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.     

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.     

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.     

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.     

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.     

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.     

Fracture response of pipelines subject to large plastic deformation under bending

Demand for long-distance offshore pipelines is steadily increasing. High internal pressure combined with bending/tension, accompanied by large plastic strains, along with the potential flaws in girth welds make the structural integrity of pipelines a formidable challenge. The existing procedures for the fracture assessment of pipelines are based on simplified analytical methods, and these are derived for a load-based approach. Hence, application to surface cracked pipes under large deformation is doubtful. The aim of this paper is to understand and identify various parameters that influence the fracture response of cracks in pipelines under more realistic loading conditions. The evolution of CTOD of a pipeline segment with an external circumferential surface crack is investigated under pure bend loading as well as bending with internal pressure. Detailed 3D elastic–plastic finite element simulations are performed. The effects of crack depth, crack length, radius-to-thickness ratio and material hardening on fracture response are examined. The results show that at moderate levels of CTOD, the allowable moment capacity of the pipe decreases significantly with increase in internal pressure. Further, the variation of CTOD with strain can be well approximated by a simple linear relationship.     

Ductile tearing tests of 316 stainless steel wide plates containing weldments

Twelve wide plate tests have been performed in recent years, as part of the European Fast Reactor development programme, to examine the influence of residual stresses on the fracture behaviour of Type 316 stainless steel plates and weldments. The first ten tests consisted of a study of through-thickness cracks and the last two tests addressed the behaviour of surface cracks. Results obtained from this series of tests are summarised and the ability of the R6 fracture assessment procedure to predict the results is reviewed. Significant amounts of ductile tearing under increasing applied load was shown to have occurred in both the plain plate and weldment through-thickness crack tests, prior to instability conditions being attained. R6 calculations were shown to conservatively underpredict instability applied loads for all through-thickness cracks tested. The through-thickness crack weldment tests indicated that residual stresses should be included in predicting initiation and the earliest stages of crack growth, but that they may be excluded in predicting the later stages of crack growth leading to instability. There was some evidence to suggest however that having to include residual stresses or not may depend on whether the section containing the crack is in contained yield or not. A wide plate test of a semi-elliptical crack in parent material, loaded in tension, indicated that the global limit load, although being slightly overpredictive in terms of applied load, was significantly more accurate than the local limit load approach.     

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.     

Rules for the assessment of interacting surface cracks under mode I load

A review of the current widely used methods for the assessment of adjacent and coplanar surface cracks under LEFM conditions shows them to be neither realistic nor always safe, having little scientific basis. Since such cracks can interact they cannot necessarily be assumed to be independent, so improved methods of accounting for the interaction are required and are provided by this work. In a companion paper a continuous method of estimating the interaction factors for two coplanar surface cracks was developed and proven. This enables realistic predictions of crack interaction and growth and is used as a basis for comparison of simpler assessment methods, the crack interaction estimation being somewhat cumbersome for general use. The assessment methods developed are thus shown to be safe. Furthermore they are realistic and straightforward to use. Cracks under tension, bending, and combined tension and bending load are considered.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.     

Compound cracks in pipes under combined bending and tension

Limit load and J-resistance curve solutions are developed for a compound crack in a pipe under combined tension and bending. The solution is based on a thick-walled cylinder assumption and the J solution can be applied with load-displacement data from one pipe test. Material resistance curves are developed for compound cracks in Type-304 stainless steel, Inconel 600 and A106 GrB base materials and a 304 stainless steel TIG weld. Tearing instability analyses are performed to assess the load-carrying capacity of pipes containing compound cracks and to evaluate nonconservatism associated with the use of C(T) specimen J-R data.     

冻融损伤混凝土梁抗力性能演化的裂缝分形特征

为分析冻融循环作用下钢筋混凝土梁受压区的表面裂缝分布特征,采用自主设计的混凝土梁反力试验架对受压区受冻融循环作用的梁进行抗弯试验,取得各级荷载作用下梁表面的裂缝图像。依据分形理论计算表面裂缝分布的分形维数,并讨论分形维数与梁的荷载、跨中挠度、屈服荷载和冻融循环次数之间的关系。研究表明,受压区冻融损伤的钢筋混凝土梁表面裂缝的分布符合分形特征,其分形维数与荷载、跨中挠度、屈服荷载和冻融循环次数之间均有一定关系,裂缝的分形维数可作为钢筋混凝土构件安全性能预测的指标。研究成果可为冻融损伤混凝土梁安全预测提供参考。     

Toroidal elasticity stress states for an end shear force acting on an elbow

The paper develops the stress fields for a 90° elbow acted upon by an end shear force which produces deformation in the plane of the elbow. There are coupled fields of stress because the elbow is held in equilibrium by an end normal force and by a bending moment. The solution makes use of a previous result for bending of an elbow under end couples.

The methods used are those of toroidal elasticity which requires the development of certain equilibrium functions and the development of certain compatibility functions.

There appears to be a total of ten fields of stress. Five initial fields of stress generate five first-order fields of stress when the method of successive approximations is applied to the general theory of toroidal elasticity.     

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.     

The interaction of two parallel non-coplanar identical surface cracks under tension and bending

In this paper, the interaction between two identical, non-coplanar, semi-elliptical surface cracks is investigated. The interacting cracks are assumed to be in an infinite plate subjected to remote tension or to pure bending loads. The stress intensity factors (SIFs) for these cracks are calculated using three-dimensional linear finite element analysis. A parametric study involving the relative horizontal and vertical separation distance between the two surface cracks is carried out for a specific crack shape and crack depth to plate thickness ratios of 0.3 and 0.2, respectively. An empirical formula is derived that relates the effects of the relative positions of these cracks to their SIFs.     

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.     

The effect of a superimposed compressive load on the failure of a stainless steel pipe containing a through-wall circumferential crack : Part I: Bending deformation is load-controlled

A net-section stress failure criterion is currently used to determine the critical size of circumferential crack for austenitic stainless steel as used in nuclear reactor piping systems. The acceptable crack size for prescribed applied loadings is determined by assuming that failure occurs when the stress across the net-section (that is the pipe cross-section area reduced by the crack area) attains a critical value, the stress at the cracked section being calculated on the basis of the undeformed pipe configuration. However, an axial compressive load can give an increased bending moment if the stress calculation is based on the deformed pipe configuration. This paper therefore examines the effect of an axial compressive load, superimposed upon a bending load, on the failure of piping due to the presence of a through-wall circumferential crack. The results show that the failure assessment should be based on the deformed configuration even when the axial load is only a small fraction of the pipe's Euler critical load, or otherwise the net-section stress approach can give non-conservative failure predictions.