Shakedown Analysis Research Articles

Effects of pavement roughness on shakedown limits under moving traffic loadings

This study proposes a lower-bound shakedown analysis method for pavement systems under moving traffic loadings considering pavement roughness. The effects of key variables, including moving speed, wavelength irregularity, and the modulus and thickness of pavements, on the shakedown limit are discussed. The calculation results indicate that pavement roughness has a considerable effect on shakedown limits, which can be overestimated in some cases when the pavement roughness is not considered. The effect of pavement roughness on the shakedown limit of a homogeneous subgrade layer gradually decreases with increasing speed. Additionally, the shakedown limit decreases with wavelength irregularity, particularly under high-speed traffic loadings. Furthermore, the effect of pavement roughness on the shakedown limit of a two-layered pavement system is more pronounced for a larger pavement modulus and thickness and a lower moving speed.

A FEM cluster-based basis reduction method for shakedown analysis of heterogeneous materials

A FEM cluster-based basis reduction method for shakedown analysis of heterogeneous materials

Mesh-free kinematic shakedown analysis of cohesive soils

A novel mesh-free solution is proposed for kinematic shakedown analysis of cohesive soils under repeating loads. For this purpose, the continuous velocity field in the mathematical expression of Koiter’s theorem is discretized by the Radial Point Interpolation Method (RPIM), as a mesh-free approach. The strain rate smoothing technique is implemented in conjunction with the RPIM to satisfy the admissibility conditions at the entire problem domain. Using the nodal integration and the discretized velocity field, the kinematic shakedown problem is expressed as a nonlinear optimization problem. The optimization problem is solved by separation of plastic and non-plastic/rigid zones using a repetitive algorithm. Eventually, the efficiency of the proposed approach is elucidated by solving examples of a strip footing resting on cohesive soil and a cohesive half space pavement under repeating loads.

Lower-bound shakedown solutions for transportation infrastructures subjected to moving harmonic loads: A focus on saturated subgrade support

As a valuable tool, the shakedown analysis technique can be used to evaluate the long-term resilience of transportation infrastructures subjected to dynamic vehicular loads. In this paper, we introduce a 3-D analytical model that faithfully replicates the geometric complexities inherent in real-world transportation systems. Rather than utilizing the total stress field, modifications to the existing shakedown limit determination methods are executed by focusing on the effective stress field within saturated subsoil. This refined methodology is specifically designed for multi-layered transportation infrastructures like road pavements and rail tracks built over saturated subgrade soils. Upon validation of both the analytical model and the enhanced method for calculating shakedown limits, a systematic investigation into the role of various parameters is conducted. These include the strength ratio, embankment height, and dynamic loading characteristics such as speed and frequency. Subsequent numerical analyses reveal that both the strength ratio and embankment height markedly influence shakedown limits within defined boundaries. Additionally, a comprehensive consideration of load speed and frequency is imperative for an accurate shakedown behavior assessment of the infrastructure in question. The findings validate that the proposed approach offers a more rigorous framework for the optimized design and safety evaluation of transportation systems comprising saturated soil layers.

Effect of boundary conditions on shakedown analysis of heterogeneous materials

Effect of boundary conditions on shakedown analysis of heterogeneous materials

Probabilistic Analysis of Dynamic Shakedown of Pavement due to Vehicular Loads and Earthquakes

Probabilistic Analysis of Dynamic Shakedown of Pavement due to Vehicular Loads and Earthquakes

Shakedown-reliability based fatigue strength prediction of parts fabricated by directed energy deposition considering the microstructural inhomogeneities

For mechanical components made by additive manufacturing (AM) techniques such as directed energy deposition (DED), micropores and other defects prevalently exist in the microstructure and they significantly reduce the reliability of these parts. To allow the influence of these microstructural features over the load bearing capacity to be considered in the design stage of AM structures, in the present paper we developed a multiscale numerical approach. The goal of this approach is to predict the failure probability of AM structures subjected to time-varied loadings, and it is realized through combining statistical homogenization, shakedown analyses, and reliability methods such as first-order reliability method and Monte Carlo simulation. Based on this strategy, we employed statistics of material parameters obtained from micromechanical models as inputs, implemented numerical tools and applied them to two exemplary structures, the plate with a hole and an aircraft bracket. Through a series of case studies carried out using different methods and under different assumptions of material randomness, the paper confirmed the robustness of the derived results and explained the mechanism of how micropores influence the structural reliability. The method developed in this paper can be a viable means for the design and optimization of metallic and metal matrix composite structures produced by AM techniques.

Effect of primary carbides on rolling contact fatigue behaviors of M50 bearing steel

It is perceived that the fatigue performance of high-strength steel is preponderantly affected by internal defects. Inclusions may be efficaciously controlled and primary carbides have been proved to be the principal type of defects in M50 bearing steels. Primary carbides in service-status M50 were systematically characterized and the rolling contact fatigue (RCF) properties were evaluated under different friction conditions in this work. An innovative method of shakedown analysis was proposed for the first time, extending the shakedown analysis of materials as an integral based on traditional shakedown theory to that of weaker internal defects. By the response of primary carbides, the RCF process was separated into a plastic instability period and an elastic shakedown period. The accumulation of plastic deformation around carbides before shakedown was proposed as the RCF damage parameter, and the influence of primary carbides was studied based on the damage criterion. The competition between surface and internal fatigue initiation was revealed via different friction coefficients. A predictive model for RCF initiation was established, which could predict the critical sizes for disparate types of primary carbides not deleterious to RCF performance, and the critical sizes under typical industrial service condition were reckoned.

A polygonal finite element method for shakedown analysis of structures

A polygonal finite element method for shakedown analysis of structures

Thermomechanical shakedown analysis considering temperature-dependent yield stress based on the primal-dual eigenstress-driven method

Many industrial structural components are loaded repeatedly in high-temperature environments. The shakedown limit under variable thermomechanical loads is a critical factor in thermomechanical elastoplastic structure design assessment. The shakedown limit is weakened due to material strength reduction at high-temperature conditions.This paper develops an efficient shakedown analysis Primal-dual Eigenstress-driven Method(PEM) algorithm for temperature-dependent yield condition, which overcomes the obstacle of the original PEM method [1] which can only deal with constant material properties. The developed temperature-dependent PEM algorithm proposes a two-level iteration algorithm to determine the shakedown limit and the time-independent residual stress field. At the inner level the theoretical framework tailored to the temperature dependence problem is derived to solve the Karush-Kuhn-Tucker (KKT) conditions of the extended temperature-dependent shakedown theorem and establish an increment algorithm for forming residual stress fields. Besides, the lower bound safe multipliers are computed to verify the residual stress field obtained by inner-level algorithm. At the outer level an efficient fixed-point algorithm is applied to search the shakedown multiplier, where the corresponding updating strategy for the yield stress with temperature is nested. For practicability, the proposed algorithm is seamlessly integrated into commercial platform ANSYS, which is applicable to industrial structures. Four numerical examples from multiple perspectives show the effectiveness of developed PEM method.

Direct method-based probabilistic shakedown analysis for the structure under multiple uncertain design conditions

Maintaining shakedown during the design stage is crucial for the engineering components under cyclic loading conditions. Due to the multiplicity of uncertainty existing in the actual operating parameters, probabilistic shakedown analysis is more conducive to dealing with risk management for critical infrastructures, compared with the traditional experience-based safety factors which mainly rely on deterministic evaluations and lack statistical information for decision-making. Aimed at reasonably considering the redundancy in structural integrity assessment against cyclic load conditions, a novel direct method-based probabilistic shakedown analysis is proposed under the new probabilistic Linear Matching Method framework (pLMM). The risk of losing the shakedown state is predicted by the physics-based estimation model, where the efficient iteration is employed to derive the reliability index. In the benchmark, the probabilistic shakedown boundary is constructed, with the additional influence of the uncertain cyclic loading pattern on the reliability fully reflected. This pLMM shakedown analysis facilitates the progress from over-reliance on the conservative safety factor towards precise risk management.

A Numerical Procedure for Shakedown Analysis of Thick Cylindrical Vessels with Crossholes under Dual Cyclic Loadings.

A modified numerical procedure for the shakedown analysis of structures under dual cyclic loadings, based on the Abdalla method, is proposed in this paper. Based on the proposed numerical procedure, the shakedown analysis of the thick cylindrical vessels with crossholes (TCVCs) under cyclic internal pressure and cyclic thermal loading was carried out. The effects of material parameters (elastic modulus and thermal expansion coefficient) and crosshole radius on the elastic shakedown limit of TCVCs are discussed and, finally, normalized and formularized. Furthermore, the obtained shakedown limit boundary formulation is compared with FEA results and is verified to evaluate the shakedown behavior of TCVCs under cyclic internal pressure and cyclic thermal loading.

Loading capacity prediction of the auxetic tubular lattice structures by multiscale shakedown analysis

Auxetic tubular structures have raised increasing research interest due to their unique and tailored properties. However, the complexity of multiaxial loads eliminates the availability of the classic fatigue evaluation method and the difficulties of connecting lattice unit design with structural capacities have postponed the study on lattices’ fatigue limit loads. To fill this research gap, we present a direct shakedown method using tetrahedron elements with the reduced Gaussian integration point to predict the biaxial bearing capacities for time-varied loadings. The rationality and precision of the approach are validated using the residual stress comparison with the incremental method. The nodal-coupling-based periodic boundary conditions that are proposed and verified facilitate the parameterization study of the tubular lattice structure design. The results reveal that the edge width of the lattice unit positively impacts the elastic shakedown loads while the curvature influences reversely. The circumferential loading capacities of the auxetic tubular lattice have a significant susceptibility to curvature. This work provides an efficient tool for quantitatively evaluating the shakedown loading capacities of auxetic tubular lattice structures and the impacts of design parameters.

Limit and shakedown analysis of double-layer tube considering different material properties

This paper is devoted to the maximum bearing capacity of double-layer tubes subjected to monotonic and cyclic loading, within the framework of static limit and shakedown analysis. The double-layer tube comprises inner and outer layers made of different materials, both obeying the von Mises criterion. The effects of different geometric dimensions (thickness and volume fraction) and material properties (elastic modulus, Poisson’s ratio, and yield stress) on the effective mechanical behavior are studied analytically and numerically. The results indicate that the bearing capacity of the double-layer tube under monotonic load reaches its maximum value when both inner and outer layers reach their elastic or plastic limits simultaneously. The computation of plastic and shakedown limits are presented, divided into different cases by considering the volume fraction of the inner and outer layers. The double-layer structure under cyclic loads may fail due to fatigue and excessive deformation mechanisms of the inner or outer layers at the first cycle. Only when both layers fail due to fatigue simultaneously, the shakedown limit of the structure reaches its maximum value. Compared to a single-layer tube, the long-term strength of the double-layer tube can be significantly improved by applying appropriate geometric dimensions and materials in fabrication.

Multi-domain optimization of cast iron components in wind turbines

The continuously rising demand for renewable energies leads to increased installations of wind turbines with higher power. While the current power-to-weight ratio of up to 20 metric tons of cast iron per megawatt is stagnating, cast iron components of modern wind turbines are facing new challenges in terms of weight, manufacturability, and castability. These challenges can be addressed by systematically using multi-domain optimization approaches to reduce component weight and increase local component utilization.In order to meet the requirements for modern cast iron components, this multi-domain approach must employ methods from casting simulation, micromechanical analysis, topology optimization, and strength assessment. Here, casting simulation is used to determine local microstructure descriptors, which are subsequently used in micromechanical shakedown analysis to estimate the local microstructure-dependent fatigue strength. In parallel to the fatigue strength estimation, topology optimization is performed iteratively in combination with a castability analysis. The component strength is evaluated using a strength assessment approach based on the previously determined local material properties in combination with the topology optimized component.In this study, the overall concept of the proposed multi-domain approach is presented and requirements for the application of such an approach are formulated. The use case of this study is a planet carrier of a wind turbine gearbox manufactured from austempered ductile cast iron ADI-GJS-1050‑6. For this use case, a weight reduction of 17% was achieved while maintaining the required stiffness, such that the microstructure variance along the component was significantly reduced. Furthermore, the potentials and limitations of the presented approach are outlined and discussed in the context of the design of heavy-section castings.

Elastic core concept for modified Bree problems considering in-phase and out-of-phase loading conditions

The elastic core concept developed from the Bree problem can be used for structural integrity evaluation of high temperature components, such as creep ratcheting and creep fatigue. The simplified inelastic analysis method of the ASME 2013 III-1 NH Code adopts the idea of core stress, which protects against creep rupture and collapse of the structure by limiting the accumulated inelastic strain. Currently, the formulas of core stress and evaluation diagram of effective creep stress parameter for Tests B-1 and B-3 adopted by the APPENDIX NH-T are developed from the classical loading Bree problem, however, the related concepts and evaluation diagrams of modified Bree problems considering generalized loading conditions have not been reported. This paper first reviews the development history of the elastic core concept and its corresponding analysis methods. Based on the modified Bree problems considering in-phase and out-of-phase loading developed by Bradford, the corresponding formulas of the core stress in the non-plastic ratcheting zone are completely proposed, and the corresponding evaluation diagrams of effective creep stress parameters are constructed for the first time. By comparing the core stress formulas and effective creep stress parameter diagrams under three loading cases, the conservatism and applicability of the classical core stress method are discussed. In addition, the differences and relations between two creep ratcheting assessment methods applicable to transient and sustained thermal load conditions respectively are clarified. The new results and conclusions presented in this paper can deepen the understanding of structural response under complex loading conditions, and provide guidance for shakedown analysis and creep ratcheting assessment of high-temperature components.

An Efficient Numerical Algorithm for Shakedown Analysis Combined with Basis Reduction

An Efficient Numerical Algorithm for Shakedown Analysis Combined with Basis Reduction

FEM shakedown analysis of structures under random strength with chance constrained programming

Direct methods, comprising limit and shakedown analysis, are a branch of computational mechanics. They play a significant role in mechanical and civil engineering design. The concept of direct methods aims to determine the ultimate load carrying capacity of structures beyond the elastic range. In practical problems, the direct methods lead to nonlinear convex optimization problems with a large number of variables and constraints. If strength and loading are random quantities, the shakedown analysis can be formulated as stochastic programming problem. In this paper, a method called chance constrained programming is presented, which is an effective method of stochastic programming to solve shakedown analysis problems under random conditions of strength. In this study, the loading is deterministic, and the strength is a normally or lognormally distributed variable.

Shakedown Analysis and Experimental Study of Thermal Barrier Coatings.

Interfacial stress–strain fields become complicated in thermal barrier coatings (TBCs) under cyclic thermal loading, which affects the stability and spalling failure of TBCs directly. The convex and concave interfacial structures of TBCs were approximated as a multilayer cylinder model, and an analytical method of TBCs for shakedown analysis was established. A series of 8-YSZ TBC specimens were prepared by the plasma spraying process, followed by isothermal and thermal shock tests. The results showed that the stability limit is significantly greater than the elastic limit, the limit for the convex model was higher than that in the concave model, the first failure occurs in the concave area, and the main failure mode of a thermal barrier coating is the appearance of cracks at the interface layer during a thermal shock test. For the coating samples prepared in this study, the stability limits were between 950 °C and 1050 °C, and the validity of the stability limit analysis model of a multilayer structure was verified.

Shakedown analysis and assessment method of four-stress parameters Bree-type problems

For pressure equipment in the industrial field, the shakedown design methods presented in the ASME and other international design codes are mainly based on the Bree problem. The original Bree problem and its related modified Bree problems usually contain two types of stress parameters, whereas design by analysis based on the stress categorisation of the codes actually includes multiple types of stress parameters. In this paper, the four-stress parameters Bree-type shakedown problems (involving mechanical membrane and bending as well as thermal membrane and bending) have been studied under three typical thermo-mechanical loading conditions for the first time to reveal the mechanisms of the four-dimensional Bree diagram. A detailed discussion on the mechanical models for various uniaxial Bree-type problems is conducted to clarify the scope of application and mechanical assumptions for each problem. A common finite element model is developed for all discussed Bree-type problems. The Linear Matching Method (LMM) is utilised to construct the shakedown boundaries of three examples under three loading conditions, and the semi-analytical parametric equations of the shakedown boundaries are derived based on the numerical results and four-dimensional ratcheting boundary theory. Using the idea of a minimum shakedown boundary, a unified and conservative shakedown assessment scheme is proposed for the four-stress parameters Bree-type problems. The newly presented results can enhance the understanding of the structural behaviour of pressure equipment for many loading conditions and guide shakedown design and ratcheting assessment of practical industrial components.