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.

RSDM: A Powerful Direct Method to Predict the Asymptotic Cyclic Behavior of Elastoplastic Structures

Mechanical engineering structures and structural components are often subjected to cyclic thermomechanical loading which stresses their material beyond its elastic limits well inside the inelastic regime. Depending on the level of loading inelastic strains may lead either to failure, due to low cycle fatigue or ratcheting, or to safety, through elastic shakedown. Thus, it is important to estimate the asymptotic stress state of such structures. This state may be determined by cumbersome incremental time-stepping calculations. Direct methods, alternatively, have big computational advantages as they focus on the characteristics of these states and try to establish them, in a direct way, right from the beginning of the calculations. Among the very few such general-purpose direct methods, a powerful direct method which has been called RSDM has appeared in the literature. The method may directly predict any asymptotic state when the exact time history of the loading is known. The advantage of the method is due to the fact that it addresses the physics of the asymptotic cycle and exploits the cyclic nature of its expected residual stress distribution. Based on RSDM a method for the shakedown analysis of structures, called RSDM-S has also been developed. Despite most direct methods for shakedown, RSDM-S does not need an optimization algorithm for its implementation. Both RSDM and RSDM-S may be implemented in any Finite Element Code. A thorough review of both these methods, together with examples of implementation are presented herein.

The role of surface topography and normal load in the initiation of ratchetting-peak friction, seizure, scuffing, and elastic shakedown

Surface topography is a critical parameter that can influence friction and wear in engineering applications. In this work, the influence of surface topography directionality on seizure and scuffing initiation during tribological interactions is explored. For this, unidirectional sliding wear experiments were carried out in immersed lubrication conditions for various normal loads. The tribological interactions were studied using EN31-60 HRC flats and SAE52100-60HRC pins in a sphere on flat configuration. The results show that, in some cases, the sliding interactions in the initial cycles lead to a high friction coefficient of up to ∼0.68 in lubricated conditions, which was termed as ‘peak friction’, and this was accompanied by scuffing. The existence of peak friction was found to be dependent on surface topography directionality, especially when the directionality in topography was parallel to the sliding direction. Continuous ratchetting was found to be the cause of peak friction which was accompanied by seizure and scuffing. When the topography directionality was perpendicular or independent of sliding direction, elastic shakedown occurred at earlier cycles and prevented peak friction initiation, scuffing and also facilitated for higher steady-state friction values.

Fatigue Damage of an Asperity in Frictionless Normal Contact with a Rigid Flat

Surface fatigue wear widely exists, and it occurs as long as a sufficient number of loading–unloading cycles are applied. Slowing down surface fatigue wear requires understanding the evolution of fatigue damage in the surface. Real surfaces are composed of many asperities; therefore, it is important to study the fatigue damage of a single asperity. A finite element model of an asperity subjected to cyclic elastic–plastic normal loading was developed under frictionless contact condition. The asperity can be either completely or partially unloaded in a loading cycle. For the sake of completeness, both cases were investigated in the present study. The multiaxial Fatemi-Socie fatigue criterion was adopted to evaluate the fatigue damage of the asperity in elastic shakedown state, which was achieved after several loading cycles. For the case of complete unloading, severe fatigue damage was confined in a subsurface ridge starting from the edge of the maximum loaded contact area. The shape and volume of the wear particles were predicted based on a fundamentally valid assumption. For the case of partial unloading, the fatigue damage was much milder. Finally, potential research directions to expand the current study are suggested.

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.