Development of a Reliable Failure Assessment Diagram by Persian Curve

Corresponding Author: Abdolrasoul Ranjbaran Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran Email: ranjbarn@shirazu.ac.ir Abstract: A conventional tool for assessing the integrity of structures containing flaws is, the Failure Assessment Diagram (FAD). The (FAD) is a combination of the limiting conditions for load, the flaw size and fracture toughness or yielding stress. The abscissa is the load ratio and the ordinate is the toughness ratio. The toughness ratio is defined as a function of the load ratio. Three options are available for this function. The (FAD) drive is based on parameters of fracture mechanics. Classical fracture mechanics contains epistemic uncertainty and is unreliable. As a result, the state of the art for the (FAD) is also unreliable. The present authors' team, conducted an extensive investigation in the past two decades, which led to the birth of the change of state philosophy that is digested in the Persian Curve. In the present paper the Persian Curve is used for development of a reliable (FAD). The validity of the work is verified via concise mathematical logics and comparison of the results with those of the others.


Introduction
A conventional tool for assessing the integrity of structures containing flaws is, the Failure Assessment Diagram (FAD), a two-dimensional construct, which is a combination of the limiting conditions for load, the flaw size and fracture toughness or yielding stress. The abscissa is the load ratio and the ordinate is the toughness ratio. The toughness ratio is defined as a function of the load ratio. Three options are available for this function. The (FAD) drive is based on parameters of fracture mechanics. Classical fracture mechanics contains epistemic uncertainty and is unreliable. As a result, the state of the art for the (FAD) is also unreliable. Toward detection and removal of uncertainty the relevant literature briefly reviewed as follow.
A number of methods are available for assessing the integrity of structures containing flaws. Milne et al. (1988), selected one of the available methods and defined a route for establishing the integrity of a structure under consideration. Zhao (1989), developed a methodology for the reliability analysis of structural steel component with pre-existing cracks. The CEGB R6 is used for assessing the critical state of the structure. A rigorous methodology for the system reliability analysis of a jacket structure under both extreme wave and fatigue conditions, is presented by (Shetty, 1992). Bloom (1995) presented a brief history of the evolution of the CEGB R6 failure assessment diagram procedure used in assessing defects in structural components. The research into elastic-plastic crack tip field expanded using the first two terms of the Williams expansion to characterize the degree of crack tip constraint, by (MacLennan, 1996). Chell et al. (1999), prepared guidelines to address proof test issues and describe procedures, in the form of road map, for implementing state of the art fracture technology into the design and analysis of a proof test. The crack tip parameters T and Q were used to quantify the crack tip constraint in mode I, by (Ayatollahi, 1998). Welding residual stresses can, in combination with operating stresses, lead to premature failure of components by failure mechanisms such as fatigue cracking and fracture. It is therefore important that welding residual stresses are accounted for properly in safety assessment codes such as the Nuclear Electric code R6 and BS 7910. This is what is done by (May, 2002). Talei-Faz (2003) presented an investigation into 69 the performance of offshore tubular components containing large defects. The significance of the residual strength of cracked tubular members is considered with respect to inspection and maintenance of structural integrity. An investigation of structural monitoring is presented by (De Leeuw, 2004). Shabakhty (2004) research subject was durable reliability of jack-up platforms with the aim to explore the possibilities for the extension of the life-time. Sisan (2005), investigated the influence of prior thermal and mechanical loading on brittle and ductile fracture behavior. A threedimensional-constraint based failure methodology based on failure assessment diagrams has been proposed using the analytical expression for constraint loss, by (Yusof, 2006). Dobmann et al. (2007), investigated new technologies for detection, classification and sizing of defects in combination with probabilistic (FAD) approaches. The fracture behavior of a hollow cylinder with internal circumferential crack under uniform tensile loading is examined by (Bach et al., 2009). Neto et al. (2008) investigated the structural integrity assessment of cracked piping of Pressurized Water Reactor nuclear reactors primary systems. An actual assessment has been carried out on cracked, circular hollow section, tubular K-joint containing a surface crack located at the crown location, using BS 7910:2005 (FAD) by (Lie and Yang, 2009). Ayala-Uraga (2009) considered the reliabilitybased assessment of deteriorating ship-shaped offshore structures. The aim of (Hosseini, 2010) study was to evaluate the effect of crack in corrosion defects on the failure pressure of natural gas transmission pipelines. Research on structural lifetime, reliability and risk analysis approaches for power plant components and systems was conducted by (Cronvall, 2011). The aim of (Elsaadany et al., 2012), was determination of shakedown boundary and fitness-assessment-diagrams for cracked pipe bends. Qu (2013) investigated the applicability of AIP579-1 PFSs (partial safety factor) and developed new PFSs which could provide enough accuracy for approximate evaluation of the safety margin. Hosseini et al. (2013), conducted experimental rupture tests to investigate the failure behavior of longitudinally oriented corrosion, crack and crack with corrosion. Ductile fracture and structural integrity of pipelines and risers was the subject of dissertation of (Kofiani, 2013). Hosseini (2014) considered crack in corrosion flaw assessment in thin-walled pipe. Fracture assessment of cracked components under biaxial loading is investigated by (Arafah, 2014). Vesga Rivera (2014) presented a combined engineering critical assessment which involved the examination of materials used to transport flue-gas and established a methodology to determine fracture toughness alongside with the (FAD) to assess the integrity of pipelines. The failure analysis and damage prevention on offshore pipelines under external loaded is studied by (Zhang, 2016). Sahu et al. (2015) worked on determination of fracture toughness curve using R-6 failure assessment method. Fracture assessments of large-scale straight pipes and elbows of various pipe diameters and crack sizes is reported by (Ainsworth et al., 2016). Orrock (2018) investigated the effects that scaling has on key structural integrity concepts, namely, stress fields, stress intensity factors and the J-integral. The influence of biaxial loading on the assessment of structures with defects is the subject of (Meek, 2017) study. Fuentes et al. (2018) provided a structural integrity assessment method for the analysis of non-metallic materials which uses BS 7910 Option 1 (FAD). Updating failure probability of a welded joint considering monitoring and inspection is investigated by (Mai, 2018). Hoh et al. (2018) worked on (FAD) analysis of fatigue test results for X65 welded joints. A study on the failure analysis of the neutron embrittled reactor pressure vessel support using finite element analysis is conducted by (Han, 2018). The title of (Sumesh and Arun Narayanan, 2018) paper is, influence of notch depth to width ratios on J-integral and critical failure load of single edge notched tensile aluminum 8011 alloy specimens. Coêlho et al. (2019) compared both British standard BS 7910 (2013) and American standard API 579/ASME FFS-1 (2016) stress intensity factor solutions by considering a series of semielliptical surface cracks located in the external surface of a pressurized hollow cylinder in the axial direction. Mechanical behavior of high strength structural steel under high loading rates is studied by (Alabi, 2019). Pillai et al. (2019) work covers the validation of standard safety assessment procedure in the new BS 7910: 2013+ A1:2015 for cracked joints. Analysis of crack behavior in pipeline system using (FAD) based on numerical simulation under XFEM is proposed by (Montassir et al., 2020).
The state of the art of the (FAD) contains epistemic uncertainty, that should be modified. The current authors research team detected the need for remedy and after extensive research (Ranjbaran, 2010;Ranjbaran et al., 2013;Ranjbaran and Ranjbaran, 2014;2017;Ranjbaran et al., 2020a;2020b;2020c;2020d;2021;Amirian and Ranjbaran, 2019;Baharvand and Ranjbaran, 2020), in the last two decades, proposed the Change of State Philosophy (CSP) which digested in the Persian Curves (PC) (Ranjbaran et al., 2020a), that is used for the development of a reliable (FAD). The validity of the work is verified via concise mathematical logics and comparison of the results with those of the others as follows.

Basic Formulation
The analysis of a phenomenon, considered as a change in the state of the system, is conducted as follows. A decreasing parameter for the system, called stiffness and the inverse of stiffness, called flexibility are considered. The contributing parameters, as shown in Fig. 1, are defined as, the system stiffness at a changed state (kSS = kS-kC), the system stiffness at intact state (kS), the change stiffness (kC), the system flexibility at the changed state (fSS = fS+fC), the system flexibility at intact state (fS) and the change flexibility (fC). The (CSP) formulation is commenced with the obvious relation between the system parameters, as shown in Fig. 2 and defined in Eq. (1): Equation (1) is rearranged to obtain the (kSS) and (kC), in Eq. (2): in which the phenomenon functions (collection of the failure function (FR) and the survive function (SR)) are defined in Eq. (3): Definition of the dimensionless phenomenon functions in a unit interval, introduced a down to earth method for human knowledge. Since obvious concepts are used, then the proposed method is free of epistemic uncertainty and the common problems in the conventional methods, such as singularity, instability and etc. Consequently, the authors decided to complete the work. To continue, the unknown parameters in Eq. (4) should be explicitly determined: The investigation for explicit definition of the aforementioned functions, is continued in the next paragraph via definition and construction of the so called, state functions.
Development of a functional (FR and SR) in terms of two functions (fS and fC) is not possible. Therefore, the phenomenon functions are customized for (kS = fS = 1) to define the destination function (D) and the origin function (O), which are collectively called the state functions and the state ratio (R) in Eq. (5). This is an artifice to define functions (D and O) in terms of only one variable (R): Consequently the (D) and (O) are defined in terms of the (R) in Eq. (6): The state functions may be considered as solution of the boundary value problems in Eq. (7), in which (min) and (max) denote minimum and maximum respectively: The (R) with one end in the infinity, as shown in Fig.  3, is not a good working parameter. Moreover, this ratio is itself a function, so it is not wise to be used as an independent variable. Therefor, the state variable (ξ  [0, 1]) with a zero value (ξ = 0) at the origin and a unit value (ξ = 1) at the destination is defined. The state variable is the phenomenon's abstract lifetime. In terms of the state variable, the boundary value problems in Eq. (7) is rewritten in Eq. (8): Investigation for construction of solution for boundary value problems in Eq. (8), led the authors to use their experience in structural mechanics, finite element method, mathematics and extensive research (Ranjbaran et al., 2020a). The results are the state functions defined in Eq. (9) The authors invite the readers from all over the world to propose new (better in some sense) state functions in place of Eq. (9).

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Now attention is paid to construction of the phenomenon functions. Via the definition of the (kSS) and the (fSS) and the crack compliance (fC) in fracture mechanics (Anderson, 2005), the authors detected a fact that, the (fC) is directly proportional to the (kS)! This detection is called "the Persian Principle of Change (PPC)". In view of this principle the (fC) is defined in Eq. (11): Note that Eq. (11) is an alternative for the whole fracture mechanics (Anderson, 2005)! Insertion of Eq. (11) into Eq. (3) concluded in the general definition for the phenomenon functions in Eq. (12): The (kS) is not explicitly known so it is not a feasible working parameter. Toward better definition and preparation for using reliable data, Eq. (12) is rewritten in Eq. (13) in terms of the positive control parameters (aM) and (b) (Ranjbaran et al., 2020a). The flexibility for translation and rotation of phenomenon functions in the (11) working box, which lets the experts to enforce their will, is provided by selection of two control parameters from calibration of reliable data: To this end the proposed formulation is mathematically in abstract form. Consequently, it is a certain universal formulation, in a sense that it is independent of geometry, coordinates, material properties, size and changing agent. Therefore, it equally applies to all natural-phenomena.

Persian Curve
As observed, the proposed formulation is derived based on logical reasoning and concise mathematics, without a need for construction of differential and integral equations, which is the paramount basis of the conventional methods of analysis in human knowledge. Consequently, the proposed formulation is reliable and free of epistemic uncertainty, because it is based on obvious and certain basis, for example the definition of flexibility as inverse of stiffness in Eq. (1).
To prepare for determination of control parameters for a phenomenon, the (FR) is renamed as Persian-Fasacurve (PF), the (SR) is renamed as Persian-Shiraz-curve (PS) and the two collectively called the Persian curve (PC), defined in Eq. (14), in which (PO) is the ordinate of the origin point (O) and (PT) is the ordinate of the truncated (end) point (T). Note that insertion of (PO = 1 and PT = 0) and (PO = 0 and PT = 1) into Eq. (14) conclude into (FR) and (SR) respectively as in Eq. (13): In comply with the vocabulary of human knowledge, the (PS) is the unified equation for capacity and reliability representing a decreasing data and the (PF) is the unified equation for the probability and fragility representing an increasing data. Moreover, in comply with the common practice in stochastic analysis the (probability) density distribution, here called the Persian-Zahedan-curve (PZ) and defined as the derivative of phenomenon functions with respect to the (ξ). The (PZ) is casted in Eq. (15), in which (D (1) ) is derivatives of (D) with respect to (ξ). In spite of the paramount role of the density distribution in stochastic theory, it has no such important role in the (CSP) and it is used only for comparison with the conventional one: In order to determine the control parameters, Eq. (14) is rearranged as in Eq. (16): Equation (16)

Failure Assessment Diagram
Persian curve is defined in terms of an abstract life time called state variable in a unit interval (ξ[0, 1]). In real phenomena, the lifetime (λ) is selected in a truncated interval (λ [λO, λT]), where (λO) is the lifetime origin and (λT) is the lifetime destination. Lifetime is a characteristic of the system. In common practice the ratio (to make it user friendly) of a predefined parameter (FY) and an intentionally defined one (FE) is expressed as lifetime (λ), as in Eq. (22) The key points ordinates on unified Persian curves are defined in Eq.
Formulation is complete now. The work is verified in the following section.

Verification
To this end, it is shown that the Persian Curve is a reliable tool for analysis of failure phenomena, conceived as a change in the structural system. Concise mathematical formulation presented in previous sections.
Note that the modified (FAD) goes through the unified Key Points (KPU). A set of test data (TEST) (Gibstein and Moe, 1986) is also shown. Comparison of results shown that the conventional results contain uncertainty and should be replaced by Persian curve.

Example 2
The Unified Persian curve is compared with the (FAD) defined in Eq. (28):

Example 3
Compare the Unified Persian curve with the (FAD) defined in Eq. (29):

Example 4
A conventional (FAD) is defined in Eq. (30). Compare the Unified Persian curve with the conventional (FAD):

Solution
The (FAD) from Eq. (30) (KIV) by (Ainsworth et al., 1999;Dai et al., 2020) and the test data (TEST) by (Gibstein and Moe, 1986) are compared with the unified Persian curve (PSU) in Fig. 9. The lifetime is mapped onto the state variable as (Lr = 2.25 ξ). The (KIV) goes through the unified key points, but over estimates the test results, therefore contain uncertainty and is not reliable.

Example 5
Compare the Unified Persian curve with the (FAD) defined in Eq. (31)

Discussion
The conventional (FAD) is based on conventional fracture mechanics, which is based on singularity of stress at crack tip. Stress singularity at crack tip is a wrong concept, then the conventional fracture mechanics and the (FAD) contain epistemic uncertainty and are not reliable. On the other hand, the proposed formulation is logical and reliable. The parameter (Lr) is defined as the ratio of the applied stress over the yield stress. The applied stress is not a property of the system. Therefore, only the yield stress is considered as the effective parameter for the system. Experience in determination of ultimate strength of structures shown that the yield stress is not enough. But the geometry also should be considered. For example, for two columns with the same cross section and yield stress, the capacity of the longer one is less than the other! The combination of the material and geometric properties, in the form of the relative slenderness ratio (λ), as in Eq. (23), is the necessary and sufficient condition. Note that, Eq. (23) is the inverse of the current definition of (Lr) when applied stress is selected as the Euler's stress. Consequently, replacement of the (Lr) with the (λ) is recommended.

Conclusion
The following conclusion obtained from this study. The failure phenomenon is conceived as the change of state of the system. Based on logical reasoning and concise mathematics, the change of state is expressed in terms of the Persian curve. The Persian curve is completely calibrated from four key points on real world data. Making use of reliable data in the literature, the unified control parameters and the unified Persian curves are explicitly expressed in term of the state variable. The unified Persian-Shiraz curve (PSU) is proposed as a reliable (FAD). Validity of the work is verified via concise mathematical logics and comparison of the results with those of the others in five examples. The proposed formulation is recommended to be used as a certain logical basis for the assessment codes of practice such as BS7910 and API 579. Based on its logical basis, there is no restriction on the application of the proposed method. What remains, is to determine the modified slenderness ratio and section capacity for structures with defect.

aM:
Control parameter at M aN: Control parameter at N b: Control parameter (power) CSP: Change of State Philosophy D: Destination function E: Initial modulus F: Force FAD: Failure assessment diagram Fy: Yield limit fS: System flexibility fC: Change flexibility FS: Dimensioned flexibility fSS: Survived flexibility FR: Failure function (FR and SR): Phenomenon functions fW: Weibull probability density function FW: Weibull cumulative distribution function : Recovery rate Kr: Toughness ratio kS: System stiffness kC: Change stiffness