Shakedown and Limit Analysis of Kinematic Hardening Piping Elbows Under Internal Pressure and Bending Moments

In this paper, the Stress Compensation Method (SCM) adopting an elastic-perfectly-plastic (EPP) material is further extended to account for limited kinematic hardening (KH) material model based on the extended Melan's static shakedown theorem using a two-surface model defined by two hardening parameters, namely the initial yield strength and the ultimate yield strength. Numerical analysis of a cylindrical pipe is performed to validate the outcomes of the extended SCM. The results agree well with ones from literature. Then the extended SCM is applied to the shakedown and limit analysis of KH piping elbows subjected to internal pressure and cyclic bending moments. Various loading combinations are investigated to generate the shakedown limit and the plastic limit load interaction curves. The effects of material hardening, elbow angle and loading conditions on the shakedown limit and the plastic limit load interaction curves are presented and analysed. The present method is incorporated in the commercial finite element simulation software and can be considered as a general computational tool for shakedown analysis of KH engineering structures. The obtained results provide a useful information for the structural design and integrity assessment of practical piping elbows.

Optimal Plastic Analysis and Design of Pile Foundations Under Reliable Conditions

In this research, in order to evaluate the plastic limit load and also plastic design parameters of the long pile foundations subjected to horizontal loads, shakedown method is applied. In carrying out shakedown analysis and design methods, large plastic deformations and residual displacements could develop in the pile foundation which might lead to the failure of the structure. For this reason, complementary strain energy of residual forces proposed as a limit condition to control the plastic deformation of the pile structure. Furthermore, considering the uncertainties (strength, manufacturing, geometry) the limit conditions on the complementary strain energy of residual forces are assumed randomly and the reliability condition was formed by the use of the strict reliability index. The influence of the limit conditions on the plastic limit load and design parameters of the long pile in cohesionless soil subjected to lateral load were investigated and limit curves for shakedown load factors are presented. The numerical results show that the probabilistic given limit conditions on the complementary strain energy of residual forces have significant influence on the load bearing limit and the design parameters of pile foundations. The formulations of the reliability based problems lead to mathematical programming which were carried out by the use of non-linear algorithm.

Plastic Limit Analysis Using the Simplified Theory of Plastic Zones

The simplified theory of plastic zones (STPZ) was mainly developed to determine strain ranges and accumulated strains in the state of shakedown at cyclic loading between prescribed levels of loading. Kinematic hardening is an indispensable feature of the STPZ. The plastic limit load, however, is defined for monotonic loading and elastic–plastic material behavior without hardening. Simply assigning a zero value or a numerically very low value of the tangent modulus when applying the STPZ is generally not possible due to arising numerical instabilities. It is, therefore, not immediately obvious how the STPZ can be used to determine the maximum load level that can be applied to a structure without developing a kinematic mechanism. This paper describes the theory and the analysis steps required and provides some illustrative examples. Typically, between one and three linear elastic analyses and some local calculations are required to provide either the exact value or at least a reasonable estimate of a range of the plastic limit load, as well as of the associated stress and strain fields and displacements that are not provided by classical limit analysis.

Determination of Transition From Thin Shell to Thick Shell Theory for Torispherical Heads With Head Dish Radius to Thickness Ratios Under 20

This paper will clarify the point of transition where the behavior of the dish of a torispherical head goes from thin wall theory (collapse failure and membrane) to thick wall (burst failure) as the head dish radius to thickness ratios (L/t) gets smaller. There are several stated ratio limits for this transition. Three separate Welding Research Bulletins WRC 364 New Design Curves for Torispherical Heads[1], WRC 444 Buckling Criteria for Torispherical Heads Under Internal Pressure [3] and, WRC 501 Design of Torispherical and Ellipsoidal Heads Subjected to Internal Pressure[4] each provide a different definition of the transition point, that being 16.67, 15 and 20 respectively. This paper will review the actual test performed for L/t ratios from 20 down to 15 (which is the lowest ratio test run) and provide the results of a numerical desktop study in lieu of actual testing. Linear elastic, elastic perfectly plastic limit load and elastic plastic limit load finite element analysis will be parametrically run across many L/t ratios and the knuckle radius will be varied across the runs. The results will be reviewed to check through wall behavior to find the transition point of thin to thick wall behavior. These will also be compared against the existing ASME BVP Section VIII Division 2 [5] formulas.

Shakedown and Limit Analysis of Kinematic Hardening Piping Elbows Under Inner Pressure and Bending Moments

The stress compensation method (SCM) for shakedown and limit analysis was previously proposed and applied to elastic-perfectly plastic (EPP) piping elbows. In this paper, the SCM is extended to account for limited kinematic hardening (KH) material model based on the extended Melan’s static shakedown theorem using a two-surface model defined by two hardening parameters: initial yield strength and ultimate yield strength. To validate the extended SCM, a numerical test on a cylinder pipe is performed. The results agree well with ones from literature. Then the extended SCM is applied to the shakedown and limit analysis of KH piping elbows subjected to inner pressure and cyclic bending moments. Various loading combinations are investigated to create the shakedown limit and plastic limit load interaction curves. The effects of the material hardening, angle of the elbow and loading conditions on the shakedown limit and plastic limit load interaction curves are presented and analysed. The present method is incorporated in the commercial software of Abaqus and can be considered as a general computational tool for shakedown analysis of KH engineering structures. The obtained results provide a useful information for the structural design and integrity assessment of practical piping elbows.

Shakedown and Limit Analysis of 45-Degree Piping Elbows Under Internal Pressure and Cyclic In-Plane Bending

This paper carries out the shakedown and limit analysis of 45-degree piping elbows subjected to steady internal pressure and cyclic in-plane closing, opening and reverse bending moments by means of the recently proposed stress compensation method (SCM). Different geometries of the piping elbows and various combinations of these applied loads are investigated to create various shakedown limit and plastic limit load interaction curves. The plastic limit loads for single internal pressure and single bending moment calculated with the SCM are compared to those calculated with the twice-elastic-slope method. Full step-by-step elastic-plastic incremental finite element analyses are utilized to verify the structural cyclic responses on both sides of the curves obtained and further to confirm the correct shakedown limit loads and boundaries. It is shown that the SCM calculates the shakedown limit load accurately and possess more than 40 times the computational efficiency of the step-by-step elastic-plastic incremental method. The effects of the ratios of bending radius to mean radius and mean radius to wall thickness of the piping elbow as well as loading conditions on shakedown limit and plastic limit load interaction curves are presented. The results presented in this work provide a comprehensive understanding of long term response behaviors of the piping elbow under the combined cyclic loading and offer some essential points to be concerned for the design and integrity assessment of piping systems.

Simple Calculations of J-Integral for Through-Wall Crack in Welded Pipes Based on Failure Assessment Diagram

In the present study, simple J-integral estimations of welded pipes with a circumferential through-wall crack (TWC) in weld zone were proposed based on the failure assessment diagram (FAD) concept using scaling factor that is defined as a ratio of plastic limit load of welded pipe to that of base metal pipe. For this purpose, the detailed 3-dimensional (3-D) finite element (FE) limit analyses for welded pipes with a circumferential TWC have been systematically carried out considering various thickness of pipe, circumferential crack lengths, strength mismatches between base and weld metal, widths of weldment and locations of TWC (weld center and interface between the weld metal and the base metal). As for loading conditions, axial tension and global bending were considered in the FE analyses. Based on the present FE results, numerical expressions on scaling factors representing the ratio of plastic limit loads of welded pipes to those of base metal pipes were derived. Finally, the new FAD-based J-estimations based on option 1 concept was proposed to predict J-integrals of welded pipes incorporating the present scaling factors, i.e. ratio of plastic limit load. Moreover, the proposed J estimations were validated by comparing with FE results using actual properties.