Shakedown Theorem Research Articles

Shakedown analysis of cavities in cohesive-frictional materials and its application to underground energy storage caverns

In this paper, a shakedown analysis is conducted for both cylindrical and spherical cavities in cohesive-frictional materials subjected to cyclic internal pressures. Based on Melan’s lower-bound shakedown theorem, theoretical shakedown conditions for a two-layered cavity system are developed using the stress and strain solutions obtained from cavity expansion and contraction analyses. The shakedown limits are investigated in relation to the dimensions of the two-layered cavity system and the elastic/plastic material parameters. Numerical validation is conducted through a comparison of the stress path of the critical element and the failure lines of the materials. In addition, closed-form solutions for special cases, namely, the analytical shakedown conditions of cavities in an unbound cohesive-frictional material and in a bound material with different boundary conditions, are explicitly expressed. Finally, the shakedown solutions are applied to solve the thickness design problem of underground compressed air energy storage in lined and unlined caverns.

Effect of plastic volumetric strains on shakedown of hardening granular soils

The granular soils are pressure dependent materials and substantial volume change occurs during loading. Some granular soils contract (volume decrease) when sheared and some dilate (volume increase), depending on their initial state. Owing to the plastic volume change that accompanied plastic deformation, the lack of normality rule arises and the granular soil exhibits a non-associated flow rule. As known the non normality has destabilizing effect on the soil behavior in the hardening regime (loss of positive definiteness of the second order work). The unstable behavior usually develops in association with dilation. A contracting soil displaying hardening and stable material behavior under drained conditions, may succumb to unstable flow type when its behavior becomes dilating. This paper deals with the extension of the static shakedown theorem to dilative granular soils in drained conditions, exhibiting pre-failure instability in the hardening regime, in the framework of non associated plasticity.

A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures

This paper presents a quasi-static formulation, based on integrated radial basis functions (iRBF) and conic programming, for direct analysis of 2D and 3D structures. The iRBF shape functions are stabilized using stabilized conforming nodal integration (SCNI) technique. Equilibrium equations, boundary conditions and yield criterion associated with the residual stresses are directly enforced at scattered nodes without any special treatment. The safety load multipliers are determined by solving the conic optimization problems resulting from static shakedown theorem. The computation cost in terms of the number of variables and optimization CPU time is kept to a minimum. Various benchmark problems in two and three dimensions are investigated, showing the efficiency of the presented numerical approach.

Dynamic shakedown analysis of two-layered pavement under rolling-sliding contact

Shakedown theory provides a rational tool to predict the long-term plastic behavior of pavement subjected to repeated traffic loads. Lower-bound shakedown solutions have been proposed in recent years to estimate the shakedown limit load under which the failure of the pavement due to excessive permanent deformation will not occur. However, for such a problem when the tire-pavement interaction is characterized by rolling-sliding contact, the dynamic responses of pavement structures were not thoroughly explored. In this paper, a numerical method for computing the shakedown limit of pavement is presented based on the lower-bound shakedown theorem. A moving Hertzian contact with sliding friction is constructed in a finite element model with infinite element boundary to calculate the dynamic elastic stress field due to rolling and sliding contact between the pavement and vehicles with different moving speeds. Shakedown analysis for both uniform subgrade and two-layered pavement has been investigated. It is found that the shakedown limit decreases with increasing vehicle moving speed. The dynamic effects of moving speed become less significant as the contact friction coefficient increases. Both the speed and the friction coefficient can affect the location of critical failure point, which tends to transit from the top layer to the bottom layer with increasing speed or decreasing friction coefficient. Eventually, the influences of frictional coefficient and material properties on shakedown limit are also investigated and the characteristics of the distribution of critical residual stress is also discussed.

A direct approach to the evaluation of structural shakedown limit considering limited kinematic hardening and non-isothermal effect

This paper presents a novel direct method for the structural shakedown analysis considering limited kinematic hardening and non-isothermal effect. The Melan's static shakedown theorem is extended to consider limited kinematic hardening material and implemented into the Linear Matching Method (LMM) shakedown module. Instead of using a specific kinematic hardening rule and an explicit back stress field, the general nonlinear hardening laws are considered by using a two-surface hardening model. A two-stage procedure is developed in the extended LMM algorithm, which can generate the limited hardening shakedown envelope and the unlimited hardening curve efficiently and accurately. Also, the material non-isothermal effect is considered during the computation process of the shakedown limit by proposing a temperature-dependent hardening factor, in place of a constant and fictitious one. To validate the extended LMM method, a numerical test on a thin cylinder pipe with temperature-independent material properties is performed, and the results match well with ones from literature. Then, a numerical study on a typical aero-engine turbine disk is conducted to investigate the influence of temperature-dependent material properties and operating conditions. Several shakedown curves considering kinematic hardening effect are derived and adequately discussed. As a result, the extended LMM shakedown module is proven to be a robust, efficient and versatile tool for practical industrial problems.

Static shakedown theorem accounting for elastic stiffness degradation during plastic yielding

The soil is modeled as coupled elastoplastic to account for the elastic stiffness degradation during plastic yielding. In such model the irreversible strain rate results from a combination of a plastic part and a coupled part of strain rates. The former arises from plastic yielding and the latter is generated by the variation in the elastic stiffness. This leads to non associated and coupled flow rules in which the outward direction of the irreversible strain rate do not coincides neither with the outward normal to the yield surface nor the outward normal to the plastic potential. In this paper an extension of the static shakedown theorem to soil is proposed in the framework of the coupled elastoplasticity.

Application of shakedown theory in track substructure design

Tracks require high performance of earth structures against excessive residual settlement under repeated train loads. In the current design standards, the quality of soil materials and compaction control are strictly prescribed based on empirical knowledge and scale tests; however, construction costs become relatively high accordingly. A more rational design method based on shakedown theory is presented here. This method makes use of the lower-bound shakedown theorem and models the soils as Mohr–Coulomb materials. By calculating the analytical load-induced dynamic elastic stress field in a three-dimensional half-space and introducing a self-equilibrated residual stress field, the maximum magnitude of surface pressure on the soil foundation (i.e. shakedown limit) against long-term residual settlement is calculated through an optimisation programme. It is found the shakedown limit is dependent on the ratio of train velocity to shear wave velocity for a given earth structure. The maximum admissible train loads for the cases with different design velocities can be predicted by modifying the quasi-static shakedown limit with an attenuation factor.

Shakedown analysis of a torispherical head with a piping nozzle under combined loads by the stress compensation method

Shakedown assessment is an important task in determining the load-bearing capacity of structures and evaluating their safety. The traditional shakedown analyses, which are based on the upper or lower bound shakedown theorem to establish the mathematical programming formulation and solve an optimisation problem, are difficult to apply in engineering practice owing to limitations of the computing scale and computational efficiency. In this study, a numerical shakedown analysis using the recently developed stress compensation method (SCM) is performed for a torispherical head with a piping nozzle, which is a typical structural component of pressure vessel equipment. The loads applied to the structural component include an internal pressure, axial force, twisting moment, out-of-plane and in-plane bending moments, and thermal loading, all of which vary independently of each other. Two- and three-dimensional strict shakedown boundaries for the torispherical head under different combinations of these loads are presented and analysed. In addition, the effect of a temperature-dependent yield strength on the shakedown domain is also investigated. These investigations demonstrate that the proposed SCM is capable of solving the practical shakedown problem for structures under complicated combined loads in industrial applications. The obtained results can provide guidance for the safe structural design of torispherical heads with piping nozzles.

Shakedown of porous material with Drucker-Prager dilatant matrix under general cyclic loadings

This paper is concerned with the shakedown limit states of porous ductile materials with Drucker-Prager matrix under cyclically repeated loads. Using the hollow sphere model and Melan’s shakedown theorem based on time-independent residual stress fields, a macroscopic fatigue criterion is derived for the general conditions of cyclic loads. First, the case of the hollow sphere subjected to pure hydrostatic loading is studied and the limit states of collapse by fatigue or by development of mechanism are derived. Then, the general case involving shear effects with any arbitrary cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The key idea is in two steps: (i) the choice of appropriate trial stress and trial residual stress fields and (ii) then maximizing the size of the load domain in the spirit of the standard lower shakedown theorem. The new macroscopic shakedown criterion depends on the porosity, the friction angle, Poisson’s ratio, the two stress invariants of the effective stress tensor and the sign of the third one. Together with the limit analysis-based yield criterion corresponding to the sudden collapse by development of a mechanism at the first cycle, it defines the safety domain of porous materials subjected to cyclic load processes. Interestingly, it is found that the safe domain is little sensitive to variations of the friction angle, however, it is considerably reduced compared to the one under monotonic loads obtained by limit analysis. Finally, a comparative study between the analytical results and numerical predictions performed by micromechanics-based finite element simulations is conducted for different porosities and friction angles.

Stress Compensation Method for Structural Shakedown Analysis

This paper presents a novel direct method called the stress compensation method (SCM) for structural shakedown analysis. Being different from the popular direct method of mathematical programming, the SCM just carries out some iterative calculations. Making full use of static shakedown theorem, the residual stress field is constructed via solving the modified global equilibrium equations. An effective and robust iteration control technique is adopted to generate a sequence of decreasing load multipliers. The numerical procedure is incorporated into the ABAQUS platform via some user subroutines. The shakedown problems for a cantilever beam, a symmetric continuous beam and a practical shell with nozzles are effectively solved and analyzed. These results are compared to the analytical solutions and those found in literatures. Both the incremental collapse mechanism and the alternating plasticity mechanism are revealed to determine the shakedown boundaries. Numerical examples show that the SCM is of numerical stability, good accuracy, high computational efficiency, and can effectively perform shakedown analysis of large-scale practical engineering structures.

Application of three-dimensional shakedown solutions in railway structure under multiple Hertz loads

Shakedown analysis is a robust approach to solve the strength problem of a structure under cyclic or repeated loading, e.g. railway structure subjects to rolling and sliding loads. A three-dimensional analytical shakedown solution is developed in this paper for the strength analysis of cohesive-frictional materials under multiple Hertz loads, which is then applied into the shakedown analysis of railway structure. Different to the pavement, the railway structure subjects to multiple wheel loads acting on the surface of two parallel rails, and then transformed to the substructure, which leads to different elastic and residual stress fields. For the application of Melan's shakedown theorem, the residual stress field in railway strucuture is rationally simplified to find the most critical plane that influence the most on the shakedown limit. The elastic stresses is then analytically derived for a single layer half-space under multiple Hertz loads, and is obtained numerically by means of the finite element technique for the multi-layered railway structure. Finally, the shakedown limit can be obtained in a direct way. Parametric study indicates how wheel loads, material properties such as frictional coefficient, friction angle and Poisson's ratio, and multi-layers influence the shakedown limit and stability condition of railway structure. It is found that the shakedown load increases with the increase of cohesion ratio, ballast frictional angle, and thickness ratio. However, the increase of the shakedown load ceases once the failure location moves from ballast layers to sub-ballast layers. This indicates that there is an optimum combination of material properties and layer thickness which provides the maximum resistance to the train loads. The obtained results give a useful reference for the engineering design of the railway structure.

Mesh-free shakedown analysis of cohesive-frictional pavement under moving traffic loads: deterministic and probabilistic frameworks

A mesh-free limit analysis approach has been modified and extended to comprise the problem of shakedown analysis of pavements. In this novel study, both deterministic and probabilistic frameworks are presented. Deterministic modelling is undertaken by the combination of Melan’s shakedown theorem, Shepard’s mesh-free technique and linear programming. A probabilistic framework is also introduced to incorporate soil’s shear strength inherent heterogeneity into the mesh-free shakedown analysis. In this context, the covariance matrix decomposition technique is used to generate the random fields of cohesion and friction angle. Thereafter, analyses have been conducted by employing Monte Carlo simulations which invoke mesh-free shakedown analysis several times. At the end of the paper, as an application of stochastic modelling, the probability of failure has been presented as a function of safety factor to quantify the geotechnical risks of pavement’s design from shakedown concept viewpoint.

A Simple Method to Estimate Numerically the Residual Stresses in Bodies Subjected to Cyclic Loads

Abstract Residual stresses in metallic materials may arise during the manufacturing process as a byproduct of the process itself, or they may be introduced on purpose to enhance the behavior of the product in service (for instance the compressive hoop stress in railroad vehicle wheels). Depending on the service regimen, the residual stresses in a body subjected to cyclic loads may either attain the stable (time-independent) state, thus leading to elasto-plastic shakedown of the body to the given loading program, or may not stabilize, resulting in the destruction of the body because of the ratcheting (alternating plasticity). The residual stresses in a considered body may be calculated using the standard incremental approach, where the final state is found through following step by step the actual loading program, or they may be estimated by the application of plastic shakedown theorems. The first approach, although yielding precise information on body behavior, may be extremely time consuming in the case of cyclic loads. The second approach, although yielding only an estimate of the final residual stress distributions, is much more time efficient. The numerical approach discussed in this article is based on Melan’s theorem and allows for a reasonably accurate estimate of residual stresses induced in an elastic plastic body made of strain hardening material as long as an upper bound of the loading program is known. The tests performed on an autofrettage problem exhibit excellent agreement with an exact analytical solution. The results for the calculations of the residual stresses in railroad rails subjected to simulated service loads are provided.

Dynamic shakedown limits for flexible pavement with cross-anisotropic materials

Shakedown solution is widely used to analyse the elastic-plastic behaviours of structures in pavement design. Previous studies only concerned the shakedown theorem under traffic loads for an isotropic material, although the material is usually anisotropic. This paper firstly proposed a numerical analysis for anisotropic materials under moving traffic load, based on Melan’s lower-bound shakedown theory. An anisotropic Finite Element–Infinite Element (FE–IE) model is used to calculate the dynamic stresses in the anisotropic material subjected to moving traffic loads with different speeds. Then, the shakedown limits for an anisotropic half-space and a two-layered pavement system are determined, respectively. It is found that the Rayleigh wave speed of the soil has a significant effect on the shakedown limits. For a cross-anisotropic half-space, the shakedown limits mainly depend on the stiffness ratio of the two layers and the Poisson’s ratio only has a small effect, although both of them significantly affect the Rayleigh wave speed. Furthermore, the shakedown limit increases with increasing cohesion ratio until it reaches a maximum value, and it gets rid of the control of shakedown condition when the moving speed exceeds the Rayleigh wave speed. For a two-layered anisotropic system, the results are similar to those in an isotropic system. Failure tends to occur on the top of the second layer instead of the first layer when the speed of moving load or the anisotropic Young’s modulus ratio increases, together with the decrease of shakedown limit.

A numerical formulation and algorithm for limit and shakedown analysis of large-scale elastoplastic structures

In this paper, a novel direct method called the stress compensation method (SCM) is proposed for limit and shakedown analysis of large-scale elastoplastic structures. Without needing to solve the specific mathematical programming problem, the SCM is a two-level iterative procedure based on a sequence of linear elastic finite element solutions where the global stiffness matrix is decomposed only once. In the inner loop, the static admissible residual stress field for shakedown analysis is constructed. In the outer loop, a series of decreasing load multipliers are updated to approach to the shakedown limit multiplier by using an efficient and robust iteration control technique, where the static shakedown theorem is adopted. Three numerical examples up to about 140,000 finite element nodes confirm the applicability and efficiency of this method for two-dimensional and three-dimensional elastoplastic structures, with detailed discussions on the convergence and the accuracy of the proposed algorithm.

Shakedown analysis for the evaluation of strength and bearing capacity of multilayered railway structures

Shakedown analysis is a robust approach for solving the strength problem of a structure under cyclic or repeated loading, e.g. railway structures subject to rolling and sliding traffic loads. Owing to the traffic loads, which are higher than the “shakedown limit”, railway structures may fail due to the excessive permanent deformation. This paper develops the analytical shakedown solutions based on Melan’s shakedown theorem, which is then applied for the evaluation of the strength and bearing capacity of multilayered railway structures. The shakedown solutions utilize the elastic stress fields obtained from the fully three-dimensional finite/infinite model, and calculate the shakedown multiplier for each layer of railway structures by means of a self-equilibrated critical residual stress field. The shakedown limits are then determined as the minimum shakedown multiplier among all layers. Parametric studies are also conducted, which indicate how the frictional coefficient, strength and stiffness of the materials, and the thickness ratio of ballast to subballast influence the shakedown limit and the stability condition of railway structures. The critical points of shakedown occur at the rail for low values of rail’s yield stress and large frictional coefficient, while they occur at the ballast layer when the frictional coefficient is relatively small. The shakedown limits are found to decrease with the increase in the strength and thickness of the ballast for a relatively small frictional coefficient. For the engineering design, there is an optimum combination of material properties and layer thickness, which provides the maximum bearing capacity of the railway structure based on this research. The results obtained from this study can provide a useful reference for the engineering design of railway structures.

Soil arching in highway piled embankments subjected to moving shakedown limit loads

Shakedown analysis provides the loading variation limit of a structure, under which the plastic deformation occurred in first loading cycles will cease to further develop and the structure will respond purely elastically to the remaining load cycles. This paper investigated the arching in highway piled embankments under moving shakedown limit loads, which was calculated by the shakedown solutions. The shakedown solutions for materials obeying Mohr–Coulomb criterion was firstly presented based on Melan’s shakedown theorem, which utilised elastic stress fields obtained from the full three-dimensional (3D) finite element (FE) method. Finally, the shakedown limit load for pure rolling condition (i.e. only normal load) was applied onto the surface of pavement as a moving vehicle load. The FE results under moving shakedown limit loads captured similar trend with those under the static loads in terms of settlement and vertical stress, but gave the normalised settlement and vertical stress 33 and 24% larger. The soil arching remained existed in the highway piled embankments under moving shakedown limit loads, but was reduced. The inclusion of moving shakedown limit loads increased the vertical stress of the embankment and thus resulted in a reduction of the maximum value of lateral earth pressure coefficient by approximately 38%.

Limit and shakedown analysis under hydrogen embrittlement condition

A modified shakedown theorem and its solving technique are presented to involve hydrogen embrittlement of steel into limit and shakedown analysis. Firstly, the shakedown theorem for hydrogen embrittled material is derived from a limited kinematic hardening shakedown theorem and hydrogen enhanced localized plasticity mechanism of hydrogen embrittlement. In the presented theorem, hydrogen’s effect is taken into account by the synergistic action of both strength reduction and stress redistribution. Secondly, a novel solving technique is developed based on the basis reduction method, in which the complicated constraints in the resulting nonlinear mathematical programming are released. At last, three numerical examples are carried out to verify the performance of the proposed method and to reveal hydrogen’s effect on the limit and shakedown load of structure. The numerical results are discussed and compared with those from literatures, which proves the accuracy and high efficiency of the introduced solving technique. It is concluded that the proposed theorem can predict the limit and shakedown load of hydrogen embrittled structure reasonably.

Shakedown for slab track substructures with stiffness variation

In this paper, shakedown analyses are carried out to predict the long-term response of slab track substructures under repeated moving train loads. The train loads are converted into a distributed moving load on the substructure surface by using a simplified track analysis. Based on Melan’s static shakedown theorem, a well-established shakedown analysis method is extended to determine shakedown limits of the slab track substructures. The influence of a linearly increasing stiffness modulus on the shakedown limits is considered by conducting finite-element analysis with a user-defined material. It is found that a rise in stiffness modulus or stiffness variation ratio can either increase or decrease the shakedown limit, depending on the competitive effects of the two mechanisms. Furthermore, the subgrade thickness determines the dominant mechanism.

Influence of graphite morphology on static and cyclic strength of ferritic nodular cast iron

High silicon alloyed nodular cast iron consists of a purely ferritic matrix and graphite nodules, mainly. Varying wall thicknesses and manufacturing conditions result in different graphite morphologies throughout a structural component. From an experimental point of view, axial fatigue and tensile tests were carried out on specimens with differently degraded graphite. From a numerical point of view, the microstructure has been modelled using a finite element (FE) approach with representative volume elements (RVE). The RVE models were built according to micrographs of fatigue specimens. The generated RVEs determine effective material properties through elasto-plastic homogenization and were subsequently analysed using a shakedown approach. In shakedown theory, the material re-enters the elastic regime after a few cycles of initial plastic deformation. This work uses the shakedown theorem to derive a lower bound estimation of the endurance limit from a non-incremental simulation. Here, the material has to be modelled elastic-perfectly plastic. The major challenge in modelling nodular cast iron is to determine suitable material parameters for the graphite and ferrite phase, revealed by parameter studies on the static and cyclic model. By using reasonable material parameters, fundamental effects, observed in the fatigue tests, were reproduced on the model level.