Method For 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

A polygonal finite element method for shakedown analysis of structures

A polygonal finite element method for shakedown analysis of structures

A novel primal‐dual eigenstress‐driven method for shakedown analysis of structures

AbstractThe shakedown load of elastoplastic structures under multiple variable loading is an important factor in structural design and integrity analysis. In classical plasticity shakedown analysis is an essential and challenging problem. Most existing methods are based on the solution of super large‐scale mathematical programming, basis reduction or mechanics insight method which have their own limitations in practical engineering problem. In the present study, the proposed method explores the Karush–Kuhn–Tucker (KKT) conditions and the physical reinterpretation of its primal‐dual variables of the Melan's lower bound theorem, and establishes the relation of the primal‐dual variables in finite element formulation. Based on the primal‐dual theory, the primal‐dual eigenstress‐driven method is proposed which is a two‐level nested algorithm: the inner level is the eigenstress‐driven incremental algorithm for constructing the beneficial residual stress field and calculating the safe multiplier and the outer level is the load multiplier descent algorithm for searching the shakedown multiplier. For the purpose of algorithm's universality, the proposed algorithm is fully integrated with the commercial finite element software ANSYS APDL without special optimization solvers, which is well suited for large‐scale practical engineering structures. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Shakedown analysis method for composites based on homogenization theory

Direct methods of plastic analysis are widely used in composites analysis to determine material strength for safety assessment or lightweight optimization design. Multi-scale processing of periodic heterogeneous composite material is needed due to its existing of microstructure. The standard method is to determine the macroscopic properties from the calculation results of microcosmic representative volume elements (RVEs) by using the homogenization theory. However, in current practice, there are some disadvantages of transforming the micro strain domain to the macro stress shakedown domain when considering multiple external loads. The domain cannot fully demonstrate the shakedown condition, and it is impossible to evaluate a known loading combination only from the knowledge of whether the load leads to the shakedown state. To overcome this disadvantage, a new comprehensive approach was proposed to enhance endurance limit strength of composites under variable loads for long term. Considering the example of in-plane strength analysis, for microcosmic RVEs, a new set of boundary condition was defined in the form of uniform strain. The boundary condition was derived from the elastic response under unit loads by using Hook’s law and stiffness matrix. The resulting elastic stress field was used later for plastic shakedown analysis. Based on the lower bound theorem of plastic mechanics, optimization programming for load factor was performed, and after proper mathematical reformulation, the conic quadratic optimization problem could be solved efficiently. Macro-stress shakedown domain can be obtained after scale-transformation of the RVE results. The bases of this stress domain are unidirectional stress in geometry space. The stress amplitude of a structure can be evaluated by this domain for determining the shakedown state in a simple and practical manner. Further, changes in the boundary condition of RVE do not affect the limit and elastic analysis. Finally, few numerical examples were presented for verification and illustration. This approach can be expanded to three dimensions and employed for more complex structures.

Shakedown solutions for ballasted track structure under multiple uniform loads

Abstract Shakedown analysis method is attractive in calculating the bearing capacity of structures under repeated loads with a large number of cycles. This paper firstly introduces an equivalent Finite Element (FE) model to simulate the ballasted track structure, whose results of elastic stress field were calibrated against those obtained by the full three-dimensional (3D) model, so as to reach an optimum balance between the accuracy and consumption of the numerical simulation. By representing the wheel loads as equivalent uniform loads acting on the ballast layer, an analytical lower-bound shakedown solution is developed to determine the capacity of a half-space subjected to multiple uniform loads, and is further extent to shakedown analysis of ballasted track structure. The required elastic stresses in shakedown analysis are derived analytically for homogenous half-space, while are acquired numerically based on the equivalent FE models. The parametric study on the basis of the proposed shakedown solution is finally performed by introducing the envelope diagrams of shakedown limit, to investigate how the properties and thickness of the materials affect the shakedown limit of ballasted track structure under multiple uniform loads, whose conclusion may provide reference for the optimized design of ballasted track structure subjected to repeated moving train loads.

Numerical limit and shakedown analysis method for kinematic hardening structure made of arbitrary inhomogeneous material

The prediction of structure’s ultimate load has a lot of practical applications in composite structure. In the present work, limit and shakedown load of structure made from arbitrary inhomogeneous material was predicted by a novel solution algorithm named Back Stress Iteration Method, where material’s kinematic hardening behavior is involved as well. Firstly, details about the proposed algorithm were introduced. Basis reduction method was employed as framework and its advantage in computational efficiency is inherited. Secondly, the influence of inhomogeneous kinematic hardening was solved by a new invented Back Stress Iteration process. Then, three examples with increasing complexity were investigated numerically for validation purpose. The results were discussed intensively with respect to the accuracy, the effect of inhomogeneity and the inhomogeneous kinematic hardening behavior to verify the rationality of the proposed method. At last, the paper was summarized and conclusions were drawn. The proposed method is promising to provide a powerful tool for the accurate prediction of arbitrary inhomogeneous structure’s load-bearing capacity, such is applicable to many kinds of composite structures.

A numerical shakedown analysis method for strength evaluation coupling with kinematical hardening based on two surface model

The current research utilizes an efficient shakedown computation technique to determine the load-bearing capacity of a structure. It can predict the strengths of a structure under arbitrary varying loads where failure form of alternate plasticity and incremental collapse will be avoided. To achieve a realistic result, a kinematical hardening material model is formulated into the shakedown analysis based on Melan's static theorem using a two-surface model so that both incremental collapse and alternating plasticity can be captured. Thus it leads to a nonlinear convex optimization problem which later reformulated to an efficient form for numerical computation. In contrast to the results of benchmark examples for a steel hollow section bar and a plate with hole, the present method is well suited to determine the limit and shakedown states of these two problems especially for a large scale case. Specifically, the present approach does not require full loading history. In order to better illustrate the computing power potential of this algorithm, the proposed technique is also utilized to determine the load-bearing capacity of a cast aluminum beam to be used in a high-speed train. In addition to constructing the plastic and shakedown domain, the proposed approach is also employed to study how kinematical hardening material model influences the feasible load domains.

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.

Special Section on Ratcheting.

The design methods for pressure vessels and piping are based on the avoidance of potential modes of failure, such as collapse, excessive deformation, and cracking. Collapse and excessive deformation from a single load application are addressed by methods such as buckling analysis, primary stress analysis, and plastic limit analysis. For instance, limit load design underlies the evaluation methods for design conditions in most existing pressure vessel design codes. When significant cyclic loading is applied, the evaluation of fatigue and ratcheting effects becomes necessary. Fatigue analysis is concerned with avoiding the initiation and propagation of cracks that could eventually cause a sudden fracture. Ratcheting is a failure mode typically associated with components that are subjected to pressure loading and simultaneously large cyclic thermal stresses. It is characterized by deformations or plastic strains that accumulate with increasing load cycles. Continued deformation can eventually render the component unserviceable and strain accumulation can accelerate fatigue cracking, which is not accounted for in the fatigue analysis. Ratcheting can occur in metals, but also in nonmetallic materials. In order to avoid ratcheting, the cyclic loads must be kept within a specific limit that depends on the level of simultaneously applied mechanical loading (shakedown limit). By maintaining the stresses below the shakedown limit, some incremental plastic deformation may occur during the initial loading cycles, but the deformation in the subsequent cycles will be stable cycling, either in the elastic range (elastic shakedown) or involving alternating plasticity (plastic shakedown). Typical simplified evaluation methods in the design codes are based on perfect (nonhardening) plasticity. Such methods and their extensions, such as direct methods of shakedown analysis, promise simple and efficient solutions and are still being developed. Due to the numerical methods that are now available, a full cyclic plastic simulation that could include work hardening is also becoming feasible. There are, however, some knowledge gaps in the application of hardening plasticity models to shakedown or ratchet analysis. For example, apparently equivalent descriptions of a plastic stress–strain curve by different plasticity models which give comparable results for static loading and even for steady cyclic response can have widely different ratcheting behavior. Depending on the plasticity model, the response may vary from guaranteed shakedown independent of loading to unexpected ratcheting under some stress states. Therefore, there is active research involving experimental studies of ratcheting and development of plasticity models to better describe the material response to various loading combinations. Simultaneously, efforts are underway to identify simple existing plasticity models that include hardening and are suitable for an engineering analysis of shakedown or ratcheting. This special section on ratcheting has a total of 12 papers which represent a good mixture of theoretical, numerical, and experimental research in this area.

Mixed moving least-squares method for shakedown analysis

The elastic shakedown theory can be used to analyze and design structures under loads that change over time. Methods that use mixed interpolations provide simultaneous information regarding the kinematics and the equilibrium without the necessity of postprocessing. Meshless methods, as suggested by the name, do not require mesh generation or node connectivity and are an alternative to classical numerical methods. However, no current method exists that combines a mixed approximation of the kinematics and equilibrium with meshless methods for an elastic shakedown problem. In this paper, a discretization of a mixed variational principle for the elastic shakedown is presented based on the moving least-squares method, where stress and velocity approximation functions are used. The optimization problem, which comes from the mixed variational principle, determines the load factor and the residual stress and velocity fields. The stresses and velocities are approximated using linear and quadratic polynomial functions, respectively. The solution to the problem is obtained through a nonlinear algorithm based on Newton’s method. The results obtained for the load factor and the residual stress field approach those obtained from the analytical solution for the tested examples.

Element Bearing Ratio Based Shakedown Analysis for Branch Pipe

Shakedown analysis is important for branch pipe because it is often damaged under various water pressure. In this paper, an element bearing ratio (EBR) based shakedown analysis method is employed for shakedown analysis of branch pipe. The EBR is used to replace the stress term in classical optimization problem in the procedure, and series of residual EBR fields can be generated by the D-value of the elastic-plastic EBR fields and the elastic EBR fields at every incremental loading step. The shakedown load is determined by performing an incremental non-linear static analysis when the yield criterion is arrived either by the elastic-plastic EBR fields or residual EBR fields. By introducing the EBR, the proposed procedure can be easily used to shakedown analysis of branch pipe with multi-material and complicated configuration. Numerical examples validate the method and demonstrate its performance.

Shakedown analysis of a slope with cyclic groundwater level

AbstractGroundwater levels vary with precipitation and other environmental impacts, rendering wetting and drying cycles in parts of a slope. It may be quite disadvantageous, as groundwater has been implicated as a crucial controlling factor in slope stability. The wetting–drying cycles can be considered as cyclic loads for the slope. In such cases, a shakedown analysis is appropriate for the long‐term safety evaluation. This paper presents an upper bound shakedown analysis method based on strong version of Koiter's theorem to evaluate slope stability. An element integration method is employed to provide limit and shakedown safety factors for the slope. For plane slip surfaces or logarithmic spiral slip surfaces, the validity of the method is verified by two documented examples. Copyright © 2009 John Wiley & Sons, Ltd.

Finite element shakedown analysis of reinforced concrete 3D frames

In the present paper, the method proposed in [Casciaro R, Garcea G. An iterative method for shakedown analysis, Comput Methods Appl Mech Eng 2002;191:5761–92] for the shakedown analysis is used for the evaluation of the shakedown safety factor of reinforced concrete 3D frames subjected to a combination of varying loads. The FEM discretization for 3D frames is obtained subdividing the frame into rods and using a 3D beam finite element. Beam end-sections are used for checking plastic admissibility and for this reason the formulation models the interaction between the axial force and biaxial bending by a piecewise linearization of the elastic domain, for generic shape reinforced concrete sections. Concrete and steel are assumed to be elasto-plastic, and, for concrete, tensile resistance is ignored. The algorithm is described in detail and some numerical results, that show the efficacy and effectiveness of the proposed method, are reported and discussed.

Performance of a p-adaptive finite element method for shakedown analysis

This paper is concerned with the application of the p-version finite element method in carrying out a shakedown analysis. In particular, an adaptive scheme, implemented within the discretized framework of a static approach and mathematical programming, is proposed. The primary motivation of the present work is to overcome the well-known locking behavior caused by the incompressibility constraint that may occur in computational plasticity, and yet is still able to process the underlying nonlinear optimization problem efficiently. Quadrilateral elements and plane problems are specifically considered. A number of examples, some being benchmarks problems, are given to illustrate suitability of the approach.

INCREMENTAL METHOD FOR UNLOADING PHENOMENON FIXATION AT SHAKEDOWN

Incremental method for shakedown analysis of the elastic perfectly plastic structures is based on the extremum energy principles and non-linear mathematical programming approach. Residual force increment calculation problem is developed applying minimum complementary deformation energy principle. The Rozen project gradient and equilibrium finite element methods were applied for solution. The Rozen optimality criterion (Kuhn-Tucker conditions) ensures compatibility of residual strains and allows plastic strain and residual displacement increment calculation without dual problem solution. The possibility to fix the structure cross-section unloading phenomenon during shakedown process was developed. The proposed technique is illustrated by annular bending plate residual force and deflection calculation examples, when the von Mises criterion is taken into account.

INCREMENTAL METHOD FOR UNLOADING PHENOMENON FIXATION AT SHAKEDOWN

Abstract Incremental method for shakedown analysis of the elastic perfectly plastic structures is based on the extremum energy principles and non-linear mathematical programming approach. Residual force increment calculation problem is developed applying minimum complementary deformation energy principle. The Rozen project gradient and equilibrium finite element methods were applied for solution. The Rozen optimality criterion (Kuhn-Tucker conditions) ensures compatibility of residual strains and allows plastic strain and residual displacement increment calculation without dual problem solution. The possibility to fix the structure cross-section unloading phenomenon during shakedown process was developed. The proposed technique is illustrated by annular bending plate residual force and deflection calculation examples, when the von Mises criterion is taken into account.

Shakedown Limits for a Metal Matrix Composite

The paper discusses a method for the evaluation of the yield values of a metal matrix composite material when subjected to a variation in temperature. A direct shakedown analysis method is described which produces a shakedown stress value as a result of a sequence of incompressible linear solutions with a spatially varying modulus. The process converges to the minimum upper bound associated with the class of displacement fields when implemented in a finite element method (Ponter and Carter [1,2]). The solutions for an Aluminum/alumina system loaded in a direction perpendicular to the direction of the fibers demonstrate the high sensitivity of this material to fluctuations in temperature. Simple approximate equations for strength values are presented in terms of nondimensional variables.

Shakedown Analysis at Finite Elastoplastic Strains and Deformations

AbstractA finite element method for shakedown analysis of structures that undergo finite elastic ‐ plastic strains and rotations is proposed The idea is the simultaneous computation of all relevant load cases coupled by a common plastic strain field.

Shakedown analysis of defective pressure vessels by a kinematic approach

In this paper, a kinematic approach and an iterative procedure, earlier proposed for limit analysis, are adopted for shakedown analysis under variable repeated loading. Reference is made to three-dimensional structures of engineering relevance, such as pressurized pipelines and vessels with fluctuating pressure and with slot damages due, e.g. to pitting corrosion. The numerical performance of the solution algorithm is investigated, and the cost-effectiveness of the proposed direct shakedown analysis method is assessed and compared to that of time-marching solutions by up-to-date codes.