Haofeng Chen

Shakedown and limit analyses for 3-D structures using the linear matching method

A recently developed method for 3-D shakedown and limit analyses is evaluated in the present paper. The shakedown and limit loads of a holed plate subjected to biaxial loading are calculated by implementing the upper bound linear matching method into the commercial FE code ABAQUS. A defective pipeline under the combined action of internal pressure and axial tension is also analysed for both shakedown and limit capacities and the results compared with a standard programming method. All the numerical examples confirm the applicability of this procedure to complex 3-D structures.

A minimum theorem for cyclic load in excess of shakedown, with application to the evaluation of a ratchet limit

Although the shakedown theorems for perfect plasticity have been known since Koiters 1960 review paper, extensions of the theory to situations where ratchetting or reverse plasticity occurs in excess of shakedown have not appeared in the literature. In this paper a generalisation of the upper bound theorem is derived which reduces to the upper bound shakedown theorem in the limiting case when the load point approaches the shakedown boundary. The new theory is used to develop a method for identifying the ratchet limit for a class of loading histories through the sequential minimisation of two functionals. A programming method, based on the Elastic Compensation method for shakedown is then derived and convergence proven. Numerical examples of the application of the method to practical problems are discussed by us in an accompanying paper.

A method for the evaluation of a ratchet limit and the amplitude of plastic strain for bodies subjected to cyclic loading

An extension of the upper bound shakedown theorem to load histories in excess of shakedown has been presented elsewhere in this issue. Here the minimisation process described therein is applied to the solutions of the ratchet limit as well as shakedown and limit load for a range of simple problems. The solutions provide an estimate of the maxima of the varying plastic strain magnitudes, which is compared with the Neuber approximate values. The position of the ratchet boundary is confirmed by comparison step-by-step analysis.

Lower and Upper Bound Shakedown Analysis of Structures With Temperature-Dependent Yield Stress

Based upon the kinematic theorem of Koiter, the Linear Matching Method (LMM) procedure has been proved to produce very accurate upper bound shakedown limits. This paper presents a recently developed LMM lower bound procedure for shakedown analysis of structures with temperature-dependent yield stress, which is implemented into ABAQUS using the same procedure as for upper bounds. According to the Melans theorem, a direct algorithm has been carried out to determine the lower bound of shakedown limit using the best residual stress field calculated during the LMM upper bound procedure with displacement-based finite elements. By checking the yield condition at every integration point, the lower bound is calculated by the obtained static field at each iteration, with the upper bound given by the obtained kinematic field. A number of numerical examples confirm the applicability of this procedure and ensure that the upper and lower bounds are expected to converge to the theoretical solution after a number of iterations.

A Direct Method on the Evaluation of Ratchet Limit

This paper describes a new Linear Matching Method (LMM) technique for the direct evaluation of ratchet limit of structure subjected to a general cyclic load condition, which can be decomposed into cyclic and constant components. The cyclic load history considered in the paper contains multi-load extremes to include most complicated practical applications. The numerical procedure is described which uses LMM state-of-the-art numerical technique to obtain a stable cyclic state of component, followed by a LMM shakedown analysis, to calculate the maximum constant load, i.e. the ratchet limit, which indicates the load carrying capacity of the structure subjected to cyclic load condition to withstand an additional constant load. This approach is particularly useful in conjunction with the evaluation of the stable cyclic response, which produces the cyclic stresses, residual stresses and plastic strain ranges for the low cycle fatigue assessment. A benchmark example of holed plate under the combined action of cyclic thermal load and constant mechanical load is presented to verify the applicability of the new ratchet limit method, through the comparison with published results by a simplified method assuming a cyclic load with two extremes. To demonstrate the efficiency and effectiveness of the method for complicated cyclic load condition with multi-load extremes, a composite thick cylinder with radial opening subjected to cyclic thermal loads and constant internal pressure is analyzed using the proposed ratchet limit method. The further verification by the Abaqus step-by-step inelastic analysis demonstrates that the proposed new method provides a general-purpose technique for the evaluation of ratchet limit, and has both the advantages of programming methods and the capacity to be implemented easily within a commercial finite element code Abaqus.

On the solution of limit load and reference stress of 3-D structures under multi-loading systems

The concepts of limit load and reference stress have been widely used in structural engineering design and component integrity assessment, especially considering multi-loading systems. The limit analysis of structures and the reference stress method (RSM) have been proven to be successful in problems pertaining to two-criteria failure assessment, creep growth, rupture damage, and more recently, elastic-plastic fracture toughness. However, the determination of limit load and reference stress of 3-D structures under multi-loading systems is not a simple task. In the present paper, a solution method for radial loading is presented, the mathematical programming formulation is derived for the upper bound limit analysis of 3-D structures under multi-loading systems, and moreover, a direct iterative algorithm used to determine the reference stress is proposed which depends on the evaluation of limit load. The penalty function method is used to deal with the plastic incompressibility condition. All the numerical examples show that the proposed radial loading path scheme is reasonable and effective. The mathematical programming method without search used here can overcome the difficulties caused by the nonlinearity and nonsmoothness of the objective function and avoid the complicated computations of incremental elastic-plastic analysis.

Numerical analysis of limit load and reference stress of defective pipelines under multi-loading systems

Abstract The concepts of limit load and reference stress have been widely used in structural engineering design and component integrity assessment, especially in Nuclear Electrics (formerly CEGB) R5 and R6 procedures. The reference stress method has been proven to be successful in problems pertaining to creep growth, rupture damage, creep buckling, and more recently, elastic–plastic fracture toughness. An approximate method of reference stress determination relies on prior knowledge of limit loads for various configurations and loadings. However, determination of the limit loads for the problems with complicated geometric forms and loading conditions is not a simple task. In the present paper, a numerical solution method for radial loading is presented, the mathematical programming formulation is derived for the kinematic limit analysis of 3D structures under multi-loading systems, and moreover, a direct iterative algorithm used to determine the reference stress is proposed which depends on the evaluation of limit load. The numerical procedure is applied to determine the limit load and reference stress of defective pipelines under multi-loading systems. The effects of four kinds of typical part-through slots on the collapse loads of pipelines are investigated and evaluated in detail. Some typical failure modes corresponding to different configurations of slots and loading forms are studied.

Linear matching method for creep rupture assessment

The recently developed linear matching method (LMM), which is easily implemented within commercial FE codes, has been successfully used to evaluate elastic and plastic shakedown loads. In this paper, the method is extended to the prediction of the creep rupture life of a structure, based upon a bounding method currently used in the life assessment method R5. The method corresponds to the requirement that, for the operating load history, the structure should shakedown where the yield stress is given by the lesser of the plastic yield stress and a high temperature rupture stress corresponding to a rupture time. A holed plate subjected to cyclic thermal load and constant mechanical load is assessed in detail as a typical example to confirm the applicability of the above procedures. The examples show that the method remains numerically stable, even when the method is inverted.

Plastic collapse analysis of defective pipelines under multi-loading systems

Abstract In this paper, a finite element mathematical programming formulation is presented for the kinematic limit analysis of 3-D rigid–perfectly plastic bodies. A numerical path scheme for radial loading is adopted to deal with complex multi-loading systems. A direct iterative algorithm is employed in solving the above optimization formulation. The numerical procedure has been applied to carry out the plastic collapse analysis of defective pipelines under multi-loading systems. The engineering situation considered has a practical importance in the pipeline industry. The effects of four kinds of typical part-through slots on the collapse loads of pipelines are investigated and evaluated. Some typical failure modes corresponding to different configurations of slots and loading forms are studied.

On Shakedown, Ratchet and Limit Analyses of Defective Pipeline

In this study, the limit load, shakedown and ratchet limit of a defective pipeline subjected to constant internal pressure and a cyclic thermal gradient are analyzed. Ratchet limit and maximum plastic strain range are solved by employing the new Linear Matching Method (LMM) for the direct evaluation of the ratchet limit. Shakedown and ratchet limit interaction diagrams of the defective pipeline identifying the regions of shakedown, reverse plasticity, ratcheting and plastic collapse mechanism are presented and parametric studies involving different types and dimensions of part-through slot in the defective pipeline are investigated. The maximum plastic strain range over the steady cycle with different cyclic loading combinations is evaluated for a low cycle fatigue assessment. The location of the initiation of a fatigue crack for the defective pipeline with different slot type is determined. The proposed linear matching method provides a general-purpose technique for the evaluation of these key design limits and the plastic strain range for the low cycle fatigue assessment. The results for the defective pipeline shown in the paper confirm the applicability of this procedure to complex 3-D structures.