Non-cyclic shakedown/ratcheting boundary determination – Part 1: Analytical approach

From the practical point of view, the classical elastic shakedown methods are not very useful for design, since in most components the stresses can safely exceed the elastic limit locally. This paper generalizes the static shakedown theorem (Melan’s theorem) to allow the analysis of plastic shakedown. Since the method is derived from a lower bound formulation in shakedown, it is very useful for the design purposes (safe). The ratchet boundary is analytically determined using the proposed method for several examples with uniform stress distributions. The numerical implementation of the method along with several examples is discussed in an accompanying paper.     

Shakedown analysis of thick-walled cylinders subjected to internal pressure with the unified strength criterion

Most previous studies on shakedown of thick-walled cylinders were based on the assumption that the compressive and tensile strengths of the materials were identical. In this paper the shakedown of an internally pressurized cylinder made of a material with a strength-difference and intermediate principal stress effects is dealt with by using a unified strength criterion which consists of a family of convex piecewise linear strength criteria. Through an elasto-plastic analysis the solutions for the loading stresses, residual stresses, elastic limit, plastic limit and shakedown limit of the cylinder are derived. It is shown that the present solutions include the classical plasticity solutions as special cases and have the ability to account for the strength-difference and intermediate principal stress effects. Finally, the influence of the two effects on the shakedown limit of the cylinder is investigated. The results show that the shakedown limit depends on the two effects and is underestimated if these effects are neglected as in the classical plasticity solution based on the Tresca criterion.     

Lower-bound shakedown analysis of spherical shells containing defects

In this paper, a lower-bound shakedown analysis of spherical shells containing defects is given based on the static shakedown theorem. The pseudo-residual stress field is simulated by the temperature parameter method. In order to get over the difficulty of large computative quantities, the yield condition is linearized, so that shakedown analysis is transformed into a linear programming problem. The shakedown and the limit loads of spherical shells containing part-through slots and gas holes are computed by this method, and the relationships between the carrying capacities of vessels and the geometric parameters of shells and defects are obtained. The effects of pits and gas holes on carrying capacities of vessels are then compard with each other.     

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.     

Shakedown limit loads for elbows under internal pressure and cyclic in-plane bending

This paper presents elastic, shakedown and plastic limit loads for 90° elbows under constant internal pressure and cyclic in-plane bending, via finite element (FE) analysis. Effects of the elbow geometry (the bend radius to mean radius ratio and the mean radius-to-thickness ratio) and of the large geometry change are systematically investigated. By normalizing the in-plane bending moment by the plastic limit load solution of Calladine, the shakedown diagram is found to be close to unity up to a certain value of normalized pressure (normalized with respect to the limit pressure) and then to decrease almost linearly with increasing normalized pressure. The value up to which shakedown limit loads remain constant depends on the elbow geometry and the large geometry change effect. Effects of the elbow geometry and the large geometry change on shakedown diagrams are discussed.     

Shakedown and limit analysis of 90° pipe bends under internal pressure,cyclic in-plane bending and cyclic thermal loading

The Linear Matching Method is used to create the shakedown limit and limit load interaction curves of 90° pipe bends for a range of bend factors. Two load cases are considered i) internal pressure and in-plane bending (which includes opening, closing and reversed bending) and ii) internal pressure and a cyclic through wall temperature difference giving rise to thermal stresses. The effects of the ratios of bend radius to pipe mean radius (R/r) and mean radius to wall thickness (r/t) on the limit load and shakedown behaviour are presented.     

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.     

Limit and shakedown analysis of nozzle/cylinder intersections under internal pressure and in-plane moment loading

A simple technique called the elastic compensation method developed previously by Mackenzie & Boyle is used in combination with full three-dimensional finite-element analysis to obtain limit and shakedown interaction diagrams for nozzle/cylinder intersections subject to combined internal pressure and in-plane nozzle moment loading. The results are compared with solutions from the literature and also with detailed elasto-plastic thin shell finite-element analysis. It is found that the simple elastic compensation procedure can provide good estimates of plastic failure mechanisms for complex three-dimensional structures. A detailed discussion of various issues which arose relates to finite-element modelling and the measures taken to improve the results are also documented.     

Direct finite element route for design-by-analysis of pressure components

In the new European standard for unfired pressure vessels, EN 13445-3, there are two approaches for carrying out a Design-by-Analysis that cover both the stress categorization method (Annex C) and the direct route method (Annex B) for a check against global plastic deformation and against progressive plastic deformation. This paper presents the direct route in the language of limit and shakedown analysis. This approach leads to an optimization problem. Its solution with Finite Element Analysis is demonstrated for mechanical and thermal actions. One observation from the examples is that the so-called 3f (3Sm) criterion fails to be a reliable check against progressive plastic deformation. Precise conditions are given, which greatly restrict the applicability of the 3f criterion.     

Integrity assessment of a 3D tubeplate using the linear matching method. Part 2: Creep relaxation and reverse plasticity

In an accompanying paper [Int J Pres Ves Pip, 2004], a 3D tubeplate in a typical AGR superheater header was analysed for the shakedown limit, reverse plasticity and ratchet limit based on the application of the linear matching method (LMM). Both the perfectly plastic model and the cyclic hardening model were adopted for the evaluation of the plastic strain range. In the present paper, based on the creep-reverse plasticity model, the accumulated creep strain during a creep dwell at the steady cyclic state is assessed by the LMM. The accumulated creep strains, the creep flow stresses as well as the elastic follow-up factors with differing dwell times are evaluated in detail. The total inelastic strain over the cycle at the steady cyclic state is calculated. By comparing these results with ABAQUS step-by-step inelastic analyses, the applicability of the methods is demonstrated.     

Integrity assessment of a 3D tubeplate using the linear matching method. Part 1. Shakedown,reverse plasticity and ratchetting

The Linear Matching Method (LMM) has been developed recently for the integrity assessment for the high temperature response of structures. In this article, a complex 3D tubeplate in a typical AGR superheater header is analysed for the shakedown limit, reverse plasticity and ratchet limit based upon the LMM. At this stage, both the perfectly plastic model and the cyclic hardening model are adopted for the evaluation of the plastic strain range. Comparisons of LMM results with other results by ABAQUS step-by-step inelastic analyses for several material models are given. Further cyclic creep-reverse plasticity analyses are presented in an accompanying paper [Int. J. Pressure Vessels Piping (2004)].     

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.     

Integrity assessment for a tubeplate using the linear matching method

A series of numerical procedures have been presented recently for the integrity assessment of structures based upon the Linear Matching Method. A typical example of a holed plate has been used to verify these procedures for the evaluation of plastic and creep behaviours of components. In this paper, a more complex 2D tubeplate at the outlet from a typical AGR heat exchanger is analysed for the shakedown limit, reverse plasticity, ratchet limit and creep relaxation based on the application of the Linear Matching Method for a thorough case study. Both a constant material yield stress and a temperature-dependent yield stress are adopted for the evaluation of the ratchet limit. For the evaluation of accumulated creep strains, flow stresses and elastic follow-up factors with differing dwell times at the steady cyclic state, a monotonic creep computation is performed, where the start-of-dwell stress is the rapid cycle creep solution at the beginning of the dwell period. An estimation of the tubeplate lifetime is then obtained by the evaluation of fatigue and creep endurances.     

On the shakedown analysis of welded pipes

This paper presents the shakedown analysis of welded pipes subjected to a constant internal pressure and a varying thermal load. The Linear Matching Method (LMM) is applied to investigate the upper and lower bound shakedown limits of the pipes. Individual effects of i) geometry of weld metal, ii) ratio of inner radius to wall thickness and iii) all material properties of Weld Metal (WM), Heat Affected Zone (HAZ) and Parent Material (PM) on shakedown limits are investigated. The ranges of these variables are chosen to cover the majority of common pipe configurations. Corresponding individual influence functions on the shakedown limits are generated. These are then combined to allow the creation of a safety shakedown envelope, which can be used for the design of any welded pipes within the specified ranges. The effect of temperature-dependent yield stress (in PM, HAZ and WM) on these shakedown limits is also investigated.     

Shakedown of complex structures according to various hardening rules

This article is concerned with a review of the literature on shakedown problems of complex structures according to various hardening rules. Although many hardening rules have been developed, shakedown analyses for hardening materials in the literature are limited to the simple kinematic hardening rules of Prager and Armstrong–Frederick. In other words, much more advanced hardening rules may not be vital to decide whether a structure will shakedown or not. Results of shakedown analyses for some complex structures are included.     

Analysis of cyclic creep and rupture. Part 2: calculation of cyclic reference stresses and ratcheting interaction diagrams

Part 1 gives the basis for the use of cyclic reference stresses for high temperature design and assessment. The methodology relies on elastic–plastic calculations for limit loads, ratcheting and shakedown. In this paper we use a commercial non-linear finite element code for these calculations. Two fairly complex and realistic geometries with cyclic loads are analysed, namely a pipe elbow and a traveling thermal shock in a pressurized pipe. The special case of start-up shut-down cycles is also discussed. Creep and rupture predictions may be made from the results. When reference stresses can be economically calculated, their use for high temperature design has the following advantages.

• •Accuracy. Limit loads, shakedown and ratcheting limits are based on detailed analysis, and do not rely on rules or judgement.
• •Efficiency. Use of shakedown and ratcheting reference stresses to predict rupture and creep strain, respectively, allowing details of time and temperature to be dealt with as material data, not affecting the analysis.
• •Factors of safety. For both low and high temperature problems, factors of safety can be determined or applied, based on the real failure boundaries.
• •Conservatism. The rupture and strain calculations reflect the limit of rapid cycle behaviour. Cycles with relaxation will be associated with longer lives.

     

基于CAD/CFD发动机气道的设计

以应用于实际设计为目的,研究了三维造型设计方法在发动机螺旋进气道设计中的应用。实现了PROE设计以及高精度的流动数值模拟,为气道的设计与改进提供了高效实用的研究手段。讨论了螺旋气道三维建模的基本过程,研究了网格尺寸对计算结果收敛性的影响。最后,以气道试验验证计算结果的准确性。三维设计表明,现代设计方法的应用有益于克服传统设计中的盲目性与局限性,提高产品的自主开发能力与设计制造质量。     

Elasto-plastic analysis as a basis for design of cylindrical pressure vessels with different end closures

The elasto-plastic investigations described in this paper have been carried out in order to study the performance of unfired cylindrical steel pressure vessels with hemispherical, torispherical, semi-ellipsoidal and toriconical end closures concave to pressure, using the finite element method. The various aspects covered by the study include spread of plastic zones, load-deflection characteristics, head-height growth characteristics, redistribution of effective stresses, effective plastic strain, limit analysis, excessive deformation and shakedown criteria of pressure vessel design. Based upon the results of this analysis, guidelines for the design of cylindrical pressure vessels are indicated in the form of various non-dimensional curves which will be useful for the better design of pressure vessels.     

On the shakedown analysis of nozzles using elasto-plastic FEA

The performed shakedown calculations of a dished end with a nozzle in the knuckle region with varying internal pressure load, and two cylinder–cylinder intersections with constant moment load and varying internal pressure load are example cases for the application of the check against progressive plastic deformation as stated in the new European UFPV standard, Annex 5.B: “Direct route for design by analysis” (DBA). To calculate the shakedown limits, Melan's (lower bound) shakedown theorem is used. In this context, the usage of the deviatoric maps of stress states to obtain proper self-equilibrating stress fields is shown. Furthermore, some problems and corresponding possible solutions for performing the shakedown check using a finite element model with shell elements are stated and shown in the examples.     

基于变分原理的三维土坡稳定分析方法研究及应用

采用变分方法研究了三维非均质土坡的稳定性．并编制了二、三维土坡稳定计算程序。将该程序应用于糯扎渡大坝的稳定计算,得到了各种工况下的安全系数及对应的最危险滑裂面．与传统的极限平衡法计算结果相比较得到了较为理想的解。说明基于变分原理的三维土坡稳定分析方法在实践应用中的合理性。