met11010050