A Study on the Elastic Compensation Method for Design by Analysis Codes

In the Design by Analysis (DBA) codes, it is important to ensure that the integrity of the structures can be maintained under the various loadings. For this purpose, elastic analyses are not sufficient and generally, elastic-plastic analyses are necessary. However, elastic-plastic analyses are still costly, especially for three-dimensional complex components. To solve this problem, the Elastic Compensation Method (ECM) has been studied, in which a conservative plastic collapse load can be obtained based on the lower bound theorem. In this paper, this investigation, including application of the ECM for combined loadings and body forces has been summarized. Application of the method on a structure consisting of dissimilar materials is also discussed.

Limit Analysis for Anisotropic Solids Using Variational Principle and Repeated Elastic Finite Element Analyses

Many structural components, such as rolled sheets, directionally 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 limit 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 inelastic FEA.

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.

Shakedown Analysis of a Finite Plate With Single Edge Notch Under Uniaxial Tension

In this paper, the shakedown load factor for a finite rectangular plate with a single edge notch at its mid-length is evaluated by a number of numerical and analytical techniques. These techniques include full elasto-plastic finite element solution coupled with line search technique and an iterative elastic technique known as the elastic compensation method which relies on successive reduction of elastic modulus in regions where elastic stresses exceed yield strength. In addition, an analytical technique proposed recently in the literature, which relies on a gradient optimization technique is employed. The shakedown load factor values obtained through different solution techniques are compared. The comparison showed discrepancies between literature solutions and the solutions presented herein. The reasons behind this discrepancy are scrutinized and certain recommendations are made to ensure validity of shakedown solutions.

Optimal Shapes for Axisymmetric Pressure Vessels: A Brief Overview

This paper describes how optimal shapes for axisymmetric pressure vessels can be established based on maximizing limit pressure. This type of problem has been rarely examined in the literature due to the difficulty of evaluating limit loads. However, the “elastic compensation method” is used to approximate the limit load using elastic analysis alone, which opens the possibility of studying shape optimization based on limit pressure. The basic procedure, using a commercial finite element analysis system, is described and three example problems are examined. The aim is to investigate how much of an increase in load-carrying capacity could potentially be achieved if nonstandard pressure vessel shapes could be employed in practice. Of course, this may not be possible, but the results described here do contribute to a better understanding of the role shape plays in providing strength to a simple pressure vessel. [S0094-9930(00)00304-8]

The elastic compensation method for limit and shakedown analysis: A review

A comprehensive review of the elastic compensation method for calculating limit and shakedown load bounds for complex structures is presented. The origins of the method in pressure vessel design by analysis is described and related methods for load and shakedown analysis considered, in particular Marriott's reduced modulus method, Seshadri's GLOSS r-node method and Ponter's modified elastic modulus method. The paper concludes with a recommendation for future work: development of an element level formulation of the method.