Plastic limit pressures for cracked pipes using finite element limit analyses

Based on detailed finite element (FE) limit analyses, the present paper provides approximations for plastic limit pressure solutions for plane strain pipes with extended inner axial cracks; axi-symmetric (inner) circumferential cracks; axial through-wall cracks; axial (inner) surface cracks; circumferential through-wall cracks; and circumferential (inner) surface cracks. In particular, for surface crack problems, the effect of the crack shape, semi-elliptical or rectangular, on the limit pressure is quantified. Comparisons with existing analytical and empirical solutions show a large discrepancy for short circumferential through-wall cracks and for surface cracks (both axial and circumferential). Being based on detailed 3D FE limit analysis, the present solutions are believed to be accurate, and thus to be valuable information not only for plastic collapse analysis of pressurised piping but also for estimating non-linear fracture mechanics parameters based on the reference stress approach.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part I: stress intensity factor and limit load solutions

The J-integral and the crack opening area are the main parameters required for a leak-before-break evaluation of a piping system. Stress intensity factor and limit load solutions have been widely used for evaluating these parameters in a simplified way. Solutions for the stress intensity factor and limit load for a pipe with a circumferential through-wall crack subjected to axial and bending loads are reviewed and compared in this study. Based on the comparisons, recommendations are then made on expressions for calculating these parameters.     

Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes

This paper firstly presents net-section limit moments for circumferential through-wall and part-through surface cracks at the interface between elbows and attached straight pipes under in-plane bending. Closed-form solutions are proposed based on fitting results from small strain FE limit analyses using elastic–perfectly plastic materials. Net-section limit moments for circumferential cracks at the interface between elbows and attached straight pipes are found to be close to those for cracks in the centre of elbows, implying that the location of the circumferential crack within an elbow has a minimal effect on the net-section limit moment. Accordingly it is also found that the assumption that the crack locates in a straight pipe could significantly overestimate the net-section limit load (and thus maximum load-carrying capacity) of the cracked component. Based on the proposed net-section limit moment, a method to estimate elastic–plastic J based on the reference stress approach is proposed for circumferential cracks at the interface between elbows and attached straight pipes under in-plane bending.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part III: estimation of crack opening area

Evaluation of the crack opening area (COA) plays a central role in the evaluation of the critical crack length for a detectable leak for piping systems. Simplified evaluation methods for the COA for a circumferential through-wall crack in a pipe subjected to axial and bending loading or their combination is reviewed in this paper. Elastic solutions are compared and recommendations are given. Plastic solutions by the reference stress method are compared with nonlinear finite element solutions. The reference stress method tends to overestimate the COA for medium or large crack angles. Considerable improvement is achieved by making empirical modifications to the limit load expressions used in the calculation of the reference stress.     

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.     

The analytical fracture behaviour of a bar in tension containing a circumferential edge crack

This paper evaluates the analytical fracture behaviour of a bar in tension containing a circumferential edge crack. The stress intensity factor and limit load solutions available in the literature are reviewed and then used to derive the compliance and J-integral functions for a test specimen. The load at which plastic necking occurs in the specimen is also evaluated for idealized material behaviour to establish the maximum fracture toughness that can be measured as a function of specimen size. The analyses enable the optimum geometry and size of a specimen for fracture toughness measurements to be deduced.     

Reference stress method for evaluation of failure assessment curve of cracked pipes in nuclear power plants

In order to obtain a precise failure assessment curve (FAC) in the R6 defect assessment procedure, it is necessary to evaluate the J-value of cracked components. The reference stress method can be used for estimating J-values. However, the accuracy of estimation depends on the limit load used for evaluating the reference stress. In this study, the applicability of several limit load solutions was investigated through comparison with the results of elastic-plastic finite element analyses (FEA). A pipe containing a circumferential surface crack was analyzed under pure bending load. Six materials used in nuclear power plants were assumed. It was shown that the reference stress method is valid for FAC evaluation. The maximum non-conservativeness caused by using the reference stress method is less than 20% compared to the results obtained by FEA.     

Local and global collapse pressure of longitudinally flawed pipes and cylindrical vessels

Limit loads can be calculated with the finite element method (FEM) for any component, defect geometry, and loading. FEM suggests that published long crack limit formulae for axial defects under-estimate the burst pressure for internal surface defects in thick pipes while limit loads are not conservative for deep cracks and for pressure loaded crack-faces. Very deep cracks have a residual strength, which is modelled by a global collapse load. These observations are combined to derive new analytical local and global collapse loads. The global collapse loads are close to FEM limit analyses for all crack dimensions.     

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.     

Plastic collapse moment of 90° long radius elbows with internal circumferential surface crack at intrados under in-plane bending

Piping elbows under in-plane bending moment are vulnerable to cracking. The crack initiates at the surface and eventually reaches through the thickness and may lead to failure. The structural integrity assessment requires knowledge of the limit load. Limit load solutions for elbows with through-wall crack configurations are available in the open literature. But solutions for surface crack are not available. This paper presents a closed form expression for the plastic collapse moment (PCM) of 90°, long radius elbows with circumferential surface cracks at the intrados, under in-plane bending moment. The expression is derived, based on the results of non-linear (geometric and material) FE analyses covering a wide range of geometries and crack sizes. These plastic collapse moments evaluated herein will help in structural integrity assessment.     

Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending

The present work presents plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on detailed three-dimensional (3-D) finite element (FE) limit analyses using elastic–perfectly plastic materials. To assure reliability of the FE limit loads, modelling issues are addressed first, such as the effect of kinematic boundary conditions and branch junction geometries on the FE limit loads. Then the FE limit loads for branch junctions under internal pressure and in-plane bending are compared with existing limit load solutions, and new limit load solutions, improving the accuracy, are proposed based on the FE results. The proposed solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and the mean radius-to-thickness ratio of the run pipe from 10.0 to 20.0.     

Post-yield fracture mechanics theory and its application to pressure vessels

A post-yield fracture mechanics theory is presented based on Bilby, Cottrell and Swinden-Dugdale model solutions for cracks in bodies of finite width and subject to stress gradients. The model solutions are in good agreement with computed elastic-plastic values of the path independent integral J up to loads of 0·8 of the collapse load in the case of a cracked plate subject to bending. As an example, the model is applied to a thermal transient in a pressure vessel. The possibility of cracking and failure of a nozzle was considered. It is shown that a semi-circular crack will grow into an extended defect. Therefore, in the main analyses, axial symmetry and infinitely long longitudinal and fully circumferential cracks were considered. The post-yield fracture mechanics solutions are presented in the form of elastic-plastic stress intensity factors. Within the geometric approximations it is shown that longitudinal cracks are more dangerous than circumferential ones and can result in general yielding across, and full penetration of, the pressure vessel wall. In addition, linear elastic fracture mechanics under-estimates the danger of deep cracks and in some circumstances can considerably overestimate the critical size of small cracks. It is shown that the addition of a residual pressure stress considerably reduces the predicted critical defect sizes.     

A review of limit load solutions for cylinders with axial cracks and development of new solutions

Limit load solutions for axially cracked cylinders are reviewed and compared with available finite element (FE) results. New limit solutions for thick-walled cylinders with axial cracks under internal pressure are developed to overcome problems in the existing solutions. The newly developed limit load solutions are a global solution for through-wall cracks, global solutions for internal/external surface cracks and local solutions for internal/external surface cracks. The newly developed limit pressure solutions are compared with available FE data and the results show that the predictions agree well with the FE results and are generally conservative.     

Limit loads for piping branch junctions under internal pressure and in-plane bending—Extended solutions

The authors have previously proposed plastic limit load solutions for thin-walled branch junctions under internal pressure and in-plane bending, based on finite element (FE) limit loads resulting from three-dimensional (3-D) FE limit analyses using elastic–perfectly plastic materials [Kim YJ, Lee KH, Park CY. Limit loads for thin-walled piping branch junctions under internal pressure and in-plane bending. Int J Press Vessels Piping 2006;83:645–53]. The solutions are valid for ratios of the branch-to-run pipe radius and thickness from 0.4 to 1.0, and for the mean radius-to-thickness ratio of the run pipe from 10.0 to 20.0. Moreover, the solutions considered the case of in-plane bending only on the branch pipe. This paper extends the previous solutions in two aspects. Firstly, plastic limit load solutions are given also for in-plane bending on the run pipe. Secondly, the validity of the proposed solutions is extended to ratios of the branch-to-run pipe radius and thickness from 0.0 to 1.0, and the mean radius-to-thickness ratio of the run pipe from 5.0 to 20.0. Comparisons with FE results show good agreement.     

A global limit load solution for plates with surface cracks under combined end force and cross-thickness bending

A global limit load solution for rectangular surface cracks in plates under combined end force and cross-thickness bending is derived, which allows any combination of positive/negative end force and positive/negative cross-thickness moment. The solution is based on the net-section plastic collapse concept and, therefore, gives limit load values based on the Tresca yielding criterion. Solutions for both cases with and without crack face contact are derived when whole or part of the crack is located in the compressive stress zone. From the solution, particular global limit load solutions for plates with extended surface cracks and through-thickness cracks under the same loading conditions are obtained. The solution is consistent with the limit load solution for surface cracks in plates under combined tension and positive bending due to Goodall & Webster and Lei when both the applied end force and bending moment are positive. The solution reduces to the limit load solution for plain plates under combined end force and cross-thickness bending when the crack vanishes.     

Limit loads and fracture mechanics parameters for thick-walled pipes

In this paper, information on plastic limit loads and both elastic and elastic-plastic fracture mechanics parameters is given for cracked thick-walled pipes with mean radius-to-thickness ratios ranging from two to five. It is found that existing limit load expressions for thin-walled pipes can be applied to thick-walled pipes, provided that they are normalized with respect to the corresponding un-cracked thick-walled pipe values. For elastic fracture mechanics parameters, FE values of the influence functions for the stress intensity factor and the crack opening displacement are tabulated. For elastic-plastic J, it is shown that existing reference stress based J estimates can be applied, provided that a proper limit load for thick-walled pipes is used.     

Limit loads for pipe bends under combined pressure and in-plane bending based on finite element limit analysis

Approximate plastic limit load solutions for pipe bends under combined internal pressure and bending are obtained from detailed three-dimensional (3-D) FE limit analyses based on elastic-perfectly plastic materials with the small geometry change option. The FE results show that existing limit load solutions for pipe bends are lower bounds but can be very different from the present FE results in some cases, particularly for bending. Accordingly closed-form approximations are proposed for pipe bends under combined pressure and in-plane bending based on the FE results.     

Analysis and experiments on the plastic limit pressure of elbows

This paper discusses the plastic limit pressure of elbows without defects and with local thinned area in the extrados. Finite element analysis (FEA) and experiments have been used. The results of FEA show that the limit load of elbows under internal pressure increases with increasing wall thickness and bend radius of the elbow. The results are consistent with the calculated results by the Goodall formula, the maximum error is 6.58%. By data fitting of FEA, an empirical formula for the limit load of elbows with local thinned area in the extrados has been proposed, which is validated by experiments.     

Effects of attached straight pipes on finite element limit analysis for pipe bends

The effect of the length of an attached straight pipe on the plastic limit load of a 90° pipe bend under combined pressure and bending is quantified, based on finite element (FE) limit analyses using elastic–perfectly plastic materials with the small geometry change option. Systematic FE limit analyses of pipe bends with various lengths of the attached pipe are performed. It is shown that the effect of the length of the attached straight pipe on plastic limit loads can be significant, and the limit loads tend to decrease with decrease of the length of the attached straight pipe. In the limiting case of no attachment, the limit loads are found to be close to existing analytical solutions.     

A simple procedure to evaluate the stress intensity factor in internally pressurized cylinders

The evaluation of stress intensity factors in internally pressurized cylinders, with both surface and sub-surface flaws, is examined. The method of analysis is based on the equivalent linear representation of the circumferential stress distribution in accordance with ASME rules, the non-linear hoop stress distribution then being conservatively approximated by the membrane and bending stresses. The stress intensity factor for an elliptical crack embedded in an elastic solid and subjected to internal pressure is considered for two conditions of load (tension and bending) and the effects are added.The results are presented in non-dimensional form to evaluate the effect on stress intensity factor of the various parameters (outside and inside radius, crack position, cylinder thickness, form of ellipse).