Melan's Theorem Research Articles

Shakedown of a plate with a circular hole: An educational problem

The shakedown limit for an infinite plate containing a circular hole and subject to oscillatory arbitrary remote loading is found, first by using Melan’s lower bound theorem, and secondly by using a finite element model. It is shown that in some cases the limit found from the Melan theorem using the solution for the residual state of stress for an over-pressurized hole provides the exact solution, specifically when the limiting factor is the reversal of the state of stress at a particular point, while in other cases the shakedown limit is rather higher.

Shakedown analysis on the integrity of cement sheath under deep and large-scale multi-section hydraulic fracturing

In the process of deep shale gas exploitation through horizontal drilling and large-scale multi-section hydraulic fracturing, the cement sheath can easily fail due to the high peak pressure and the frequent and violent fluctuations. In this paper, firstly, the deformation characteristics of the cement was studied based on the uniaxial and triaxial compression experiments at different temperature conditions. Then, mechanical model of cement sheath was set up. Based on Melan's shakedown theorem, shakedown load of the cement sheath under the periodic alternating loads is obtained and then converted to the internal pressure of the casing. Results shows that: (1) Compared with that at 25 °C, the compression resistance and elastic modulus of cement reduced significantly under 130 °C. The cement exhibits obvious plastic flow characteristics and can be regarded as a rational elastic-plastic material. (2) During the cyclic loading and unloading process, new plastic strains are continuously generated, the width of hysteresis loops is continuously reduced, and the hysteresis loop becomes denser, which indicates that the accumulation rate of plastic deformation is continuously decreasing. (3) The shakedown load is the smaller of the loading and unloading yield pressures, and reverse yielding tend to occur in the depths. (4) The shakedown load of the cement sheath has a positive linear and non-linear relationship with the cohesive force and the friction angle, respectively. (5) the shakedown load has a positive linear relationship with the external pressure. The external pressure, as the most influence of the calculation result of the shakedown load, accounts for close to 80%. The research could provide valuable theoretical and engineering guidance for the design and construction of shale gas well cement.

Limit analysis of conical and parabolic domes based on semi-analytical solution

The evaluation of limit loads of masonry domes has received increasing interest especially due to the importance of historical buildings where domes mainly are one of the most relevant structures. The limit design is used to obtain the safety assessment and the design guidance for restoration and transformation toward preservation and reuse of historical heritage. In the following paper, we present a formulation of the limit analysis based on the semi-analytical approach that starts on Melan's theorem. The self-equilibrated Melan's residual is obtained through the discretization of the analytical form of the equilibrium equation of the spherical dome. The procedure provides a finite-dimensional map of the eigenstress of the structure. Furthermore, the superimposition of the elastic solution to actual loads, obtained by finite element calculation, completes the admissible stress evaluation. Such amissible stress is introduced into the maximization algorithm, based on the lower bound theorem, which results in the collapse load. The same approach is used to get the safety assessment under prescribed load that allows checking the safety of prescribed load pattern and geometry.

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.

A direct method-based strength evaluation of the cast aluminum beam used in a high-speed train

In this paper, the authors elaborate how numerical techniques developed from the direct method can be used to predict the load-bearing capacity of the aluminum casting beam structure which is presently used in the Chinese high-speed train CRH5A. The numerical method introduced in this paper is formulated based on the Melan's static theorem; thus, it determines the strength of the cast aluminum beam under both monotonic and cyclic loads without following the entire load profile. In addition to constructing the plastic and shakedown limits of the component, the proposed approach is also employed to study how stiffeners' thickness on the key areas influences the feasible load domains. Based on the intensive sensitivity analysis, an optimal thickness has been determined which gives the highest strength-to-weight ratio. To this end, the paper confirms that the direct method is a viable means for designing structures used in the rolling stock.

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.

Shakedown analysis of elastic-plastic structures considering the effect of temperature on yield strength: Theory, method and applications

According to the extended Melan's static theorem, theoretical and numerical aspects of the stress compensation method (SCM) are presented to perform shakedown analysis of elastic-plastic structures considering the effect of temperature on yield strength. Instead of constructing a mathematical programming formulation, this developed method consists of the two-level iterative scheme. The inner loop constructs the statically admissible self-equilibrating stress field, while the outer loop evaluates a sequence of decreasing load factors to approach to the shakedown limit multiplier. The yield strength considering temperature effect is updated based on the current temperature at each outer iteration, and the yield conditions are checked at all Gauss points. The numerical procedure is well incorporated into ABAQUS finite element code and used for calculating the shakedown limits of structures considering yield strengths as different functions of temperature under complex thermomechanical loading system. The method is validated by some plane stress and axisymmetric numerical examples with theoretical and numerical solutions, and subsequently applied to solve the practical shakedown problem of a pipe with oblique nozzle. The results demonstrate that the developed method is stable, accurate and efficient, and can effectively evaluate the shakedown limit of an elastic-plastic structure where the yield strength of material varies with temperature.

Ratcheting in pressurized pipes and equipment: A review on affecting parameters, modelling, safety codes, and challenges

AbstractThe current study has attempted to comprehensively review ratcheting response of materials involving various influential parameters such as loading spectra, thermal cycles, stress levels, stress raisers, strain rate, and visco‐plasticity and material types with a focus on pressurized pipes and equipment. The mechanism of deformation, types, and the rate of progress over stages of ratcheting were discussed. Safety design codes and procedures were highlighted for reliable design of pressurized pipes against progressive ratcheting and to safeguard the system against catastrophic failure at which both mechanical and thermal ratcheting were integrated. Boundaries and demarcation of ratcheting‐shakedown zones developed based on Bree's diagram were employed to assess plastic deformation accumulation over stress cycles. Shakedown and ratcheting boundaries were discussed through methods developed on the basis of Melan's theorem over last few decades. Ratcheting response of materials was reviewed through descriptions of parametric models and kinematic hardening rules involving various variables and parameters. Interaction of ratcheting with creep and low‐cycle fatigue has promoted progressive damage in materials. Pressurized pipes subjected to thermal cycles and external bending cycles were evaluated by numerical solutions along with kinematic hardening rules to assess ratcheting over stress cycles.

Three-dimensional shakedown analysis of ballasted railway structures under moving surface loads with different load distributions

Shakedown of ballasted railway structures was analyzed based on Melan's shakedown theorem, in which the wheel/rail contact was approximated by Hertz (circular and elliptical contact areas), uniform and trapezoidal load distributions, separately. The shakedown solutions incorporating to the three-dimensional finite element model calculated shakedown multiplier by means of a self-equilibrated critical residual stress field. The shakedown multiplier for multi-layered ballasted railway structure was determined as the minimum one among all layers. The results showed that elliptical Hertz and uniform load yielded the largest and smallest shakedown limits, respectively, with the maximum difference of approximately 64%. The shakedown limits always occurred at ballast layer for relatively small frictional coefficient, whilst occurred at rail for low rail's yield stress with large frictional coefficient. As expected, the shakedown limits decreased as the ballast stiffness and thickness increased, especially for relatively small frictional coefficient; while increased with raising rail's yield stress. The material properties and thickness should therefore be optimally designed so as to provide a maximum resistance to the structure failure and reduce the material costs.

Solution validation technique for optimal shakedown design problems

Modern computer technology allows conjoining shakedown theory, optimization and ever stricter standardized design requirements in a single mathematical problem formulation. However it raises a question of reliability: easily achieved solution should not be taken for granted but should be adequately assessed. This paper focuses on the physical validation technique for optimal shakedown design problem solution in the aspect of Melan theorem (statics) and residual deformation compatibility (kinematics). For that purpose Rosen gradient projection method is used. Optimization problem of bending circular, symmetric plate at shakedown, which is subjected by a variable repeated load, is considered for illustration of the validation technique.

Shakedown of discrete systems involving plasticity and friction

Abstract In associated plasticity, systems subjected to cyclic loading are sometimes predicted to shake down , meaning that, after some dissipative cycles, the response goes back to a purely elastic state, where no plastic flow occurs. Frictional systems show a similar behaviour, in the sense that frictional slips due to the external loads may cease after some cycles. It has been proved that, for complete contacts with elastic behaviour and Coulomb friction, Melan's theorem gives a sufficient condition for the system to shake down, if and only if there is no normal-tangential coupling at the interfaces. In this paper, the case of discrete systems combining elastic-plastic behaviour and Coulomb friction is considered. In particular, it is proved that Melan's theorem still holds for contact-wise uncoupled systems, i.e., the existence of a residual state, comprised of frictional slips and plastic strains, is a sufficient condition for the system to shake down.

Ellipsoidal load-domain shakedown analysis with von Mises yield criterion: A robust optimization approach

Summary The static formulation of elastic shakedown analysis, based upon Melan's lower-bound theorem, can essentially be viewed as a robust optimization problem. This paper discusses an advantage that is enjoyed by taking this perspective. Specifically, we assume the von Mises yield criterion and an ellipsoidal load domain. In this setting, the shakedown analysis problem, which is viewed as robust second-order cone programming, can be recast as semidefinite programming. Copyright © 2016 John Wiley & Sons, Ltd.

An extended shakedown theory on an elastic–plastic spherical shell

The article analyses one of possible optimization methodologies for ideal elastic–plastic structures at shakedown and its application for shallow spherical shells having prescribed geometry and affected by a variable repeated load (VRL) – a system of external forces that may vary independently of each other. The paper accepts that only time-independent upper and lower bounds of variations in external forces are given. The pronounced effect of external forces, i.e. in this context, possible histories of variations in forces, is not examined (the unloading phenomenon of cross-sections is ignored in the course of plastic deformation). The discussed concept of the structure at shakedown refers to the Melan theorem related to statically allowable admissible internal forces. Thus, for the discretization of the spherical shell, with the help of an assumption about small displacements, the equilibrium finite element method based on internal force approximation is applied. The limit axial force of the cross-section is supposed to be constant within the bounds of the finite element, and only the optimal distribution of limit internal forces among elements, according to the selected criterion, is in need of search. The article presents a discrete mathematical model for determining the optimal allocation problem with strength and stiffness requirements. The model conforms to the limit axial force of the shallow spherical shell of the variable repeated load and takes into account ultimate and serviceability limit states of EC3 with corresponding reliability levels. Structural optimization methods refer to extreme energy principles of mechanics and are illustrated with the numerical examples of spherical shell optimization.

Ratchet Limit Solution of a Beam With Arbitrary Cross Section

The classical approaches in shakedown analysis are based the assumption that the stresses are eventually within the elastic range of the material everywhere in a component (elastic shakedown). Therefore, these approaches are not very useful to predict the ratcheting limit (ratchet limit) of a component/structure in which the magnitude of stress locally exceeds the elastic range at any load, although in reality the configuration will certainly permit plastic shakedown. In recent years, the “noncyclic method” (NCM) was proposed by the present authors to predict the entire ratchet boundary (both elastic and plastic) of a component/structure by generalizing the static shakedown theorem (Melan's theorem). The fundamental idea behind the proposed method is to (conservatively) determine the stable and unstable boundary without going through the cyclic history. The method is used to derive the interaction diagrams for a beam subjected to primary membrane and bending with secondary bending loads. Various cross-sections including rectangular, solid circular and thin-walled pipe are investigated.

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.

Lower Bound Methods in Elastic-Plastic Shakedown Analysis

Several methods were proposed in recent years that allow the efficient calculation of elastic and elastic-plastic shakedown limits. This paper establishes a uniform framework for such methods that are based on perfectly-plastic material behavior, and demonstrates the connection to Melan's theorem of elastic shakedown. The paper discusses implications for simplified methods of establishing shakedown, such as those used in the ASME Code. The framework allows a clearer assessment of the limitations of such simplified approaches. Application examples are given.

Application of the Shakedown Theory to Brittle-Matrix Fiber-Reinforced Cracked Composite Beams Under Combined Traction and Flexure

The cracking behavior of a composite beam with multiple reinforcing fibers under periodic traction-flexure is analyzed through a fracture mechanics-based model, where the edge-cracked beam section is exposed to external loads and crack bridging reactions due to the fibers. Assuming a rigid-perfectly plastic bridging law for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging forces are obtained from compatibility conditions. Under general load paths, shakedown conditions are explored by making use of the Melan's theorem, here reformulated for the discrete problem under consideration, where crack opening displacement at the fiber level plays the role of plastic strain in the counterpart problem of an elastic-plastic solid. The limit of shakedown is determined through an optimization procedure based on a linear programming technique.

Frictional systems under periodic loads — History-dependence, non-uniqueness and energy dissipation

Nominally static contacts such as bolted or shrink-fit joints typically experience regions of microslip when subjected to oscillatory loading. This results in energy dissipation, reflected as apparent hysteretic damping of the system, and also may cause the initiation of fretting fatigue cracks. Early theoretical studies of the Hertzian contact problem by Cattaneo and Mindlin were confirmed experimentally by Johnson, who identified signs of fretting damage in the slip annulus predicted by the theory.For many years, tribologists assumed that Melan's theorem in plasticity could be extended to frictional systems — i.e. that if there exists a state of residual stress associated with frictional slip that is sufficient to prevent periodic slip in the steady state, then the system will shake down, regardless of the initial condition. However, we now know that this is true only if there is no coupling between the normal and tangential loading problems, as will be the case notably when contact occurs on a symmetry plane.For all other cases, periodic loading scenarios can be devised such that shakedown occurs for some initial conditions and not for others. The initial condition here might be determined by the assembly protocol — e.g. the order in which a set of bolts is tightened — or by the exact loading path before the steady cycle is attained. This non-uniqueness of the steady state persists at load amplitudes above the shakedown limit, in which case there is always some dissipation, but the dissipation per cycle (and hence both the effective damping and the susceptibility to fretting damage) depends on the initial conditions. This implies that fretting fatigue experiments need to follow a well-defined assembly protocol if reproducible results are to be obtained. We shall also present results showing that when both normal and tangential forces vary in time, the energy dissipation is very sensitive to the relative phase of the oscillatory components, being greatest when they are out of phase.With sufficient clamping force, 'complete' contacts (i.e. those in which the contact area is independent of the normal load) can theoretically be prevented from slipping, but on the microscale, all contacts are incomplete because of surface roughness and some microslip is inevitable. In this case, the local energy dissipation density can be estimated from relatively coarse-scale roughness models, based on a solution of the corresponding 'full stick' problem.

Calculation of a lower bound ratchet limit part 2 – Application to a pipe intersection with dissimilar material join

In an accompanying paper in this issue a lower bound method based on Melan's theorem was derived and implemented into the Linear Matching Method ratchet analysis procedure. This paper presents a ratchet analysis of a pipe intersection subject to cyclic thermo-mechanical loading using the proposed numerical technique. This work is intended to demonstrate the applicability of the lower bound method to a structure commonly seen in industry and also to better understand the behaviour of this component when subjected to cyclic loading. The pipe intersection considered here has multiple materials with temperature dependent properties. Verification of the results is given via full elastic-plastic analysis in Abaqus.

Calculation of a lower bound ratchet limit part 1 – Theory, numerical implementation and verification

It is important to be able to calculate the ratchet limit of a component when performing integrity assessments of plant components. This paper details the addition of a lower bound ratchet limit calculation to the Linear Matching Method. The extension of Melan's theorem into the alternating plasticity region is explained, followed by its implementation into the Linear Matching Method calculation procedure. Finally, the convergence properties of this method are analysed by the analysis of a plate with a central hole subject to cyclic thermal and mechanical loading.