Elastic Compensation Method Research Articles

A computed tomography-based limit analysis approach to investigate the mechanical behavior of the human femur prone to fracture

The paper proposes a refined CT-based FE modelling strategy that implements a limit analysis numerical procedure, namely the Elastic Compensation Method (ECM), to estimate a lower bound to the collapse load of a human femur. In particular, the model geometry was obtained from CT images by segmentation of a fresh-frozen human cadaveric femur that was discretized with second-order tetrahedral 3D finite elements. A yield criterion of Tsai–Wu-type, expressed in principal stress space, was adopted to model the bone tissues for which the strength limit values in tension, compression and shear are computed locally from the femoral density distribution also derived from CT images. The developed CT-based numerical technique showed the ability to predict, at least for the examined femur for which the experimental collapse load is available, a lower bound to the collapse load. The proposed approach seems a promising and effective tool that could be adopted into clinical practice to predict the fracture risk of human femur starting from patient-specific data given by medical imaging.

Sequential elastic adaptive NS-FE analyses for lower-bound limit load determination of plane-strain structures

The paper proposes the sequential elastic analyses run within the node-based smoothed finite element (NS-FE) framework incorporating an automatic adaptive mesh scheme to converge the lower-bound collapse load limit of plane-strain structures. The approach is familiar to practical designers as it solely involves a series of linear elastic analysis solves. Each of which performs a modified version of elastic compensation method that appropriately considers the modulus variations of some critical members. The nonlinear constitutive formulations describing intrinsic material properties can be directly accommodated. The NS-FE method simply adopts the low-order displacement functions, whilst the mesh adaptive algorithm employs the newest-bisection criteria with the novel modulus variation error indicators. A number of numerical examples and benchmarks involving the plane-strain structures with vertex and curve geometries illustrate the accuracy and efficiency of the proposed method. The approach overcomes the challenges associated with stress singularity and volumetric locking phenomena in an incompressibility condition. The collapse load solution converges to the lower-bound limit at modest computing efforts, where the final NS-FE layout (namely fine meshes over the areas developing high modulus variation rates and at the same time coarse meshes for those undergoing elasticity) depicts the failure lines of structures and hence mechanisms.

Determination of Collapse Load of Engineering Structures using Iterative Node-based Smoothed Finite Element Analysis Method

This paper presents an iterative limit analysis scheme modeled within a three-node node-based smoothed finite element framework incorporating a so-called modified elastic compensation method to determine the ultimate bearing capacity of ductile structures. More explicitly, the approach iteratively reduces those elemental Young’s modulus associated with a high-stress intensity such that the inelastic behavior of the elastic-perfectly plastic material model is taken into account. The approach provides good accuracy and convergence of solution through a series of standard linear elastic analyses that make possible the application in large-scale engineering structures, even the one with complex geometric configurations, and hence eminently accessible to structural engineers. As a result, complex inelastic analyses are eliminated and computational cost is significantly reduced. Several benchmark examples, one of which reported, illustrates the efficiency and robustness of the iterative limit analysis approach in tracing the load multipliers of in-plane ductile structures from the beginning until convergence of collapse load solution or failure state occurs. The yield areas at the ultimate state gradually appear in the graph of elastic modulus distribution.

An iterative elastic SBFE approach for collapse load analysis of inelastic structures

The paper proposes a novel numerical approach that incorporates the use of a modified elastic compensation method, within a polygon scaled boundary finite element framework, to determine the maximum load capacity of structures at plastic collapse. The distinctive feature of the proposed scheme is its effective computational ability in performing a series of successive elastic analyses by systematically adjusting elastic material properties of structures up to failure. The quadtree structural discretization within a polygon scaled boundary finite element platform enables model construction of sophisticated geometries at modest computing effort and thus the effective analysis of large-scale structures. The approach overcomes the challenges associated with stress singularity and locking phenomena under incompressibility conditions, even in the presence of high-order nonlinear yield loci. The robustness and accuracy of the proposed scheme are validated through a number of benchmarks and available practical engineering applications in 2D and 3D spaces. These illustrate the influences of some key algorithmic parameters, and the satisfaction of a lower-bound limit given by the present analysis method for a sufficiently fine discretization.

Ultimate load prediction of MMNCs structures

Abstract The paper proposes a nonlocal limit analysis procedure to evaluate the load bearing capacity of metal matrix nanocomposites (MMNCs) structural elements. The promoted procedure rephrases and extends, in a nonlocal context, a static limit analysis numerical method, known in literature as elastic compensation method. Due to the ductility and the intrinsic nonlocal behaviour of MMNCs materials, a nonlocal elastic-perfectly plastic constitutive behaviour is hypothesized with the further hypothesis that nonlocality belongs only to the elastic phase. An associative von Mises type yield condition together with a nonlocal elasticity model of integral type are adopted. A nonlocal formulation of the finite element method is used throughout and a numerical application is presented and critically discussed with the aim to test the capability of the method in capturing the main features of the novel nonlocal limit analysis approach.

A modified algorithm of linear matching method for limit analysis

Linear matching method has been widely used for the numerical analysis of limit and shakedown. It has been proved theoretically that linear matching method could offer the monotonically reducing sequence of upper bound. Nevertheless, it still remains open whether linear matching method can obtain the conversed and reliable lower bound or not. Thus, an elastic compensation method is used generally for the evaluation of lower bound, but limit analysis using linear matching method and elastic compensation method needs double iterative computations. Moreover, the convergence can be checked only after the computation is finished because linear matching method and elastic compensation method cannot be performed simultaneously. From this, we propose a simple method in order to improve the numerical solution of lower bound by linear matching method. The Young’s modulus varying spatially is determined in every iteration such that not only the stress state lies on the yielding surface but also the strain state does not exceed a certain value of reference strain, leading to the evaluation of lower bound based on the strain state but not the stress one. The proposed method can improve the numerical solution of lower bound by linear matching method without any affection on the upper bound. ANSYS UserMat is used for implementing the current method. The limit analysis is performed like the general elastic finite element analysis in ANSYS. Some numerical examples are considered in order to confirm the effectiveness of proposed approach. Numerical examples showed the validity and improvement of numerical accuracy of our approach. It should be mentioned that our approach can predict the lower bound and upper one simultaneously within the framework of only linear matching method without using the elastic compensation method.

Two‐dimensional lower bound analysis of offshore pile foundation systems

SummaryThe objective of this paper is to present a simplified method to determine the pile foundation system capacity based on the lower bound theorem of plasticity. The motivations for determining the lower bound capacity are the following: (1) to evaluate the accuracy of solutions based on the upper bound method; (2) to provide a conservative and efficient solution to the system capacity; and (3) to provide information about load distribution among individual piles at the verge of failure for the pile system. The failure mechanisms for a single pile and for the pile system are assumed to be two‐dimensional. For a typical long offshore pile, the upper and lower bound analyses produce identical lateral capacities. A simplified failure surface for loads at the single pile head is proposed and verified through analysis of 16 case study piles. With this proposed failure surface for a single pile, the lower bound failure load of the pile foundation system is obtained using the elastic compensation method enhanced with the linear matching method. Comparing with the existing upper bound and finite element solutions, the proposed lower bound method is capable of accurately and efficiently predicting the ultimate capacity of a pile foundation system. Copyright © 2015 John Wiley & Sons, Ltd.

Contribution to pressure vessels design of innovative methods and comparative application with standardized rules on a realistic structure – Part I

Design of pressure vessels, which are subjected to various natures of loading, must prevent damage mechanisms occurrence. For a load applied or maintained with a given intensity, primary failure modes can appear, such as gross plastic deformation, plastic instability or buckling. For design-by-analysis, the reference methodology is based on an elastic stress calculation. During the last decade, studies have shown that this ingenious procedure could provide conservative design limits. They can become actually overly conservative in a context of increasing complexity of geometry and loading modelling. In parallel, technological and theoretical developments enabled limit analysis to be considered as an interesting design methodology. This is suggested in standards and codes (EN 13445, CODAP, Boiler and Pressure Vessels Code) since the early 2000's. In this first of two companion papers, a set of standardized and innovative procedures is introduced. These approaches rely on various concepts, such as elasticity, incremental elastoplasticity, or elastic compensation (Modified Elastic Compensation Method, Linear Matching Method). Each methodology is presented on theoretical aspects, eventually adapted so as to take into account safety margins. They are then applied on a model inspired from a real industrial reactor, using Abaqus. Results are compared to reference data from codes, in terms of accuracy and computing time. A final assessment underlines practical benefits that could be expected.

Failure analysis of FML plates with cutouts: Experimental and finite element approaches

In this paper, the experimental and numerical failure analysis of Fiber Metal Laminates (FML) with dif-ferent types of cutouts, were investigated. Fiber metal laminates are types of materials which are consist of com-bination of metal sheets, especially aluminum ones, with fiber reinforced epoxy layers. At first, specimens were manufactured and elliptical and circular cutouts were cre-ated in them. Then subjected to in-plane tensional loading to carry out the behaviors, also, finite element analysis was carried out with ABAQUS commercial software. RIKS method was used in this analysis. Finally, experimental and finite element results were compared with results of elastic compensation method. DOI: http://dx.doi.org/10.5755/j01.mech.20.1.3530

Peak load prediction of multi-pin joints FRP laminates by limit analysis

The paper presents a wide range of numerical findings aimed at validating a design methodology founded on the application of two Finite Element based limit analysis numerical procedures, namely the Linear Matching Method and the Elastic Compensation Method. Such methodology is here used to determine upper and lower bounds to the peak load multiplier of multi-pin joints composite laminates. In modeling the composite orthotropic material a Tsai–Wu type yield criterion is utilized, while the real stacking sequence of the laminate is considered applying the limit analysis procedures at lamina (layer) level. The obtained results on real prototypes, by comparing numerical tests against experimental (laboratory) ones, seem to prove the ability of the proposed approach to bracket the real peak load value of the examined structural elements.

Mechanically fastened joints in composite laminates: Evaluation of load bearing capacity

Mechanical fasteners, commonly used in many advanced engineering applications dealing with composite laminates, play the main role of transferring loads between the linked structural elements. The present study, focused on pinned-joints, explores the possibility of applying a direct method for evaluating the joint’s strength as well as for predicting some of the more common joint’s failure mechanisms. A limit analysis numerical approach for statically loaded pinned-joint orthotropic laminates in plane stress conditions is proposed. Two well known numerical procedures for limit analysis likewise having common roots, namely the Linear Matching Method and the Elastic Compensation Method, are utilized to evaluate upper and lower bounds to the joint collapse load. Both methods are rephrased assuming for the material in use a Tsai–Wu type yield surface. The results obtained are compared and plotted against some available experimental findings. Some final remarks draw attention to the potentialities and the limits of the proposed approach.

Upper and Lower Bound Limit Loads for Thin-Walled Pressure Vessels Used for Aerosol Cans

The elastic compensation method proposed by Mackenzie and Boyle is used to estimate the upper and lower bound limit (collapse) loads for one-piece aluminium aerosol cans, which are thin-walled pressure vessels subjected to internal pressure loading. Elastic-plastic finite element predictions for yield and collapse pressures are found using axisymmetric models. However, it is shown that predictions for the elastic-plastic buckling of the vessel base require the use of a full three-dimensional model with a small unsymmetrical imperfection introduced. The finite element predictions for the internal pressure to cause complete failure via collapse fall within the upper and lower bounds. Hence the method, which involves only elastic analyses, can be used in place of complex elastic-plastic finite element analyses when upper and lower bound estimates are adequate for design purposes. Similarly, the lower bound value underpredicts the pressure at which first yield occurs.

Limit analysis of structures containing flaws based on a modified elastic compensation method

The elastic compensation method (ECM) based on the conventional linear elastic finite element method is a simple and effective method for structural limit analysis. However, for structures containing flaws, computational errors of the ECM are relatively greater and numerical singularity is often caused before a better solution can be achieved. Firstly, the present paper employs the Banach's contraction mapping theorem in mathematical analysis to discuss the convergence problem of the iterative method of the ECM. The discussions demonstrate that only when iterative elastic modulus sequences are contraction mappings can a good limit load solution be obtained. Secondly, a modified elastic compensation method (KtECM) is proposed, in which iterative elastic modulus sequences of structural main load-carrying elements can satisfy the condition of contraction mapping. Moreover, an adjustable factor related with structural stress concentration factor (Kt) is introduced to define a rational nominal stress and to dynamically balance the computational precision and CPU time consumed. Finally, we perform limit load analyses for several representative structures containing flaws. These solutions obtained by the KtECM are compared with results from the analytical method, the elastic-plastic analysis method (EPAM) and the ECM. It reaches the conclusion that the KtECM can provide good estimations of plastic limit loads for structures containing flaws. The KtECM has the advantages of simplicity, high efficiency and convenience for engineering applications.

Modified elastic compensation method for limit analysis of complex structures

The elastic compensation method (ECM) for struc- tural limit analysis is a simple and effective method for simple structures. However, relatively greater computational errors and divergence are often caused for complex structures. Addressing on these problems, the present paper employs the fixed point theorem in Banach space to discuss the convergence problem of the ECM for lower bound limit load calculations. It can be pointed out that a good limit load solution can be achieved only when the iterative elastic modulus sequences satisfy the condi- tion of contraction mapping. Based on this idea, a modified elastic compensation method (KtECM) is proposed. The KtECM adopts an iterative method, in which the iterative elastic modulus sequences of the main load-carrying elements in a structure satisfy the condition of contraction mapping. At the same time, an adjustable factor λ related with the stress con- centration factor (Kt) of structures is used to define a rational nominal stress. Limit loads of several complex structures are calculated by different methods. It reaches the conclusion that the KtECM can provide a good estimation of plastic limit loads for complex structures and preserves the advantages of simplic- ity, high efficiency and convenience for engineering applica- tions. The adjustable factor λ can make a balance between computational precision and time.

Limit analysis based on a modified elastic compensation method for nozzle-to-cylinder junctions

To address the computational difficulties of the elastic compensation method (ECM) for complex structures such as nozzle-to-cylinder junctions, this paper develops a modified elastic compensation method (MECM). This method improves the precision of the ECM while preserving the advantages such as simplicity and high efficiency. Limit loads are calculated for three representative examples. The calculated solutions are compared with results from the elastic-plastic analysis method and the twice-elastic slope method. It is found that the MECM can provide a good estimation of plastic limit loads for complex structures.

LIMIT ANALYSIS FOR ANISOTROPIC SOLIDS USING VARIATIONAL PRINCIPLE AND REPEATED ELASTIC FINITE ELEMENT ANALYSES

Many structural components, such as rolled sheets, directtonally solidified superalloys and composites, are made of anisotropic materials. The knowledge of limit load is useful in the design and the sizing of these components and structures. This paper presents the extension of the modified mα - method to anisotropic materials. Mura’s variational principle is employed in conjunction with repeated elastic finite element analyses (FEA). The secant modulus of the discretized finite elements in the reference direction in successive elastic iterations is used to estimate the plastic flow parameter for the anisotropic components. The modified initial elastic properties are adopted to ensure the “elastic” stress fields satisfy the anisotropic yield surface. Using the notion of “leap-frogging to iimit state,” improved lower-bound limit loads can be obtained. The formulation is applied to two anisotropic components, and the limit load estimates are compared with those using elastic compensation method and inelasti...

Simplified lower bound limit analysis of transversely loaded thin plates using generalised yield criteria

In this paper a simple method of deriving lower bound limit loads for thin plates is presented. This method is based on the method of elastic compensation, an iterative elastic technique, which has recently been extended to allow the analysis of structures using thin shell finite element analysis using generalised yield criteria. Here the method is modified to allow analysis of plates, including the effects of transverse shear. The elastic compensation method, combined with generalised yield criteria, is implemented using the finite element numerical analysis technique. Convergence studies are carried out and limit loads are obtained for a range of geometries, boundary conditions and loading. The calculated limit loads are compared with results available in the literature and with new elasto-plastic results and show that the method can be used to quickly obtain practical results.

Upper and lower bound limit and shakedown loads for hollow tubes with axisymmetric internal projections under axial loading

In this paper, the elastic compensation method proposed by Mackenzie and Boyle is used to estimate the upper and lower bound limit (collapse) loads and the upper and lower bound shakedown loads for hollow tubes with axisymmetric internal projections subjected to axial loading. The method is based on an iterative elastic analysis procedure and the application of lower and upper bound limit load theorems. Four different geometries with a range of stress concentration factors (from low to high) are considered. Elastic-plastic finite element predictions for collapse and shakedown pressure are found to be within these upper and lower bound estimates. The method is particularly useful because it is founded on an iterative elastic approach and does not require extensive and complex elastic-plastic finite element computations.

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 Koiter's 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.

125 Elastic Compensation法の応用に関する一検討(GS01 シミュレーション・不安定変形)

125 Elastic Compensation法の応用に関する一検討(GS01 シミュレーション・不安定変形)