A unified formulation of small-strain corotational finite elements: I. Theory

Geometric nonlinearities often complicate the analysis of systems containing largedeflection members. The time and resources required to develop closed-form or numerical solutions have inspired the development of a simple method of approximating the deflection path of end-loaded, large-deflection cantilever beams. The path coordinates are parameterized in a single parameter called the pseudo-rigid-body angle. The approximations are accurate to within 0.5 percent of the closedform elliptic integral solutions. A physical model is associated with the method, and may be used to simplify complex problems. The method proves to be particularly useful in the analysis and design of compliant mechanisms

Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms

Simulation of springback

The beam flexure is an important constraint element in flexure mechanism design. Nonlinearities arising from the force equilibrium conditions in a beam significantly affect its properties as a constraint element. Consequently, beam-based flexure mechanisms suffer from performance tradeoffs in terms of motion range, accuracy and stiffness, while benefiting from elastic averaging. This paper presents simple yet accurate approximations that capture the effects of load-stiffening and elastokinematic nonlinearities in beams. A general analytical framework is developed that enables a designer to parametrically predict the performance characteristics such as mobility, over-constraint, stiffness variation, and error motions, of beam-based flexure mechanisms without resorting to tedious numerical or computational methods. To illustrate their effectiveness, these approximations and analysis approach are used in deriving the force-displacement relationships of several important beam-based flexure constraint modules, and the results are validated using finite element analysis. Effects of variations in shape and geometry are also analytically quantified.

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Characteristics of Beam-Based Flexure Modules

Advanced Issues in springback

Numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented. The values are given in tabular form with up to six significant figures. The numerical technique used for evaluating the elliptic integrals is described.

Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals

On constitutive modeling for springback analysis

On the modelling of the bending–unbending behaviour for accurate springback predictions

The importance of an accurate material modeling for the accuracy and reliability of sheet forming simulations has become increasingly evident during the last years. More advanced material models have, however, to be supported by novel methods for material characterization. The recent eight parameter yield functions Yld2000-2d and BBC2003 demand, besides data from the ordinary uniaxial tensile tests, also equibiaxial data. In the present paper a Viscous Pressure Bulge (VPB) test is described. The test yields the equibiaxial stress point and r-value, as well as a plastic hardening curve for large values of plastic strain. The test setup is based on an ARGUSS™ optical measuring system, and provides the desired result data in a very smooth and easy way. In order to verify the results from the current test, comparisons have been made with compression tests performed at Corus RD&T and hydraulic bulging tests performed at RWTH in Aachen. A discussion on how to determine the equibiaxial yield stress and how to transform the biaxial stress-strain curve to an effective stress-strain curve is included in the paper.

A viscous pressure bulge test for the determination of a plastic hardening curve and equibiaxial material data

An experimental method that frequently has been used for the determination of material hardening parameters is the three-point bending test. The advantage of this test is that it is simple to perform, and standard test equipments can be used. The disadvantage is that the material parameters have to be determined by some kind of inverse approach. The test has then been simulated by means of the Finite Element Method, and the material parameters have been determined by finding a best fit to the experimental results by means of a Response Surface Methodology. An alternative method is the tensile/compression test of a sheet strip. In practice such a test is very difficult to perform, due to the tendency of the strip to buckle in compression. In spite of these difficulties some successful attempts to perform cyclic tension/compression tests have been reported in the literature. However, a few writers have reported that there are substantial differences between hardening parameters determined from bending tests and those determined from tensile/compression tests. The purpose of the present study is to try to understand the background of these differences, to find out the influence on predicted springback, and to determine which of the two methodologies for hardening parameter identification is the most suitable one.

Kjell Mattiasson

Papers

Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals

On constitutive modeling for springback analysis

On the modelling of the bending–unbending behaviour for accurate springback predictions

A viscous pressure bulge test for the determination of a plastic hardening curve and equibiaxial material data

On the identification of kinematic hardening material parameters for accurate springback predictions