1 Introduction

Fitness-for-service (FFS) can be used for the assessment of defective (or prone to failure) items of a process plant that is in service [1]. Among the benefits of FFS, plant safer operational conditions with lower maintenance and operating costs can be mentioned [2]. After performing an FFS assessment, the equipment may remain in operation (if safe or fit), be set to repair, be reclassified, altered, or removed.

LBB, as a part of the FFS assessment, helps the user to decide whether or not to continue with the cracked equipment. It ensures no catastrophic failure due to unstable rapid crack extension unless fluid leakage through the crack is detected beforehand. Figure 1 illustrates the LBB assessment diagram, where the ordinate is the normalized crack depth, \(a/t\), and the abscissa is the crack length \(2c\). There is a critical crack length, \(2{c}_{\mathrm{crit}}\), from which the crack growth becomes unstable. This line and the ligament instability line divide the diagram into four regions, namely, safe, leak, dangerous, and break regions. Cracks with lengths smaller than the critical one will steadily increase (path 1 in Fig. 1), penetrating the entire wall thickness, \(2{c}_{p1}\). Since \(2{c}_{p1}< 2{c}_{\mathrm{crit}}\), the vessel content will leak through the cracked wall without rupturing the vessel (safe or leak region). When the crack propagates along path 2, the crack size (\(a/t\), \(2{c}_{p2}\)) is located in the dangerous or break region; thus, LBB is not the case.

Fig. 1
figure 1

Concept of LBB

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Irwin [3] showed that for an axial crack length smaller than twice the thickness of a cylinder, the radial crack driving force would be greater than the axial one, resulting in a through-wall crack up to a specific length, with no risk of cylinder failure.

According to ASME-Sec VIII, DIV.3 Boiler and Pressure Vessel Code [4], a single-wall component's critical crack depth greater than the wall thickness will lead to LBB. In fact, for crack depths greater than 80% of the wall thickness, known as very deep cracks, LBB failure mode will happen in case of the occurrence of both following conditions:

  1. (1)

    The crack at its 80% wall thickness depth to be below the critical flaw size using the failure assessment diagram (FAD) from API 579-1/ASME FFS-1 standard, and

  2. (2)

    the remaining ligament (distance from the crack tip to the free surface that the crack is approaching) is less than the quantity \({({K}_{\mathrm{Ic}}/{\sigma }_{y})}^{2}\).

API 579-1/ASME FFS-1 and BS 7910 [5] are internationally recognized FFS standards that deal with LBB. The former provides three “Levels” for assessing crack-like flaws, which successively require increasing evaluating skills. As the most sophisticated analysis, Level 3 requires specific input data and provides a better estimation of structural integrity. It allows the implementation of five different methods, such as the material’s specific FAD method according to BS 7910’s Option 2 or the elastic–plastic analysis similar to Option 3 of BS 7910 [6]. As depicted in Fig. 2, FAD relates two failure modes of brittle fracture and plastic collapse of a defective component. The collapse ratio parameter, \({L}_{r}=\frac{{\sigma }_{\mathrm{ref}}}{{\sigma }_{y}}\), is used for plastic collapse assessment, while the toughness ratio parameter, \({K}_{r}=\frac{{K}_{\mathrm{I}}}{{K}_{\mathrm{mat}}}\), is used to assess brittle fracture. In these ratios, \({\upsigma }_{\mathrm{ref}}\) is the reference stress, \({\sigma }_{y}\) is the yield strength of the material, \({K}_{\mathrm{I}}\) is the stress intensity factor, and \({K}_{\mathrm{mat}}\) is the material fracture toughness. According to Level-2 of API 579-1/ASME FFS-1 standard, \({L}_{r}\) and \({K}_{r}\) are related by the following formula:

Fig. 2
figure 2

The failure assessment diagram (FAD) [2]

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$$\begin{aligned} {K}_{r}& =\left(1-0.14{\left({L}_{r}^{P}\right)}^{2}\right)\left(0.3+0.7\mathrm{exp}\left[-0.65{\left({L}_{r}^{P}\right)}^{6}\right]\right)\nonumber \\ & \qquad \mathrm{ for }\; {L}_{r}^{P}\le {L}_{r\left(\mathrm{max}\right)}^{P}. \end {aligned}$$
(1)

Here, the extent of the FAD on the \({L}_{r}^{P}\) is determined as \({L}_{r(\mathrm{max})}^{P}={\upsigma }_{\mathrm{f}}/{\sigma }_{y}\), where \({\upsigma }_{\mathrm{f}}\) is the flow stress. In Annex 2E of API 579-1/ASME FFS-1, there are several equations for estimating the flow stress, one of them being \({\upsigma }_{\mathrm{f}}={\sigma }_{y}+69\,\mathrm{ Mpa}\) which is used in this work.

Despite the standards as mentioned earlier, performing a robust LBB assessment is not a regular and straightforward practice in the oil, gas, and petrochemical industries. The reasons mainly refer to the flammability and toxicity of the fluids as well as the free-to-air openness of the components. The former causes safety and environmental consequences, while the latter influences the leak detection capacity since vapor cloud detection will be affected by wind and weather conditions. Insulation of components can also increase the difficulty of leak detection [6].

LBB analysis has already been done on a spherical helium storage vessel [7], a vertical cylindrical oil storage tank [8], an ammonia storage tank [9], an aluminum beverage cans [10], an offshore structure [11], subsea pipelines [12], aerospace components [13], gas and oil pipelines [14, 15], and pressure vessels [16,17,18,19]. These studies needed to follow clearly established LBB procedures such as those presented in API 579-1/ASME FFS-1 or BS 7910 standards. A mix of different sources was often used, which could lead to inconsistent results [6]. Thus, there is still room for further study of LBB assessment in oil and gas industries.

The power of modeling and simulation to support the design and optimization of industrial components [20,21,22] and the developing of new [23,24,25] have been widely recognized. Numerical methods in particular, finite element analysis (FEA), have already been used to calculate stress intensity factors (SIFs) [26,27,28,29,30]. Different approaches for extracting SIFs using FEA can be categorized as field variable (displacement or stress-based) and energy release methods [31]. Among energy release methods, J-integral is widely accepted as a quasi-static fracture mechanics parameter since the knowledge of the field near the crack tip is not required. Indeed, its path independence allows calculation along a contour remote from the crack tip.

Besides, FEA is a strong tool for calculating SIF, particularly for geometrically complex components subject to complicated transient loading from pressure and/or thermal effects. To the best of our knowledge, an illustrative numerical model for LBB assessment and addressing uncertainties, which can be attributed to different sources, such as crack length measurement precision, internal pressure fluctuations, and uncertainties in mechanical properties of the base material, has hitherto not been presented in the literature. These two issues will not only guide engineering specialists to benchmark an LBB analysis but also will help those scholars who work on numerical methods for fracture analysis.

Therefore, we first customize a three-dimensional FEA for an LBB assessment of a pressure vessel in the refinery, which could precisely calculate the SIF at crack tips. The accuracy of the presented numerical solutions is then investigated via comparison with the API 579/ASME FFS-1 standard analytical solutions. Determining the maximum (limiting) through-thickness defect size that satisfies the LBB requirements is also discussed. This limiting size is useful to establish the acceptance/rejection limit in order to predict whether a specific cracked region of the vessel will leak or break. Secondly, we perform a reliability analysis to evaluate the probability of failure taking into consideration the uncertainties in the influencing factors. We aim to determine whether a pressure vessel is safe or not.

2 LBB Assessment

2.1 Geometry and Nomenclature of Postulated Flaw

An axial through-thickness defect in the cylindrical shell of a pressure vessel, as illustrated in Fig. 3, is considered. The value of \(2c\) is the length of the crack, \({R}_{i}\) is the inner radius, and \(t\) is the wall thickness of the cylinder.

Fig. 3
figure 3

Geometry of an axial through-thickness crack in a cylinder

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3 Reference Stress, SIF, and Material Toughness

3.1 Reference Stress

According to Annex-9C of API 579-1/ASME FFS-1, through-wall membrane and bending stress for a cylinder with a through-wall crack in the longitudinal direction subjected to internal pressure loading can be obtained using the following formulas:

$${\sigma }_{\mathrm{ref}}=\frac{{P}_{\mathrm{b}}+{\left({P}_{\mathrm{b}}^{2}+9{\left({M}_{t}\cdot {P}_{\mathrm{m}}\right)}^{2}\right)}^{0.5}}{3},$$
(2)

where membrane stress, \({P}_{\mathrm{m}},\) and bending stress, \({P}_{\mathrm{b},}\) are given by:

$$\begin{aligned} & {P}_{m}=\frac{p{R}_{i}}{t}\quad \mathrm{ and } \quad {P}_{b}=\frac{p}{2} (\mathrm{for \; internal \; crack})\;\mathrm{ and }\nonumber \\ & {P}_{b}=\frac{-p}{2} (\mathrm{for \; external \; crack}) \end {aligned}$$
(3)

and the surface correction factor, \({M}_{t}\), is defined as one of the equations below:

$${M}_{t}={\left(\frac{1.02+0.4411{\lambda }^{2}+0.006124{\lambda }^{4}}{1+0.02642{\lambda }^{2}+1.533({10}^{-6}){\lambda }^{4}}\right)}^{0.5}\mathrm{ with }\; \lambda =\frac{1.818c}{\sqrt{{R}_{i}t}},$$
(4)

which is recommended for all assessments, or either of the approximate expressions:

$$\left\{\begin{array}{ll}{M}_{t}={\left(1+0.3797{\lambda }^{2}-0.001236{\lambda }^{4}\right)}^{0.5} & {\rm for}\; \lambda \le 9.1\\ {M}_{t}=0.01936{\lambda }^{2}+3.3 & {\rm for}\; \lambda >9.1\end{array},\right.$$
(5)

or the upper bound expression:

$${M}_{t}={\left(1+0.4845{\lambda }^{2}\right)}^{0.5}.$$
(6)

The load ratio can then be computed as the abscissa of the FAD using the reference stress and the yield strength, \({\sigma }_{y},\) by:

$${L}_{\mathrm{r}}=\frac{{\sigma }_{\mathrm{ref}}}{{\sigma }_{y}}.$$
(7)

3.1.1 SIF

For an axial through-thickness defect, the stress intensity factor attributed to the internal pressure, \({K}_{\mathrm{I}}^{P},\) is as follows:

$${K}_{\mathrm{I}}^{P}=\frac{p{R}_{o}}{t}{G}_{p}\sqrt{\pi c},$$
(8)

where the influence coefficient, \({G}_{p}\), is calculated using the following equations:

$${G}_{p}=\frac{{A}_{0}+{A}_{1}\lambda +{A}_{2}{\lambda }^{2}+{A}_{3}{\lambda }^{3}}{1+{A}_{4}\lambda +{A}_{5}{\lambda }^{2}+{A}_{6}{\lambda }^{3}} \quad \mathrm{ with } \; \lambda =\frac{1.818c}{\sqrt{{R}_{i}t}},$$
(9)

in which parameters \({A}_{0}-{A}_{6}\) are tabulated in Annex-9B of API 579-1/ASME FFS-1.

Eventually, to determine ordinate of the FAD assessment point, the toughness ratio is calculated using the following formula, assuming zero stress intensity from the secondary and residual stresses:

$${K}_{r}=\frac{{K}_{\mathrm{I}}^{P}}{{K}_{\mathrm{mat}}},$$
(10)

where \({K}_{\mathrm{mat}}\) denotes the material toughness.

3.1.2 Material Toughness

According to the ASME Codes, Section VIII, Divisions 1 and 2, material’s toughness requirements are established based on toughness exemption curves characterized by A, B, C, and D. Category D represents materials with the highest expected toughness, category A materials have the lowest expected toughness, and categories B and C represent materials with intermediate expected toughness values. The static fracture toughness consistent with these curves is given by

$$\begin{aligned} {K}_{\mathrm{IC}}&& ={\sigma }_{ys}\left(\sqrt{3}+\left(\sqrt{3}-\frac{27}{{\sigma }_{y}}\right)\cdot \mathrm{tan}h\left[\frac{\left(T-\left(D-75\right)\right)}{C}\right]\right)\nonumber \\ && \left(ksi\sqrt{in} , ksi,^\circ{\rm F} \right), \end{aligned}$$
(11)

where \(T\) represents the assessment temperature, \(C=66^\circ{\rm F}\) and

\(D=\left(\begin{array}{l}114^\circ{\rm F} {\text{for ASME Exemption Curve A}}\\ 76^\circ{\rm F} {\text{for ASME Exemption Curve B}}\\ 38^\circ{\rm F} {\text{ for ASME Exemption Curve C}}\\ 12^\circ{\rm F} {\text{for ASME Exemption Curve D}}\end{array}\right)\). However, some options exist to estimate the fracture toughness for carbon and low alloy steels for the purpose of FFS assessment. One conservative option is the lower bound fracture toughness estimation for ferritic material in accordance with Annex 9F given by:

$${K}_{\mathrm{IC}}=33.2+2.806 \; \mathrm{exp} \; \left[0.02\left(T-{T}_{\mathrm{ref}}+100\right)\right]\left(ksi\sqrt{in} ,^\circ{\rm F} \right),$$
(12)

where for exemption curve D, \({T}_{\mathrm{ref}}=-19\mathrm{^\circ{\rm F} }\mathrm{ or}-28\mathrm{^\circ{\rm C} }\) (Table 9.2 of API 579-1/ASME FFS-1). In-service degradation of toughness is ignored here, but it can be considered in accordance with Annex 9F, paragraph 9F.4.6 or 9F.4.7, if applicable.

4 Finite Element Analysis

4.1 Governing Equations and Discretization

The governing equations for linear elastic solids are readily available in many standard FEM books; however, a summary is presented here. The strong form of the governing equations for an isotropic linear elastic solid with domain \(\Omega\) and boundary \(\Gamma\) can be summarized as follows:

$$\frac{\partial {\sigma }_{ij}}{\partial {x}_{j}}+{f}_{\mathrm{i}}=0 \quad \mathrm{ in} \; \Omega ,$$
(13a)
$${u}_{i}={\overline{u} }_{i}\quad \mathrm{ on }\; {\Gamma }_{u},$$
(13b)
$${t}_{i}={\sigma }_{ij}{n}_{j}={\overline{t} }_{i}\quad \mathrm{ on }\; {\Gamma }_{t},$$
(13c)
$$\Gamma_{u} \cup \Gamma_{t} = \partial \Omega \quad {\text{ and }}\quad \Gamma_{u} \cap \Gamma_{t} = \emptyset ,$$
(13d)
$${\sigma }_{ij}={C}_{ijkl}{\varepsilon }_{kl}\quad \mathrm{ in} \; \Omega ,$$
(13e)
$${\varepsilon }_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\quad \mathrm{ in} \; \Omega ,$$
(13f)

where \({x}_{i}\) (\(i=1, 2, 3\)) is the rectangular Cartesian coordinates, \({\sigma }_{ij}\) is the Cauchy stress tensor, \({f}_{\mathrm{i}}\) is the body force, \({\overline{u} }_{i}\) and \({\overline{t} }_{i}\) are the given displacement and traction on the essential and natural boundaries, \({C}_{ijkl}\) is the fourth-order elasticity tensor, \({\varepsilon }_{ij}\) is the strain tensor, and \({n}_{\mathrm{j}}\) is component of the unit normal vector outward to the boundary.

Assuming infinitesimal virtual displacement \(\delta {u}_{i}\) and the corresponding virtual strain \({\delta \varepsilon }_{ij}\) in such a manner that:

$$\delta {u}_{i}\in {\Pi\; \Pi }=\left\{\delta {u}_{i}|\delta {u}_{i}\in {C}^{0},\delta {u}_{i}=0 \; \mathrm{ on } \; {\Gamma }_{u}\right\},$$
(14)

by using the principle of virtual work we have:

$${\int }_{\Omega }\left({{\delta \varepsilon }_{ij}}^{T}{C}_{ijkl}{\varepsilon }_{kl}\right)\mathrm{d} \Omega -{\int }_{\Omega }{{\delta {u}_{i}}^{T}f}_{i}\mathrm{d} \Omega -{\int }_{{\Gamma }_{t}}{\delta {u}_{i}}^{T}{\overline{t} }_{i}\mathrm{d}S=0,$$
(15)

which is the weak form of the governing equation described by Eq. (13a).

Using the basis functions, \({\varvec{N}}\), at element \(e\), \({\varvec{u}}\) field is approximated as follows:

$${{\varvec{u}}}_{h}={{\varvec{N}}}^{\mathrm{T}}{{\varvec{u}}}^{e},$$
(16)
$${\partial }_{j}{{\varvec{u}}}_{h}={\partial }_{j}{{\varvec{N}}}^{\mathrm{T}}{{\varvec{u}}}^{e}={{\varvec{B}}}^{\mathrm{T}}{{\varvec{u}}}^{e}={\boldsymbol{\varepsilon}},$$
(17)

where \({\varvec{B}}\) contains the spatial derivatives of the basis functions. The discrete system of Eq. (15) is eventually expressed as follows:

$$\left[{\varvec{K}}\right]\left[{\varvec{U}}\right]=\left[{\varvec{F}}\right]$$
(18)

where \({\varvec{U}}\) is the nodal displacement vector

$${\varvec{K}}=\sum_{e}{\int }_{{\Omega }_{e}}{({\varvec{B}})}^{\mathrm{T}}{\varvec{C}}{\varvec{B}}\mathrm{d}{\Omega }_{e},$$
(19)

is known as the stiffness matrix and

$${\varvec{F}}=\sum_{e}{\int }_{{\Omega }_{e}}{{\varvec{N}}}^{\mathrm{T}}{\varvec{f}}\mathrm{ d}{\Omega }_{e}+\sum_{{\varvec{e}}}{\int }_{{{\Gamma }_{t}}_{e}}{{\varvec{N}}}^{\mathrm{T}}{{\varvec{t}}}_{\Gamma }{\mathrm{ds}}_{e},$$
(20)

is the force vector. In Eqs. (19 and 20), the subscript, \(e\), in \({\Omega }_{e}\), \({{\Gamma }_{t}}_{e}\), and \({{\Gamma }_{D}}_{e}\) denotes the \({e}\)th finite element where \(\Omega =\bigcup_{e}{\Omega }_{e}\).

4.2 J-Integral Method

The J-integral is the strain energy release rate per unit fracture surface area or the rate of change of the total potential energy for a crack extension in linear elastic materials under quasi-static conditions. It is originally a two-dimensional line integral, as discussed here; nonetheless, an extension to three-dimensional is available in [32, 33]. The theoretical concept of the J-integral was developed by Rice [34] and Cherepanov [35] independently in the mid-1960s. Assuming quasi-static conditions, the J-integral is defined in two dimensions as follows:

$$J=\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }{\varvec{n}}\cdot {\varvec{H}}\cdot {\varvec{q}} d \Gamma \quad \mathrm{ with }\; {\varvec{H}}=W{\varvec{I}}-{\varvec{\sigma}}.\frac{\partial {\varvec{u}}}{\partial {\varvec{x}}},$$
(21)

where \({\varvec{H}}\) is the Eshelby’s energy–momentum tensor, \(W\) is the strain energy function, \(\Gamma\) is the integration line surrounding the crack tip in the counterclockwise direction from the lower to the upper face of the crack (the limit \(\Gamma \to 0\) indicates that \(\Gamma\) shrinks onto the crack tip), \(\mathrm{d}\Gamma\) is an infinitesimal arc length segment of \(\Gamma\), \({\varvec{q}}\) is a unit vector in the virtual crack extension direction, \({\varvec{n}}\) is the outward normal to \(\Gamma\), and \({\varvec{I}}\) is the identity matrix. Equation (21) can be extended as follows:

$$J = \mathop {\lim }\limits_{{{\Gamma } \to 0}} \mathop \int \limits_{{\Gamma }}^{{}} n_{i} e_{i} .\left[ {W\delta_{ij} \left( {e_{i} \otimes e_{j} } \right) - \sigma_{lk} \frac{{\partial u_{l} }}{{\partial x_{k} }}} \right].q_{k} e_{k} d{\Gamma }$$
$$\begin{aligned} &&= \mathop{\lim }\limits_{{{\Gamma } \to 0}} \mathop \int \limits_{{\Gamma }}^{{}} Wn_{i} q_{k} \delta_{ij} e_{i} .\left[ {\left( {e_{i} \otimes e_{j} } \right).e_{k} } \right]d{\Gamma }\\ && - \mathop \smallint \limits_{{\Gamma }}^{{}} n_{i} \sigma_{lk} \frac{{\partial u_{l} }}{{\partial x_{k} }}q_{k} e_{i} .e_{k} d{\Gamma } \end{aligned}$$
$$\begin{aligned} &&=\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{i}{q}_{k}{\delta }_{ij}{e}_{i}\cdot ({e}_{j}\cdot {e}_{k}){e}_{i}\mathrm{d}\Gamma \\ && -{\int }_{\Gamma }{n}_{i}{\sigma }_{lk}\frac{\partial {u}_{l}}{\partial {x}_{k}}{q}_{k}{{e}_{i}\cdot e}_{k}\mathrm{d}\Gamma \end{aligned}$$
$$\begin{aligned} &&=\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{i}{q}_{k}{\delta }_{ij}{\delta }_{jk}\mathrm{d}\Gamma \\ && -{\int }_{\Gamma }{n}_{i}{\sigma }_{lk}\frac{\partial {u}_{l}}{\partial {x}_{k}}{q}_{k}{\delta }_{ik}\mathrm{d}\Gamma \end{aligned}$$
$$\begin{aligned} &&=\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{i}{q}_{i}\mathrm{d}\Gamma -{\int }_{\Gamma }{\sigma }_{lk}{n}_{k}\frac{\partial {u}_{l}}{\partial {x}_{k}}{q}_{k}\mathrm{d}\Gamma \end{aligned}$$
$$\begin{aligned} && =\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{i}{q}_{i}\mathrm{d}\Gamma -{\int }_{\Gamma }{t}_{l}\frac{\partial {u}_{l}}{\partial {x}_{k}}{q}_{k}\mathrm{d}\Gamma , \end{aligned}$$
(22)

where \({\delta }_{ij}\) is the Kronecker symbol. For the case that \({x}_{1}\) and \({x}_{2}\) are coordinate directions parallel and normal to the crack ligament line (i.e., \(={1{\varvec{e}}}_{1}\)), Eq. (22) becomes:

$$\begin{aligned} J && =\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{1}\mathrm{d}\Gamma -{\int }_{\Gamma }{t}_{l}\frac{\partial {u}_{l}}{\partial {x}_{1}}\mathrm{d}\Gamma \\ &&=\underset{\Gamma \to 0}{\mathrm{lim}}{\int }_{\Gamma }W{n}_{1}\mathrm{d}\Gamma -{\int }_{\Gamma }{t}_{l}\frac{\partial {u}_{l}}{\partial {x}_{1}}\mathrm{d}\Gamma , \end{aligned}$$
(23)

knowing that \({n}_{1}\mathrm{d}\Gamma =\mathrm{d}{x}_{2}\), and omitting the limit symbol, we have the simple form of

$$J={\int }_{\Gamma }W\mathrm{d}{x}_{2}-{\int }_{\Gamma }{t}_{i}\frac{\partial {u}_{i}}{\partial {x}_{1}}\mathrm{d}\Gamma ,$$
(24)

which is widely used in the literature.

Following [32], one can rewrite Eq. (21) in the form of

$$\begin{aligned} & J ={\int }_{\Gamma }{\varvec{n}}\cdot {\varvec{H}}\cdot {\varvec{q}} \mathrm{d}\Gamma ={\int }_{\Gamma }{\varvec{n}}\cdot {\varvec{H}}\cdot {\varvec{q}} \mathrm{d}\Gamma \\ & -{\int }_{\mathrm{C}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma -{\int }_{{\mathrm{C}}_{+}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma -{\int }_{{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \\ & +{\int }_{{\mathrm{C}}_{+}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma +{\int }_{{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \end{aligned}$$
(25)

where \(\overline{{\varvec{q}} }\) is a sufficiently smooth weighting function within the region enclosed by the closed contour \(\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}\) and has the value \(\overline{{\varvec{q}} }={\varvec{q}}\) on \(\Gamma\) and \(\overline{{\varvec{q}} }=0\) on \(\mathrm{C}\), which makes the second term in the right hand side of Eq. (21) to be zero. Moreover, \({\varvec{m}}\) is the outward normal to the domain enclosed by the closed contour, as shown in Fig. 4. \({\varvec{m}}=-{\varvec{n}}\) on \(\Gamma\); and \({\varvec{t}}={\varvec{m}}\boldsymbol{ }\cdot {\varvec{\sigma}}\) is the surface traction on the crack surfaces \({\mathrm{C}}_{+}\) and \({\mathrm{C}}_{-}\). Thus, Eq. (25) can be rewritten in the form of:

$$\begin{aligned} J &=-{\int }_{\Gamma }{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma -{\int }_{\mathrm{C}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \\ & \qquad -{\int }_{{\mathrm{C}}_{+}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma -{\int }_{{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \\ &\qquad +{\int }_{{\mathrm{C}}_{+}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma +{\int }_{{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma , \end{aligned}$$
(26)

and since \(\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}\) are piecewise smooth curves, we have

Fig. 4
figure 4

Contour for the evaluation of the J-integral in 2-D (a) closed contour C + C_ ++ Γ + C_− encloses a domain A that includes the crack tip region as Γ → 0 (b)

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$$\begin{aligned} J & =-{\oint }_{\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma \\ & +{\int }_{{\mathrm{C}}_{+}{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \\ & =-{\oint }_{\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma \\ &+{\int }_{{\mathrm{C}}_{+}{\mathrm{C}}_{-}}{\varvec{m}}\cdot \left(W{\varvec{I}} -{\varvec{\sigma}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\right)\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma \\ &=-{\oint }_{\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma -{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma . \end{aligned}$$
(27)

It is noteworthy to mention that \({\int }_{{\mathrm{C}}_{+}{\mathrm{C}}_{-}}{\varvec{m}}\cdot \left(W{\varvec{I}}\right)\cdot \overline{{\varvec{q}}} \mathrm{d}\Gamma\) = 0 because \({\varvec{m}}\perp \overline{{\varvec{q}} }\) on crack faces.

Using the divergence theorem, we convert the closed contour integral in Eq. (27) into the domain integral as follows:

$$\begin{aligned} J&&=-{\oint }_{\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}}{\varvec{m}}\cdot {\varvec{H}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma -{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma \\ && =-{\int }_{A}\left(\frac{\partial }{\partial {\varvec{x}}}\right)\cdot \left({\varvec{H}}\cdot \overline{{\varvec{q}} }\right)\mathrm{dA}-{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma , \end{aligned}$$
(28)

where \(A\) is the domain enclosed by the closed contour \(\mathrm{C}+{\mathrm{C}}_{+}+\Gamma +{\mathrm{C}}_{-}\), and it includes the crack-tip region when \(\Gamma \to 0\).

\(W\) is a function of the mechanical strain, i.e., \(W=W({\varvec{\varepsilon}})\), thus

$$\frac{\partial W}{\partial {\varvec{x}}}=\frac{\partial W}{\partial{\varvec{\varepsilon}}}:\frac{\partial{\varvec{\varepsilon}}}{\partial {\varvec{x}}}=\boldsymbol{ }{\varvec{\sigma}}:\frac{\partial{\varvec{\varepsilon}}}{\partial {\varvec{x}}},$$
(29)

substituting Eqs. (13a) and (29) into Eq. (28) gives

$$\begin{aligned} J &=-{\int }_{A}\left[H:\frac{\partial \overline{{\varvec{q}}}}{\partial {\varvec{x}} }+\left(\frac{\partial W}{\partial {\varvec{x}}}-\frac{\partial{\varvec{\sigma}}}{\partial {\varvec{x}}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}-{\varvec{\sigma}}:\frac{{\partial }^{2}{\varvec{u}}}{\partial {{\varvec{x}}}^{2}}\right).\overline{{\varvec{q}} }\right]\mathrm{dA}\\ & \qquad -{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma \end{aligned}$$
$$\begin{aligned} &=-{\int }_{A}\left[H:\frac{\partial \overline{{\varvec{q}}}}{\partial {\varvec{x}} }+\left({\varvec{\sigma}}:\frac{\partial{\varvec{\varepsilon}}}{\partial {\varvec{x}}}+{\varvec{f}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}} -{\varvec{\sigma}}:\frac{\partial{\varvec{\varepsilon}}}{\partial {\varvec{x}}}\right)\cdot \overline{{\varvec{q}} }\right]\mathrm{dA}\\ &-{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\cdot \overline{{\varvec{q}}}\mathrm{d}\Gamma \end{aligned}$$
$$\begin{aligned}=-{\int }_{A}\left[H:\frac{\partial \overline{{\varvec{q}}}}{\partial {\varvec{x}} }+\left({\varvec{f}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}\right)\cdot \overline{{\varvec{q}} }\right]\mathrm{dA}-{\int }_{{\mathrm{C}}_{+}+{\mathrm{C}}_{-}}{\varvec{t}}\cdot \frac{\partial {\varvec{u}}}{\partial {\varvec{x}}}.\overline{{\varvec{q}}}\mathrm{d}\Gamma . \end{aligned}$$
(30)

To evaluate Eq. (30) in a FE model, defining the domain in terms of rings of elements surrounding the crack tip is common practice. The first contour consists of those elements directly connected to crack-tip nodes.

J-integral can be directly related to the SIF through the following formula

$$\begin{aligned} & \left \{ \begin{array}{ll} & J=\frac{{K_{\mathrm{I}}}^{2}}{E} \qquad \text{for isotropic homogeneous plane stress linear elasticity}\\ & J=\frac{{K_{\mathrm{I}}}^{2}}{E}(1-{\upsilon }^{2}) \quad \text{for isotropic homogeneous plane strain linear elasticity} \end{array}. \right. \end{aligned}$$
(31)

4.3 Abaqus Implementation

The J-integral calculation is available in ABAQUS/CAE software; thus, it is utilized in this work. The linear elastic material model is assumed, and the structural analysis is performed using a linear general static step. Due to symmetry, only half of the cylinder is modeled, and the appropriate symmetric boundary condition is applied on the axial cut surface, as illustrated in Fig. 5. To decrease the computational cost, the top head is not modeled. The design pressure is applied on the entire internal surface of the vessel. As shown in Fig. 5a, the rigid body motion in axial and radial directions is prevented by constraining the top cut surface and the bottom center point of the vessel, respectively.

Fig. 5
figure 5

Boundary conditions and general view of the model with crack location (a), crack tip and crack front (b), and partitioning of the crack zone (b, c)

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The crack is modeled as a seam using a partitioned face which is initially closed but can get open during the analysis. The partitioned geometry of the model and types of meshes are shown in Fig. 6. The center of the circular crack area has been meshed using the wedge meshing technique (15-node quadratic triangular prism elements) and the rest with 20-node quadratic brick elements.

Fig. 6
figure 6

Partitioning of the crack zone (a) and tubular surface around the crack front (b) section view around the crack front (c)

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5 Case Study

As Fig. 7 shows, a vertical lug-supported pressure vessel with a cylindrical shell and two 2:1 elliptical heads at the top and bottom is studied. The vessel contains gaseous hydrocarbon, and a longitudinal through-thickness crack is almost at the shell’s mid-length. The FE model includes 253,586 elements, while the element size around the crack front is about 0.2 mm. The summary of design data and material properties is listed in Table 1.

Fig. 7
figure 7

General view of the studied pressure vessel

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Table 1 Summary of design data and material properties used in this analysis
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5.1 Results and Discussion

The analysis results obtained based on formulas suggested by API 579-1/ASME FFS-1 are summarized in Table 2. The assessment point \(\left({L}_{r},{K}_{r}\right)=(0.487, 0.491)\) is located within the safe region of the FAD curve, which means LBB happens. The values of \({K}_{\mathrm{I}}^{P}\) calculated by FEA are presented in Table 3 for different contours around the crack tips and locations along the crack front. Though it is widely accepted that the first contour does not provide good results because of numerical singularities, even in this case, one can see the path independency of the J-integral. Arrangement of contours is illustrated in Fig. 8.

Table 2 LBB results summary
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Table 3 The values of stress intensity factors (MPa√m) obtained by FEA
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Fig. 8
figure 8

Contours arrangement around the crack front

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The approximated value of the \({K}_{\mathrm{I}}^{P}\) from the FEM analysis takes the value between 23.3 and 24.4 MPa \(\sqrt{m}\) for different points and paths around the crack tip. Taking into account safety and correction factors which are considered in the standard, the FEM results show good agreement with the analytical ones proposed by API 579-1/ASME FFS-1.

The maximum (limiting) through-thickness defect size in the wall, which satisfies the LBB requirements, is estimated as 110 mm. The corresponding assessment point is (0.53, 0.952). This limiting size is obtained by the incremental procedure described in API 579-1/ASME FFS-1, both analytically and numerically, and continued until the calculated assessment point is located on the FAD curve. The trajectory of assessment points from the present toward the limiting defect size is shown in Fig. 9. Once again, a good agreement between numerical and analytical results is observed.

Fig. 9
figure 9

Trajectory of assessment points, obtained through both analytical and numerical methods, leading to the limiting through-thickness defect size in the shell, which satisfies the LBB

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Next, we compute the failure probability at which the assessment point lies on the FAD or in the unacceptable region. The limit state function (\(G\left(x\right)=0\)) defines the safety criterion of the functional performance for the intended application. The probability of failure is as follows:

$${P}_{\mathrm{f}}=\underset{G(x)\le 0}{\overset{}{\int }}{I}_{G\le 0}\left(X\right){f}_{x}\left(X\right)\mathrm{ d}x,$$
(32)

where \({f}_{x}\left(X\right)\) is the joint probability density function of the input variables. \({I}_{G\le 0}\) is a failure indicator with binary output. In the failure state, which occurs when the assessment point exceeds the curve of FAD, \({I}_{G\le 0}=1\); meanwhile, in the safety state, \({I}_{G\le 0}=0\).

We considered the uncertainties in the crack length, the internal pressure, the minimum yield strength, and the material toughness. At the same time, a normal distribution is assumed for each variable, with mean values as reported in Table 1 and a coefficient of variation of ten percent. The thickness, \(t,\) of the vessel wall and its inner radius, \({R}_{i},\) are kept at the prescribed values. Figure 10 depicts a scatter of the assessment points at \({R}_{i}/t=50\), considering the uncertainties in the random variables. In the figure, it can be seen that some points exceed the safety margin.

Fig. 10
figure 10

Scatter plot of the assessment points due to the uncertainty at \({R}_{i}/t=50\)

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The variation of the probability of failure and the corresponding reliability index against values of \({R}_{i}/t\) are presented in Fig. 11. The probability of failure is firstly steady. Then, it increases sharply with the increase of \({R}_{i}/t\). According to reliability analysis, the probability of failure of a pressure vessel could be more than 50% when \({R}_{i}/t\) is \(\ge\) 67.71. Contrary, the reliability probability is around 99.99% for \({R}_{i}/t\le\) 30.

Fig. 11
figure 11

Probability of failure and reliability index at different radius to thickness ratios

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6 Concluding Remarks

The J-integral method through a three-dimensional finite element analysis was developed to numerically calculate the stress intensity factors (SIFs) for a cracked pressure vessel within the leak-before-break (LBB) assessment framework. The formulation was derived, and the maximum (limiting) through-thickness flaw size was calculated using the incremental procedure. Then, the errors in measuring the exact length of the crack during inspections, the internal pressure fluctuations due to operational conditions of the vessel, and uncertainties in characterizing the mechanical properties of the base material, including its minimum yield strength and toughness, were quantified. We found that:

  • J-integral was domain independence (error less than 0.5% in the calculation of SIF) for different contours around the crack tip.

  • The J-integral method provides a consistent SIF value with the one analytically calculated by the API 579-1/ASME FFS-1 standard (for limit crack size, the error is 4.8%).

  • The probability of failure of the studied pressure vessel was more than 50% for \({R}_{i}/t\) \(\ge\) 67.71.

  • In contrast, the reliability probability was around 99.99% when \({R}_{i}/t\le\) 30.

Extending this study to the axial and circumferential surface flaws, which are postulated in different locations of the pressure vessel and corner of the nozzle under thermomechanical loadings, makes the scope of our future studies.