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.     

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.     

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.     

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.     

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.     

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.     

Limit loads of circumferentially flawed pipes and cylindrical vessels under internal pressure

Published limit load formulae for circumferential defects overestimate the burst pressure for penetrating defects in pipes by the factor two in the short crack limit, because they only consider axial stress. Therefore, a class of limit load solution is discussed which takes the triaxial state of stress into account. The solutions for pressure loaded crack faces are improved analytically. Primal–dual limit analysis with the finite element method is used to adjust all solutions to numerical results. Limit loads are obtained for circumferential cracks of all sizes in thick-walled cylinders.     

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.     

Elastic fracture properties of all-steel gas cylinders with different axial crack types

All steel cylinders are being used for on-board storage of compressed natural gas in vehicles. Typical maximum fill pressure for these cylinder is 25.85 MPa (3750 psi). These cylinders are subjected to fluctuating pressures, due to the refueling operation. In order to establish a relevant test method to ensure leak before break failure performance in the event of a through-wall cracking, the finite element stress analysis of the design containing various defects has to be firstly carried out to get some theoretical basis for the establishment of the test method. External and internal axial semi-elliptical surface cracks are modeled. Crack front regions are modeled using singular elements, whereas the rest of the cylinder is modeled using twenty-node hexahedron elements. Not only the cylindrical body but also the neck and transition areas of the cylinder are considered in the modeling. Slender cracks with approximately 10 times the wall thickness of the cylinder, which often appear in the engineering application of all steel gas cylinders, are considered. The crack depths varied from 25% to 100% of the wall thickness. Analysis is also carried out for the cylinder with through-wall axial cracks, which have similar crack lengths with external and internal surface cracks. The cylinders are assumed to be in the elastic deformation state. Stress intensity factor, K_I, and crack mouth opening displacement, CMOD, as the functions of internal pressure, crack size, location (external verdus internal) and shape (elliptical versus straight-fronted), are established. Calculated results are compared with published results. Deep axial external cracks are found to be more severe than axial internal surface cracks having similar crack lengths. Crack driving force for a semi-elliptical through-wall crack is found to be significantly less than that of a straight-fronted through-wall cracks, which have the same crack length. So, the establishment of a relevant test method to ensure leak before break failure performance in the event of through-wall cracking is of high practical value for the engineering design and application of these cylinders.     

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.     

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.     

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.     

Remaining life assessment of welded pipes containing cracks

The presence of flaws, their size and location in the welded region affects the safe service life of pressure pipes operating at elevated temperature. A remaining life assessment was required to determine the probable remaining safe life of pressure pipes in a high-temperature steam distribution system in which crack-like defects had been identified in many welds. The crack-like defects indicated by NDE lacked the required information to fully identify their locations within the pipe wall. The effect of not knowing complete details of the location of the identified cracks was studied by considering a 360° circumferential crack with initial radial dimension a₀. The crack positions evaluated were: an internal surface crack, an external surface crack, and embedded cracks at various distances from the internal wall of the pipe. Crack growth was modeled using time-dependent fracture mechanics and a newly developed computer code.     

Stress intensity factors for internal and external cracks in pressurised thick-walled cylinders

Stress intensity factors for both internal and external semi-circular and semi-elliptical surface cracks in internally pressurised thick-walled cylinders of radius ratios between 2 and 3 are presented for a wide range of crack sizes. These solutions were obtained using the boundary integral equation (BIE) method for three-dimensional numerical stress analysis. Hoop strain distributions at the outer circumference of the cylinder are also presented for some external cracks, and shown to be useful for experimentally monitoring crack growth.     

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.     

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.     

A weight function approach to stress-intensity factors for half-elliptical surface cracks in cylindrical pressure vessels subjected to thermal shock

Root-mean-square averaged (RMS-averaged) stress-intensity factors were calculated for internal half-elliptical surface cracks in cylindrical vessels using a weight function method. The weight function was derived based on an approximate crack surface displacement representation. Stress-intensity factors for longitudinal half-elliptical inner surface cracks subjected to polynomial stress distribution have been presented and compared favorably with the existing numerical solutions. Superposition of the polynomial stress-intensity factors has provided an extremely efficient solution to the thermal shock crack problems. The crack geometries analyzed were R_i/R₀ = 10/11 and 4/5, a/c = 0·3333 and 0·8, the ratio of crack depth to wall thickness ranged from 0 to 0·8. The results, as well as the proposed method, offer a very powerful and economic way for the safety assessment of pressure vessels subjected to complex and varying load conditions.     

Evaluation of stress intensity factors for internally pressurised cylinders with surface flaws

The aim of this paper is to examine the different methods currently available for evaluating stress intensity factors in cylinders subjected to internal pressure and containing surface cracks. The techniques of analysis involve the use of both two- and three-dimensional models. For a given situation, the most pertinent choice of analysis is not yet clearly established because of the difficulty and uncertainty in interpreting the currently available experimental data. Consequently, it is of interest to review and compare the different analytical techniques now available, particularly with reference to their underlying assumptions.     

Different fracture performance of all-steel and all-aluminium cylinders with axial cracks

The linear elastic and the nonlinear elasto-plastic fracture mechanics analysis on all-metal (all-steel and all-aluminum) cylinder with different axially oriented cracks were carried out using the three-dimensional finite element method and the experimental method. The crack mouth opening displacement CMOD and the crack driving forces (K_I for elastic deformation state and J_I for elasto-plastic deformation state) of the all-steel cylinder and the all-aluminum cylinder containing axial deep cracks, were obtained. Through analysis of the calculated CMOD and crack driving forces for the all-steel cylinder and the all-aluminum cylinder with cracks, whose sizes are often met, respectively, in the engineering applications, the fracture behaviors of the two kinds of all-metal cylinders are compared. The CMOD for the two kinds of all-metal cylinders with external axial cracks were measured by an experimental method and good agreements between the calculated CMOD and tested CMOD were reached. Some CMOD and crack driving force expressions about the crack sizes, internal pressure and location along the crack front are obtained.     

Shakedown and limit analyses for 3-D structures using the linear matching method

A recently developed method for 3-D shakedown and limit analyses is evaluated in the present paper. The shakedown and limit loads of a holed plate subjected to biaxial loading are calculated by implementing the upper bound linear matching method into the commercial FE code ABAQUS. A defective pipeline under the combined action of internal pressure and axial tension is also analysed for both shakedown and limit capacities and the results compared with a standard programming method. All the numerical examples confirm the applicability of this procedure to complex 3-D structures.