Elastic shakedown limit of a steel lattice girder

This paper presents a solution for the problem concerning the behaviour of a steel lattice girder subjected to dynamic load pulses. The theory of shakedown is used in the analysis. It is assumed that such loads cause a non-elastic response which includes dissipation of energy causing deformations and residual forces developed in the structural members of the girder. At a certain intensity of these forces, the girder can react to subsequent load pulses without further dissipation of energy, behaving in the elastic region after shakedown. This condition is referred to as adaptation of the structure to assumed cyclic loading. Elastic shakedown limit is determined through a direct analysis of the girder's dynamic behaviour, i.e. by checking if energy dissipation decreases with loading cycles. This gives the number of load applications after which no further increase of the energy dissipation is observed. The existing permanent deformations persist and residual forces remain in the same state. The analysis takes into account the possibility that compressed members can buckle which may result in non-elastic, longitudinal and transverse vibrations of these members. Non-linear geometry of members is taken into account. Then a perfectly elastic-viscoplastic model of the material is used. The main goal is to determine the state of the non-elastic movements of the girder joints and the residual internal forces developed in the girder members after each load application. The values obtained in this way serve as the basis for describing the next loading cycle. It is possible to use the approach presented in the paper to evaluate the effects of accidental loads. Then it is checked whether a small number of repetitions of accidental load would result in exceeding the serviceability limit state criteria of the maximum permanent deformation or displacement and/or strain amplitudes. If so, the magnitude of accidental load is greater than the elastic shakedown limit. Some examples are given to illustrate the application of the theory of shakedown.

Generation of Shakedown Boundaries of Thin-Walled 90-Degree Scheduled Pipe Bends Determined via Large and Small Displacement Finite Element Models

The present research investigates the effect of employing large displacement in finite element modelling on the generated shakedown (SD) boundaries of thin-walled 90-degree scheduled pipe bends. A recently developed methodology termed: Shakedown Limit - Plastic Work Dissipation (SDLimit-PWD) method generates the SD boundaries via employing the large displacement in the FE simulations. Additionally, a well-established direct non-cyclic technique termed: Shakedown Direct Noncyclic Technique (SD_DNT) generates the SD boundaries via employing the small displacement formulation in the FE simulations. Comparing the SD boundaries generated via both methods illustrated marked increase in the generated SD domains due to employing large displacement.

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.

Effect of Moisture Content on the Shakedown Limits of Base Course Materials

This paper summarizes the results of a laboratory testing program that was conducted to determine the effects of moisture content on the shakedown limits of unbound granular base materials. Two different types of granular base materials were investigated in this study, namely limestone and sandstone. Multi-stage repeated load triaxial tests were performed on these materials. The results of the tests were analyzed within the framework of the shakedown theory. The results indicate that the moisture content had an influence on the slope of the elastic and plastic shakedown limits lines. The effect of the moisture content was more pronounced on the slope of the elastic shakedown limit line, however. The moisture content affected the intercept of the elastic and plastic shakedown limits lines more significantly than the slope of these lines. The limestone material exhibited greater decrease in the intercept of the elastic and plastic shakedown limits with increase in moisture content compared with the sandstone material. This was explained by the limestone’s finer gradation.

Investigating Structural Response of Pressure Reducing Valve of Supercritical Steam Generator System Under Cyclic Moments, Thermal Transient, and Pressure Loadings

Pressure reducing valve (PRV) located before the start-up vessel (SUV) is an essential component that decreases the pressure and temperature of the supercritical state steam by using spray water before it flows into the SUV. The PRV is kept closed during normal operation but opened during start-up and shutdown events, which could initiate thermal fatigue defects due to significant temperature changes. In addition to the thermal shock and internal pressure, system bending and torsional moments may be imparted on the PRV, threatening its integrity. To reinforce these concerns, cracks on the inside surfaces of the PRV have often been reported during planned maintenance activities in nuclear power plants. This research aims at analysing cyclic plasticity of the PRV subjected to cyclic moments, thermal and pressure loadings by means of an advanced direct numerical technique known as the Linear Matching Method (LMM). The cyclic moments are comprised of in-plane and out-of-plane bending, and torsion which are applied to an inlet branch pipe of the PRV. The cyclic thermal load is obtained from the transient heat transfer analysis using real operational data. Two different pressures, which are high and low pressures, are applied to internal surfaces of the PRV body and outlet pipe respectively. The analysed results construct a structural response boundary such as a shakedown limit boundary. The obtained structural response boundary is validated by full cyclic incremental analysis referred to as the step-by-step analysis. The analysed results have demonstrated that the plastic collapse limit is identical to the shakedown limit. Moreover, the results provide engineers with a safe load bearing capacity domain which otherwise requires evaluating structural integrity of the PRV subjected to the complicated cyclic loading condition using detailed assessments and analyses.

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.