Relevance of plastic limit loads to reference stress approach for surface cracked cylinder problems

To investigate the relevance of the definition of the reference stress to estimate J and C* for surface crack problems, this paper compares finite element (FE) J and C* results for surface cracked pipes with those estimated according to the reference stress approach using various definitions of the reference stress. Pipes with part circumferential inner surface cracks and finite internal axial cracks are considered, subject to internal pressure and global bending. The crack depth and aspect ratio are systematically varied. The reference stress is defined in four different ways using (i) a local limit load, (ii) a global limit load, (iii) a global limit load determined from the FE limit analysis, and (iv) the optimised reference load. It is found that the reference stress based on a local limit load gives overall excessively conservative estimates of J and C*. Use of a global limit load clearly reduces the conservatism, compared to that of a local limit load, although it can sometimes provide non-conservative estimates of J and C*. The use of the FE global limit load gives overall non-conservative estimates of J and C*. The reference stress based on the optimised reference load gives overall accurate estimates of J and C*, compared to other definitions of the reference stress. Based on the present findings, general guidance on the choice of the reference stress for surface crack problems is given.     

Validation of surface crack stress intensity factors of a tubular K-joint

Tubular K-joints are encountered widely in offshore structures, and the prediction of damaged joints depends very much on the accuracy of stress intensity factor solutions (SIFs). No parametric equations and very few results have been proposed and published in the literature for estimating the SIFs of any K-joints subjected to complex loading conditions. In this paper, a mesh generation method proposed previously for the Y-joint and T-joint has been extended to the K-joint. This method is realized by dividing the K-joint into several sub-zones with each zone consisting of different types of elements and mesh densities. This method has a distinct advantage of controlling the mesh quality, and most importantly the aspect ratio of the elements along the crack front. When the mesh of all the sub-zones has been generated automatically and completely, they are merged to form the complete model. The two most commonly used methods, namely the J-integral and displacement extrapolation, are used to evaluate the SIF values along the crack front of a typical K-joint. To validate the accuracy of these computed SIFs, a full-scale K-joint specimen was tested to failure under fatigue loading conditions. The standard alternating current potential drop (ACPD) technique was used to monitor the rate of crack propagation of the surface crack located at the hot spot stress region. Using the given material parameters C and m, the experimental SIFs were deduced, and they are found to be in good agreement with the computed SIFs obtained from the generated models. Hence, the proposed finite element models are both efficient and reliable.     

Contour integral calculations for generalised creep laws within abaqus

This paper reports time-dependent C(t)-integral calculations for generalised creep laws using the user subroutine CREEP within the general-purpose FE code, abaqus [abaqus Version 5.8. User's Manual (1999)]. The evaluation of the strain energy density rate function is emphasised. As examples, the external circumferential cracked cylinder subject to mechanical loading and combined mechanical and thermal loading was considered for two different generalised creep laws, the θ-projection creep law and a secondary–tertiary creep law. The resulting abaqus C(t)-integral results were then compared with the results from another FE code, bersafe [bersafe User's Guides (1990)], to check reliability of the abaqus calculations. Excellent agreement between these results provides confidence in the abaqus C(t)-integral calculation for generalised creep laws.     

KI–T estimation for embedded flaws in pipes – Part II: Circumferentially oriented cracks

This paper, in parallel to the investigation on axially embedded cracks reported in the companion paper, presents a numerical study on the linear-elastic K_I and T-stress values over the front of elliptical cracks circumferentially embedded in the wall of a pipe/cylindrical structure, under a uniform pressure applied on the inner surface of the pipe. The numerical procedure employs the interaction-integral approach to compute the linear-elastic stress-intensity factor (SIF) K_I and T-stress values for embedded cracks with practical sizes at different locations in the wall of the pipe. The parametric study covers a wide range of geometric parameters for embedded cracks in the pipe, including: the wall thickness to the inner radius ratio (t/R_i), the crack depth over the wall thickness ratio (a/t), the crack aspect ratio (a/c) and the ratio of the distance from the centerline of the crack to the outer surface of the pipe over the pipe wall thickness (e_M/t). The parametric investigation identifies a significant effect of the remaining ligament length on both the T-stress and K_I values at the crack-front location (denoted by point O) nearest to the outer surface of the pipe and at the crack-front location (denoted by point I) nearest to the inner surface of the pipe. The numerical investigation establishes the database to derive approximate functions from a nonlinear curve-fitting procedure to predict the T-stress and K_I values at three critical front locations of the circumferentially embedded crack in a pipe: points O, I and M. The proposed T-stress and K_I functions utilize a combined second-order polynomial and a power-law expression, which presents a close agreement with the T-stress and K_I values computed from the very detailed finite element models. The comparison between the circumferentially embedded crack and the axially embedded crack indicates that both the T-stress and K_I values at crack-front points O and I in a circumferential crack equal approximately 50% the T-stress and K_I values at the corresponding front locations in an axial crack with the same crack depth ratio, the same crack aspect ratio and the same pipe wall thickness to the inner radius ratio.     

Elastic–plastic J and COD estimates for axial through-wall cracked pipes

This paper proposes engineering estimation equations of elastic–plastic J and crack opening displacement (COD) for axial through-wall cracked pipes under internal pressure. On the basis of detailed 3D finite element (FE) results using deformation plasticity, the plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the mean radius-to-thickness ratio, the normalised crack length, and the strain hardening. On the basis of these results, the GE/EPRI-type J and COD estimation equations are proposed and validated against 3D FE results based on deformation plasticity. For more general application to general stress–strain laws or to complex loading, the developed GE/EPRI-type solutions are re-formulated based on the reference stress (RS) concept. Such a re-formulation provides simpler equations for J and COD, which are then further extended to combined internal pressure and bending. The proposed RS based J and COD estimation equations are compared with elastic–plastic 3D FE results using actual stress–strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimates.     

Analysis of interacting semi-elliptical surface cracks in finite thickness plates under remote bending load

Interaction effects of two coplanar self-same shallow and deep semi-elliptical surface cracks in finite thickness plates subjected to remote tension have been previously investigated by Sethuraman et al. Using the finite element based force method. In the present study, the effect of remote bending load on interacting semi-elliptical surface cracks in a finite thickness plate is analyzed. Stress intensity factors are evaluated along the entire crack front using a modified force method based on the three-dimensional finite element solution. The line spring model has also been used to evaluate stress intensity factors at the deepest point of a crack using shell finite element analysis. Parametric studies involving a wide ranges of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.2≤a/c≤1.0), crack depth ratio (0.2≤a/t≤0.9), relative crack location (0.2≤2c/d≤0.9) and normalized location on the crack front (0≤2ϕ/π≤2) are carried out for numerical estimation of crack interaction factors. Due to the crack interaction, the stress intensity factor distribution is observed to be asymmetric along the crack front. The interaction is also observed to cease when the distance between two cracks is more than five times the crack width (i.e. 2c/d less than 0.2) irrespective of crack shape aspect ratio. Finally, an empirical relation is proposed for the evaluation of crack interaction crack interaction factors for the range of parameters considered. For the ranges considered, the proposed empirical relation predicts the crack interaction factors at the inner and outer surface points of the crack within ±4% of the three-dimensional finite element solutions.     

KI–T estimations for embedded flaws in pipes – Part I: Axially oriented cracks

This study reports a numerical investigation on the linear-elastic K_I and T-stress values over the front of elliptical cracks axially embedded in the wall of a pipe/cylindrical structure, under a uniform pressure applied on the inner surface of the pipe. The numerical procedure employs an interaction integral approach to compute the linear-elastic stress intensity factor (SIF) K_I and T-stress values from very detailed crack-front meshes. The verification study confirms the accuracy of the adopted numerical procedure in computing the K_I values based on existing results for external axial surface cracks in the wall of a cylindrical structure. The parametric investigation covers a wide range of geometric parameters including: the wall thickness to the inner radius ratio of the pipe (t/R_i), the crack depth over the wall thickness ratio (a/t), the crack aspect ratio (a/c) and the crack location measured by the ratio of the distance from the centerline of the crack to the outer surface of the pipe over the pipe wall thickness (e_M/t). Subsequent efforts develop, from a nonlinear curve-fitting procedure, a new set of equations to estimate the T-stress and K_I values at three critical front locations of the axial elliptical cracks: the crack-front point O nearest to the outer surface of the pipe, the crack-front point I nearest to the inner surface of the pipe and the crack-front point M on the centerline of the axial crack. These equations combine a second-order polynomial with a power-law expression to predict the pronounced variations in the T-stress and K_I values with respect to the geometric parameters. The coefficients of the new K_I and T-stress equations either take a constant value or incorporate the linear variation with respect to the pipe wall thickness over the inner radius ratio, t/R_i. The proposed equations demonstrate a close agreement with the finite element (FE) results, which indicate very strong dependence of the T-stress and K_I values at point O and point I on the corresponding ligament lengths, e_O and e_I.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part II: J-integral estimation

Evaluation of the J-integral plays a central part in evaluation of the critical crack length for unstable fracture for piping systems. Simplified evaluation methods for the J-integral for a circumferential through-wall crack in pipes subjected to axial and bending loading or their combination is reviewed in this paper. Use of the LBB.ENG2 method and a similar approach based on the η-factor concept were found to result in significant underestimation of the J-integral for small and medium crack angles. On the other hand, the reference stress method based on the solutions for stress intensity factor and limit load recommended in the companion paper (Part I) provides solutions which agree well with the available non-linear finite-element solutions and can be utilized as a powerful tool for J-integral evaluation for arbitrary materials, not restricted to simple power-law hardening.     

Influences of some materials on J-estimation methods for pipes with circumferential surface cracks

The approximate calculation methods (SC.ENG1 and SC.ENG2) for the J-integral for pipes with circumferential surface cracks are discussed and three-dimensional elastic–plastic finite element models for circumferentially surface-cracked pipe are conducted to evaluate the accuracy of these methods for different pipe materials used in China. The numerical studies verify that the SC.ENG2 method provides more accurate estimates of J than SC.ENG1. Based on three-dimensional elastic–plastic fracture analysis, the distribution of the local J-integral along the front of a circumferential constant-depth internal surface crack is investigated and the influences of different pipe materials with different yield plateaux on J-integral values are discussed. The validity of SC.ENG1 and SC.ENG2 J-integral estimation methods for pipe steel materials with different yield plateaux used in China are examined in detail and the SC.ENG2 method is found to provide reasonable estimates of J for materials with yield plateaux.     

Creep crack growth of seam-welded P22 and P91 pipes with artificial defects. Part II. Data analysis

This paper examines several methods for assessing experimental creep and fatigue crack growth data obtained on P22 (2.25Cr1Mo) and P91 (9Cr1MoVNb) axially notched, seam-welded pipes tested at 565 and 625 °C, respectively [Creep crack growth of seam-welded P22 and P91 pipes with artificial defects—part I: experimental study and post-test metallography. Second International HIDA Conference, Advances in Defects Assessment in High Temperature Plant, MPA, Stuttgart, Germany, 4–6 October, 2000]. The overall objective of this work is to identify the nature of any correlation between component and conventional testpiece creep crack growth rates and thereby provide a supplementary tool for structural integrity analysis. Creep crack growth rate of the notch located in the heat-affect-zone of the weld was assessed in terms of both stress intensity factors, K_I, and the C^*-integral. To estimate the C^*-integral, reference stresses were developed by deriving limit load solutions which reconcile the different collapse loads of the axially notched pipes. Both minimum and average creep rate laws were utilised in the analysis to accommodate the strain rate in the C^* relation. Each test was examined independently, but the general conclusion from each analysis was the same, in that C^*-integral, rather than the stress intensity factor, gave better correlation with respect to conventional data generated using compact tension (CT) specimens. The assessment of creep crack growth demonstrates one particular aspect of interest. In terms of the C^* based correlation of creep crack growth rates, the analysis was found to be relatively independent of the stress state and correlates well with CT specimen data when appropriate reference stresses are used. In addition, cracking in the tested pipes was observed to occur between plane stress and plane strain conditions, inferring that both creep ductility and ligament straining contribute towards the failure mechanism.     

Finite element calculation of contour integral parameters for cracked P91 pipe weld

Abstract

This paper reports the results of elastic–creep finite element (FE) analyses of a P91 steel pipe weld with two external ‘type IV’ circumferential cracks, subjected to internal pressure and end (system) load using creep properties obtained at 650°C. Numerical contour integral calculations have been performed to obtain both transient and stationary high temperature fracture mechanics parameters. A mesh sensitivity analysis was performed in order to ensure the accuracy of FE analyses in the transient creep stage. The effects of load magnitude, the material mismatch near the crack surfaces and the crack depth on the stationary creep C* contour integrals have been investigated, and corresponding analytical correlations are presented.     

J-integral and local damage fracture analyses for a pump casing containing large weld repairs

Three-dimensional J-integral and two-dimensional Local Approach finite element studies are described for postulated crack-like defects in a large repair weld to the casing of a light water reactor circulation pump. The repair weld residual stress field is simulated and plant operating pressure and thermal transient loads are applied. Crack tip constraint effects are quantified through detailed analysis of the cracked structure and compact tension fracture toughness specimens. Fracture initiation crack sizes are shown to be larger than conceivable fabrication defects that are detectable using modern ultrasonic inspection techniques. The Local Approach study demonstrates the benefits of quantifying crack tip constraint conditions, compared with conventional J-estimation schemes and cracked body J-integral analysis. The method of introducing the crack into the finite element model is shown to have a large effect on calculated crack tip fracture parameters; a slowly developing crack in the residual stress field being more benign.     

Use of local and global limit load solutions for plates with surface cracks under tension

Some available experimental results for the ductile failure of plates with surface cracks under tension are reviewed. The response of crack driving force, J, and the ligament strain near the local and global limit loads are investigated by performing elastic-perfectly plastic finite element (FE) analysis of a plate with a semi-elliptical crack under tension. The results show that a ligament may survive until the global collapse load is reached when the average ligament strain at the global collapse load, which depends on the uniaxial strain corresponding to the flow stress of the material and the crack geometry, is less than the true fracture strain of the material obtained from uniaxial tension tests. The FE analysis shows that ligament yielding corresponding to the local limit load has little effect on J and the average ligament strain, whereas approach to global collapse corresponds to a sharp increase in both J and the average ligament strain. The prediction of the FE value of J using the reference stress method shows that the global limit load is more relevant to J-estimation than the local one.     

The shape of a surface crack in a plate based on a given stress intensity factor distribution

A new method is developed for the evaluation of a crack shape based on a given stress intensity factor (SIF) distribution for a surface crack under Mode-I loading conditions. The SIF distribution along the crack front is investigated using a direct simulation technique, in which the effect of crack closure on fatigue crack growth is considered. Then a SIF distribution function is chosen based on the numerical results. Crack shape (and the SIF) is achieved based on the given SIF distribution function using a numerical iterative procedure. Empirical SIF equations for surface cracks in plates subjected to tension and pure bending fatigue load are determined by systematic curve fitting of the numerical results. The depth ratio and the aspect ratio are considered in the ranges of 0.1–0.9 and 0.2–1.2, respectively. The aspect-ratio variation of surface cracks under fatigue loading is predicted. The application of the new method to predict the shape of a surface crack in plates subjected to tension and pure bending and comparisons of the results obtained with the predictions of the empirical equations proposed by Newman and Raju are presented.     

Stress intensity factor solutions for arbitrarily shaped surface flaws in reactor pressure vessel nozzle corners

Extensive experimental results—based on the frozen stress photoelasticity technique for extracting stress intensities—for nozzle corner cracks in ITV and BWR geometries were reported by Smith et al.¹ Based on the above experimental studies, it was conjectured that if the crack shape inserted into a finite element model is not a real one, or if the inner fillet (for shallow flaws) or the outer boundary shape (for moderate to deep flaws) is improperly approximated, the obtained numerical results for stress intensity factors may differ significantly from the physical behaviour at the nozzle-vessel junction. On the other hand, almost all the numerical analyses published to date, based on finite elements, boundary integral equations or alternating techniques, considered only quarter-circular nozzle corner cracks.This paper presents stress intensity factor solutions for naturally shaped nozzle corner cracks in pressurised ITV and BWR vessels. Several actual crack geometries observed in the experimental work of Smith et al., cited above, are studied using the three-dimensional hybrid crack-element approach of Atluri et al.⁸ and Atluri and Kathiresan⁹ wherein the stress intensity factors and their variation along an arbitrarily shaped 3-D crack front are directly computed. In order to be able to compare the present results with the photoelastic experimental results (wherein the Poisson's ratio of the material is 0·5), some of the present numerical results are obtained for $\text{ν ? 0·5}$.In addition, some new solutions for stress intensity factors for pressurised thin (outer to inner radii ratios of $\text{～1·1}$) cylindrical vessels with belt-line flaws of semi-elliptical shapes of various aspect ratios and depth ratios are presented. Cases of surface flaws in the meridional direction—as well as in the circumferential direction—of the vessel are treated.     

Elasto-plastic analysis of a cracked ductile cylindrical pressure vessel

The deformation of a long cylindrical pressure vessel made of strain hardening material with a long longitudinal crack is analyzed under internal pressure by an FEM code. Stress distributions, the shape of the crack opening, the plastic zone, the J-integral and the CTOD are calculated.The $\text{J}\text{CTOD}$ ratio is found to be practically independent of the amount of pressure load, as well as of the crack height.Comparisons are made with a fully plastic strip yield model which seems to give a reasonable estimate of crack opening. This model suggests that the CTOD is linearly dependent on the radius of cylinder. This implies that the size of the pressure vessel is an important parameter when considering the tolerance of flaws.     

Stress intensity factors for circumferential cracks in pressure vessel door closures

Bandlock pressure vessel door closures and many other commonly used engineering geometries can be simplified to that of a tube with an axisymmetric internal projection. When the projection is axially loaded, the maximum stress is seen at the start of the fillet between the tube and the projection, and this is the site of potential fatigue crack growth. Stress intensity factors for circumferential cracks in tubes with axially loaded axisymmetric internal projections are presented for a wide range of geometries, governed by five parameters: tube diameter, projection depth and length, fillet radius and crack length. The dominant mode is shown to be mode I. Results are presented to allow the variation of Y_I, with fillet radius, tube diameter, projection depth, projection length and crack length, to be determined. Y_I is shown to vary widely, between 0.59 and 12.65, and is seen to be strongly dependent on projection depth. Y_I is dependent on fillet radius for short crack lengths, but for crack lengths approaching half the tube wall thickness, it becomes independent of fillet radius. Y_I results for the simplified loaded projection are compared with results from a finite element analysis of a typical pressure vessel door closure, and the results are seen to correlate to within 3.2%.     

A re-examination of entrainment constant and an explicit model for flame heights of rectangular jet fires

Flame heights of buoyant turbulent jet fires produced by rectangular nozzles whose aspect ratio varied from 1:1 to 1:71 are investigated experimentally in this work. The change of the entrainment constant parameter C₁ with aspect ratio is discussed based on the comprehensive data obtained. It is found the value of C₁ does not need to be transformed from 0.179 to 0.444 with an increase in aspect ratio from axisymmetric one to linear one as proposed previously in the classic correlation due to limited data, a change which might be misleading. It is revealed to in fact change little with rectangular fire source aspect ratio and can be constantly taken as 0.185. A new explicit model to predict flame heights for given heat release rates of rectangular jet fires is then proposed, which is shown to be in good agreement with the measured values for different source aspect ratios.     

Finite element based evaluation of stress intensity factors for interactive semi-elliptic surface cracks

A finite thickness plate with two coplanar self-same shallow and deep semi-elliptical surface cracks subjected to remote tensile surface traction is considered for fracture analysis. Based on three-dimensional (3D) finite element solutions, stress intensity factors (SIFs) are evaluated along the entire crack front using a force method. The line spring model has also been used to evaluate crack depth point SIFs using shell finite element analysis. A wide range of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.3≤a/c≤1.2), crack depth ratio (1.25≤t/a≤6), relative crack location (0.33≤2c/d≤0.9) and normalized location on the crack front (0≤2φ/π≤2) are considered for numerical estimation of crack interaction factors. SIFs evaluated at the depth point using the force method from the 3D finite element results are compared with SIFs evaluated using the line spring model. Finally, using finite element results, an empirical relation is proposed for the evaluation of crack interaction factors. For the ranges considered, the proposed empirical relation predicts crack interaction factors at critical locations within ±2% of the 3D finite element solutions.     

Experimental determination of the C1 parameter for a plate with a surface crack and submitted to bending

This article deals with the experimental determination of the C¹ parameter for a surface crack in a plate submitted to a bending moment.The first part describes the experimental device and the data acquisition performed during the creep test. From this data acquisition, a proposition is made to determine C¹.In the second part, a finite element calculation series is performed to confirm the proposition. It concerns 3D creep calculation on plates with a surface defect in which the C¹ parameter is determined and correlated to the crack mouth opening angle rate. It is shown that the proposition has a small dependence on the creep law of the material and can be easily generalized to any surface defect geometry.