A numerical method for lower bound limit analysis of 3-D structures with multi-loading systems

The determination of lower bound limit load of 3-D structures is by no means an easy task, especially for complex configurations and loading systems. In our previous work, a numerical method of upper bound limit analysis for 3-D structures with multi-loading systems was proposed. This method combines FEM and mathematical programming technique in an iterative procedure. In the present article, on the basis of the nature of the iterative procedure for upper bound limit analysis, the statically admissible stress fields, which satisfies the equilibrium equation and boundary conditions, are constructed using some intermediate variables obtained by upper bound limit analysis procedure. Moreover, a mathematical programming formulation is set up for the static limit analysis of 3-D structures under multi-loading systems and a direct iterative algorithm used to determine the lower bound limit load multiplier is proposed, which depends on the static theorem of plasticity. The numerical examples are given to demonstrate the applicability of the procedure.     

On relating the reference stress, limit load and the ASME stress classification concepts

This paper discusses the relationship between the concepts of reference stress, limit load and the ASME stress classification framework, as it applies to common mechanical components and structures. The relationship is based on the underlying notion of load-control.

The reference stress method, which has been used mainly in the UK, attempts to correlate creep deformations in a structure with the results of a uniaxial creep test. It has been observed that the reference stress is relatively insensitive to material parameters characterizing creep behaviour. The method has been widely used in the design and life assessment of nuclear as well as conventional components. Specifically, it has been applied to problems pertaining to creep growth, rupture damage, creep buckling, and more recently, elastic-plastic fracture toughness.

Determination of the reference stress is not always a simple task. An approximate method of its determination relies on prior knowledge of limit loads for various configurations and loadings. The drawback, of course, is that there is a reliance on the available catalogue of limit loads. Rather than attempting to ‘fit someone into a garment’, this paper discusses a new and a direct procedure for reference stress determination using the GLOSS R-Node method. The reference stress is determined for some pressure component configurations. The ideas are extended to the ASME stress classification framework, and the various stress categories are discussed.     

On relating redistributed elastic and inelastic stress fields

The classical lower bound theorem in plasticity states that the load required to create equilibrium stresses in a structure that are below yield will always be less than or equal to the collapse load. Recent advances in determination of lower bound limit loads involve repeating elastic analyses after systematic modification of elastic moduli. The intention is to obtain lower bound limit loads from stress fields that would progressively approach a state similar to one at plastic limit. The gradual transformation of statically admissible stress fields from elastic to limit state can be compared to those corresponding to power-law creep indices, ranging from one to infinity. This paper attempts to investigate the possibility of establishing such relationships on a one-to-one basis, by considering standard component configurations.     

The linear matching method applied to the high temperature life integrity of structures. Part 1. Assessments involving constant residual stress fields

Design and life assessment procedures for high temperatures are based on ‘expert knowledge’ in structural mechanics and materials science, combined with simplified methods of structural analysis. Of these, R5 is one of the most widely used life assessment methods internationally with procedures based on reference stress techniques and shakedown calculations using linear elastic solutions. These have been augmented by full finite element analysis and, recently, the development of a new programming method, the linear matching method (LMM), that allows a range of direct solutions that include shakedown methods and simplified analysis in excess of shakedown. In this paper, LMM procedures are compared with calculations typical of those employed in R5 for cyclic loading problems when the assumption of a constant residual stress field is appropriate including shakedown and limit analyses, creep rupture analysis and the evaluation of accumulated creep deformation. A typical example of a 3D holed plate subjected to a cyclic thermal load and a constant mechanical load is assessed in detail. These comparisons demonstrate the significant advantages of linear matching methods for a typical case. For a range of cyclic problems when the residual stress field varies during the cycle, which include the evaluation of plastic strain amplitude, ratchet limit and accumulated creep strains during a high temperature dwell periods, the corresponding LMM and R5 procedures are discussed in an accompanying paper.     

Analysis of two simplified methods (R6 & GE-EPRI) for circumferential crack stability in leak-before-break applications

The Ainsworth method (R6 Option 2) and the GE-EPRI method are compared and analysed. These J-estimation schemes allow us to assess at low cost, crack stability in piping systems, which is an important step in Leak-Before-Break applications. The reliability, flexibility and self-consistency of the two methods are examined by focusing on the effect of geometry, nature of loading and superposition of tension and bending. The GE-EPRI method, based on Finite Element results, is used to check the validity of the hypothesis on which the simplifications in the R6 method for through-wall circumferentially cracked pipes rely. The R6 method appears to be easier to use for the treatment of secondary loads. It is shown that, the Ainsworth Failure Assessment Line, derived in the single load case, is still valid for combined proportional tension and bending loads, provided the appropriate limit load formula is chosen.     

Numerical schemes for defining reference stress in components containing defects

In R6 and R5, the standards of integrity assessments of components containing defects established by the former Central Electricity Generating Board (CEGB) in the UK, the basic procedure in calculating the crack driving force parameters, J-integration and C*-integration is to define the reference stress, σ_ref, on the defective section. Three methods of directly defining the reference stress, σ_ref [multiple steady-state creep analysis (MSCA) method, creep module-modified iteration (CMMI) method and elastic module-modified iteration (EMMI) method], are suggested in this paper, differing from the common methods in which the limit loads of the components are first calculated and then the σ_ref is convertibly determined. The basic conceptions and calculational steps of these methods are discussed and compared in the context, and the validity is demonstrated by several examples.     

Analysis of cyclic creep and rupture. Part 1: bounding theorems and cyclic reference stresses

Cyclic loading on structures can produce failures not readily predicted by conventional static analysis. Ratcheting or incremental distortion leads to structural failure, and complicates the problems of creep and fatigue prediction. Predicting shakedown, ratcheting, accelerated creep and rupture, for cyclic loading, are the objectives of cyclic stress analysis.Limit load, shakedown and ratcheting analyses provide a comprehensive basis to understand static structural behaviour for ductile inelastic materials, subject to variable loading but excluding inertial dynamic effects. From them we can predict the following failure modes:

• –Plastic collapse.
• –Failure to shakedown.
• –Ratcheting.
• –Accelerated creep and rupture.

This is achieved with a generalisation of the reference stress concept. Conventionally, and for steady loading, the limit load reference stress is the lowest yield stress for which the structure does not collapse. For cyclic loading two definitions are available. The more conservative is the lowest yield stress for which the structure shakes down (behaves elastically). The less conservative is the lowest yield stress for which the structure does not ratchet. They have different meanings and uses.Explaining and justifying the use of cyclic reference stresses to bound creep and rupture is the objective of Part 1. Part 2 gives examples illustrating a range of structural behaviours. The methodology of these papers involves so-called approximate methods at one level, that of inferring limiting or conservative time-dependent behaviour from time-independent elastic–plastic cyclic analyses. The elastic–plastic cyclic analyses themselves are straightforward if tedious. Some ideas and a new analysis technique are available to reduce the trial-and-error.     

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.     

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.     

Linear matching method for creep rupture assessment

The recently developed linear matching method (LMM), which is easily implemented within commercial FE codes, has been successfully used to evaluate elastic and plastic shakedown loads. In this paper, the method is extended to the prediction of the creep rupture life of a structure, based upon a bounding method currently used in the life assessment method R5. The method corresponds to the requirement that, for the operating load history, the structure should shakedown where the yield stress is given by the lesser of the plastic yield stress and a high temperature rupture stress corresponding to a rupture time. A holed plate subjected to cyclic thermal load and constant mechanical load is assessed in detail as a typical example to confirm the applicability of the above procedures. The examples show that the method remains numerically stable, even when the method is inverted.     

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.     

Three-dimensional stress analyses of mitred elbows

Three-dimensional elastic stress analyses of a mitred elbow have been performed by the application of the MARC finite element program. Seven basic unit load cases have been considered. Results of these reference load cases may be applied in any combination to stress analyses of the same elbow subjected to more complicated loading conditions.

According to results from these analyses, when a mitred bend is subjected to in-plane loads, it exhibits behaviour similar to that of a smooth elbow under the same loading conditions. Results from out-of-plane load cases show some deviation from the behaviour of a smooth bend. These discrepancies cast some doubts on the conventional method of stress analysis of a mitred bend.     

A comparative study of usefulness for pad reinforcement in cylindrical vessels under external load on nozzle

The main purpose of this paper is to perform a comparative study of strength behavior for cylindrical shell intersections with and without pad reinforcement under out-of-plane moment loading on nozzle. Three pairs of full-scale test vessels with different d/D ratios were designed and fabricated for testing and analysis. A three-dimensional nonlinear finite element numerical analysis was also performed. The maximum elastic stress for each vessel under per unit moment on nozzle is provided. The plastic limit moment on nozzle is obtained by load–displacement and load–strain curves for each test vessel. The results indicate that the effect of pad reinforcement on decreasing maximum elastic stress and increasing plastic limit load is obviously effective. The study results will serve as the available data for understanding the usefulness of pad reinforcements and as the basis for developing an advanced design method by limit analysis for pad-reinforced cylindrical vessels under external loads on nozzle.     

Plastic limit loads of nozzles in cylindrical vessels under out-of-plane moment loading

The purpose of this paper is to study the plastic limit moment of nozzles in cylindrical vessels with different d/D ratios under out-of-plane moment loading. Three full size test models were designed and fabricated. A 3D nonlinear finite element numerical simulation was also performed. A twice-elastic-slope plastic moment on the nozzles was obtained approximately by use of load–displacement and load–strain curves. The results show that plastic loads determined by test and numerical simulation methods are in good agreement. The results can serve as a basis for developing an advanced design guideline by limit analysis for cylindrical vessels with a nozzle under external loads.     

Creep crack growth in welded components — a numerical study and comparison with the R5 procedures

In the present study, creep crack growth (CCG) in a circumferentially welded low alloyed pipe is numerically investigated for a number of different combinations of weldment constituent material properties. A creep ductility based damage model describes the accumulation of creep damage ahead of the crack tip where a constraint parameter and the creep strain rate perpendicular to the crack plane are used as characterising parameters. It is assumed that a fully circumferential creep crack, located in the heat affected zone with a depth of one quarter of the pipe thickness, is growing at a constant rate from the outer surface towards the inside. The numerical results reveal that not only the properties of the zone containing the crack, but also the deformation properties of the surrounding material influence the CCG behaviour. This influence can be noted on the characterising parameters used for the CCG rate predictions as well as on the CCG rate itself. The mismatch influence on corresponding C^* values is, however, marginal. This indicates that determination of the CCG rate in weldments, based on the C^* value only, may result in uncertain estimates.The numerically investigated cases are also assessed by use of the R5 procedures for the sake of comparison. Considering the stress re-distribution, due to the mismatch effect, the CCG rate is determined for the different weldment configurations. The comparison shows that the assumption of plane strain or plane stress conditions in the R5 analysis is essential for the agreement of the results between R5 and the two-parameter approach. Assuming plane stress conditions at the crack tip results in a relatively good agreement for the axial stress dominated cases investigated. However, for the hoop stress dominated cases, the R5 procedures predict higher CCG rates by an order of magnitude.     

冬季低温条件下原水输水管道应力分析

为研究近期在冬季低温时上海原水输水管出现开裂的原因,运用弹性力学法计算得到原水输水管道内部应力理论解;采用有限元数值分析,得到管道在冬季低温条件下的内部应力分布规律;通过现场检测,得到上海某原水输水管道的应力实测值。对比分析理论计算、数值模拟和现场检测结果可知,冬季温度降低导致的管道应力变化是管道破裂的因素之一,但并非主要原因,管道长期受荷载作用导致的管道老化才是管道产生裂痕和破裂的主要原因。研究成果对于原水输水管道的管理与维护有一定参考价值。     

A simplified ratcheting limit method based on limit analysis using modified yield surface

Two effective approaches for obtaining ratchet boundaries of a structure undergoing cyclic loads are presented. The approaches use limit analysis of a structure whose yield surface is modified according to the cyclic load. In the first approach, Uniform Modified Yield (UMY) surface is used. UMY approach reduces the Mises-based cylindrical yield surface by Mises stress of the cyclic stress amplitude. UMY method was slightly conservative, and sometimes overly conservative, especially at high ratio of cyclic load to primary steady load. Conservatism, caused by the assumption that the modified yield surface remains isotropic, is eliminated by considering anisotropic Load Dependent Yield Modification approach, LDYM. This approach reduces yield strength based on relative orientation of steady primary and cyclic stress tensors. This work assumed elastic perfect plastic material behavior, with no strain hardening for both original and modified yield surfaces. Ratchet boundaries of several structures, published in literature, were obtained using UMY and LDYM approaches and verified against published data and results of conventional methods. Numerical procedures for UMY and LDYM approaches are extremely fast relative to conventional numerical schemes, and are not restricted by complex geometry or loading.     

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.     

Prediction of creep failure life of internally pressurised thick walled CrMoV pipes

Creep continuum damage finite element analyses were performed for a typical internally pressurised main steam thick-sectioned plain pipe geometry subjected to a range of axial loading conditions. The creep failure lives, for a range of loading levels, were obtained. The stress distributions showed the presence of a skeletal point or representative rupture stress. Life estimates were made using alternative stress forms and these were compared with the failure lives predicted using continuum damage analyses. For the particular ferritic material properties used in the analyses, the failure life of the pipe can be obtained accurately using the steady-state reference rupture stress or the skeletal point rupture stress.