Stress-resultant Space Research Articles

Simulation of sheet metal forming processes by presenting a bending-dependent inverse isogeometric methodology

Recently, eliminating the gap between design and formability analysis of sheet metal parts has been studied to simulate sheet metal stamping processes. In this regard, a transfer-based inverse isogeometric formulation has been proposed. This method has various advantages such as solving the governing equations in two-dimensional networks without any concern about the convergence; however, it neglects the bending effect which is a major contributor in die/punch profile radii. The present work aims to consider the bending effects by introducing a bending-dependent inverse isogeometric formulation. The developed model deals with the minimization of potential energy, deformation theory of plasticity, classical plate theory, and considering a yield criterion in stress-resultant space. In addition to all advantages of the transfer-based inverse isogeometric formulation, one major benefit of this study is that the bending effects are included with a slight increase in the computation time. This methodology allows for accurately predicting the effects of changing die/punch profile radii and initial sheet thickness on the formability of the final part by presenting a new material updating process. To assess the credibility of this approach, an experimental setup and forward FEM software have been utilized to form a rectangular box. The results acquired by the developed method and those achieved by experiment and forward FEM reveal acceptable accuracy in the presented model. Also, strains and thicknesses predicted by the developed method, membrane inverse isogeometric model, and forward FEM for nine different values of punch radius to the sheet thickness ratio have been compared. Considering forward FEM as a reference method, the average of calculated error in the presented model for prediction of thickness at the middle of punch radius zone is around half of that in the membrane model. In solving the studied problems, the presented model requires only slightly more computation time (around 2%) than the membrane inverse isogeometric model and much less computation time than forward FEM. Therefore, the presented method is a valuable inverse forming solver especially when the bending effects are significant.

Simulating blast effects on steel beam-column members: Methods

A beam-column element is formulated and implemented within a nonlinear explicit dynamic finite element code for simulating explosive effects in steel skeletal structures composed of wide-flange members. Blast damage is encapsulated by a bounding surface plasticity model in stress-resultant space. Bounding surfaces are formed by fitting numerical failure data to a basis of real-valued spherical harmonics: the data are obtained through numerical experiments in which a shell element-based model of a member is exposed to a blast and then statically loaded to failure for various combinations of proportional moment-thrust loading. Code validation and examples are presented in a companion paper [1].

An inelastic Timoshenko beam element with axial–shear–flexural interaction

An enhanced beam element is proposed for the nonlinear dynamic analysis of skeletal structures. The formulation extends the displacement based elastic Timoshenko beam element. Shear-locking effects are eliminated using exact shape functions. A variant of the Bouc–Wen model is implemented to incorporate plasticity due to combined axial, shear and bending deformation components. Interaction is introduced through the implementation of yield functions, expressed in the stress resultant space. Three additional hysteretic degrees of freedom are introduced to account for the hysteretic part of the deformation components. Numerical results are presented that demonstrate the advantages of the proposed element in simulating cyclic phenomena, in which shear deformations are significant.

Inelastic Cyclic Model for Steel Braces

A beam--column element that can accurately model the inelastic cyclic behavior of steel braces is presented. A bounding surface plasticity model in stress-resultant space coupled with a backward Euler algorithm is used to keep track of spread of plasticity through the cross section. Deterioration of cross-section stiffness due to local buckling is accounted for by a damage model. The proposed formulation has been implemented in a large deformation analysis program and is shown to be capable of predicting with accuracy the experimentally observed inelastic behavior of a variety of members subjected to reversed cyclic loading and a subassemblage under simulated seismic conditions.

Evolution of stress-resultant loading and ultimate strength surfaces in cyclic plasticity of steel wide-flange cross-sections

Abstract The evolution of the shape of limit surfaces during plastic excursions is a key component of the inelastic response of structural members. However, stress-resultant-based computational plasticity formulations rarely account for any change in shape of the loading or ultimate strength surfaces. This research documents the evolution of the shape, size, and position of the loading and ultimate strength surfaces in stress-resultant space for wide-flange steel shapes subjected to cyclic proportional and nonproportional loading having a combination of axial force and either strong- or weak-axis flexure. The evolution is based on numerical integration of the cross-section stresses assuming a cyclic uniaxial stress–strain curve with a trilinear backbone that accounts for the Bauschinger effect. Surface evolution is investigated both with and without residual stresses. The results indicate that the evolution in shape of the loading surface, while relatively path-independent, is substantial, particularly at high levels of plastic excursion, where the loading surface essentially vanishes. The evolution in shape of the ultimate strength surface, in contrast, is mild. The research also documents the relative magnitudes of the components of cyclic plastic deformation. Recommendations are provided for the estimation of the direction of cyclic plastic flow through the calculation of gradients to the limit surfaces within the assumption of using an associated flow rule in stress-resultant space.

Stress-Resultant Plasticity for Frame Structures

Two versions of a bounding surface plasticity model implemented in stress-resultant space and applicable to the analysis of steel, reinforced concrete, or composite beam-columns are discussed. One is a two-surface model appropriate for steel members with a finite elastic region. The second, which is developed for reinforced concrete and composite steel-concrete members, employs a single outer bounding surface with an infinitely small loading surface that is degenerated to a point. Plasticity-based assumptions employed in the formulation of these models are reviewed and predicted plastic flow directions are evaluated against data from more fundamental fiber-type analyses of the beam-column cross sections. Results of these comparisons support the use of Mroz's kinematic rule in stress-resultant space and lead to recommended improvements in the bounding surface formulation.

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.

A Hardening Rule Between Stress Resultants and Generalized Plastic Strains for Thin Plates of Power-Law Hardening Materials

For power-law hardening materials, a stress resultant constitutive law of incremental plasticity nature for thin plates is constructed. The yield function in the stress resultant space is approximated in quadratic form and an equivalent stress resultant is defined. One of two parameters in the yield function is analytically determined based on the concept of complementary potential surface. The other is determined by the least square method to fit the complementary potential surface of Chou et al., (1991). In analogy to the work of Hill (1979), the equivalent work-conjugate generalized plastic strain rate is derived. Finally, the hardening rule between the equivalent stress resultant and generalized plastic strain is obtained based on the results of Chou et al. (1991) for power-law materials under proportional straining conditions.

Exact ilyushin yield surface

The approximate Ilyushin yield surface is widely used in structural calculations to represent full plasticity in stress-resultant space. The exact version of the surface has not been used, because the parametric form in which it was described by Ilyushin was not amenable to calculation. This paper presents a reparametrization of the Ilyushin yield criterion for thin plates which produces a simpler (though still exact) form which is suitable for use in practical formulations. A method is outlined by which the exact Ilyushin yield criterion can be used in such calculations, without the need to use approximate yield functions. A method for calculating positions on the surface corresponding to a linear multiple of any set of stress-resultants is presented. Various features of the exact yield surface are described.

A yield surface equation for doubly symmetrical sections

A four-parameter yield surface equation for the yield strength of doubly symmetrical steel sections (wide-flange, thin-walled circular tube, thin-walled box, rectangular and solid circular sections) under axial force combined with biaxial bending is presented. The proposed yield surface equation represents a smooth and convex surface in the three-dimensional stress-resultant space. Extensive comparisons have been made with the results of exact yield surfaces, providing the confirmation of the validity of the proposed equation. The proposed equation meets all special conditions and is easy to implement in a computer-based structural analysis. It is a useful limiting surface equation for most doubly symmetrical sections, and is therefore recommended for general use.