Development of the large increment method for elastic perfectly plastic analysis of plane frame structures under monotonic loading

The displacement-based finite element method dominates current practice for material nonlinear analysis of structures. However, there are several characteristics that may limit the effectiveness of this approach. In particular, for elastoplastic analysis, the displacement method relies upon a step-by-step incremental approach stemming from flow theory and also requires significant mesh refinement to resolve behavior in plastic zones. This leads to computational inefficiencies that, in turn, encourage the reconsideration of force-based approaches for elastoplastic problems.One of these force algorithms that has been recently developed is the large increment method. The main advantage of the flexibility-based large increment method (LIM) over the displacement method is that it separates the global equilibrium and compatibility equations from the local constitutive relations. Consequently, LIM can reach the solution in one large increment or in a few large steps, thus, avoiding the development of cumulative errors. This paper discusses the extension of the large increment methodology for the nonlinear analysis of plane frame structures controlled by an elastic, perfectly plastic material model. The discussion focuses on the power of LIM to handle these nonlinear problems, especially when plastic hinges form in the frame and ultimately as the structure approaches the collapse stage. Illustrative planar frame examples are presented and the results are compared with those obtained from a standard displacement method.     

Elasto-plastic vibrational analysis of tapered bars under uniform axial loading considering shear deformation and rotary inertia

The analysis of structures in the plastic regime is important in order to develop an adequate and competitive design in engineering. This paper presents a study of the small-amplitude free vibration of tapered bars under pre-stress in the post-elastic regime due to a uniform axial loading. The plastic behavior is taken into account using an energy approach. The method does not require an iterative procedure, unlike conventional methods used in plasticity. The Timoshenko beam theory and the dynamic version of the principle of virtual work are used to derive the eigenvalue problem. The solution is carried out using beam finite elements. The results are validated using 3D finite element software and results from the open literature. A variety of numerical results are given in order to analyze the influence of plastic behavior for various bar geometries and material parameters. The combined effect of the stiffening due to the axial loading and the plastic softening may produce an increase or decrease of the natural frequencies as the tensile load increases. The plastic softening effect is seen to be pronounced for short bars and for high taper ratios. In addition, axial normal modes are more affected than bending modes.     

Lower bound shakedown analysis by the symmetric Galerkin boundary element method

In this paper, the static shakedown theorem is reformulated making use of the symmetric Galerkin boundary element method (SGBEM) rather than of finite element method. Based on the classical Melan’s theorem, a numerical solution procedure is presented for shakedown analysis of structures made of elastic-perfectly plastic material. The self-equilibrium stress field is constructed by linear combination of several basis self-equilibrium stress fields with parameters to be determined. These basis self-equilibrium stress fields are expressed as elastic responses of the body to imposed permanent strains obtained through elastic–plastic incremental analysis. The lower bound of shakedown load is obtained via a non-linear mathematical programming problem solved by the Complex method. Numerical examples show that it is feasible and efficient to solve the problems of shakedown analysis by using the SGBEM.     

Derivation of geometric stiffness and mass matrices for finite element hybrid models

An unifying approach in deriving the geometric stiffness and mass matrices for finite element hybrid models is presented. The variational formulation is based on a modified Reissner Principle. Numerical verification is illustrated through a simple beam example. The element interchangeability in the finite element method and an alternative way of deriving the loading vector for the hybrid stress model are also discussed.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Analytical elasto-creep model of interfacial thermal stresses and strains in trilayer assemblies

A two-dimensional model has been developed for thermal stresses, elastic strains, creep strains, and creep energy density at the interfaces of short and long trilayer assemblies under both plane stress and plane strain conditions. Both linear (viscous) and non-linear creep constitutive behavior under static and cyclic thermal loading can be modeled for all layers. Interfacial stresses and strains are approximated using a combination of exact elasticity solutions and elementary strength of materials theories. Partial differential equations are linearized through a simple finite difference discretization procedure. The approach is mathematically straightforward and can be extended to include plastic behavior and problems involving external loads and a variety of geometries. The model can provide input data for thermal fatigue life prediction in solder or adhesive joints. For a typical solder joint, it is demonstrated that the predicted cyclic stress–strain hysteresis shows shakedown and a rapid stabilization of the creep energy dissipation per cycle in agreement with the predictions of finite element analysis.     

Physics-based modeling for partial slip behavior of spherical contacts

A physics-based modeling approach for partial slip behavior of a spherical contact is proposed. In this approach, elastic and elastic–plastic normal preload and preload-dependent friction coefficient models are integrated into the Cattaneo–Mindlin partial slip solution. Partial slip responses to cyclic tangential loading (fretting loops) obtained by this approach are favorably compared with experiments and finite element results from the literature. In addition to load-deformation curves, tangential stiffness of the contact and energy dissipation per fretting cycle predictions of the models are also provided. Finally, the critical assumptions of elastically similar bodies, smooth contact surface and negligible adhesion, and limitations of this physics-based modeling approach are discussed.     

A crack tip tracking algorithm for cohesive interface element analysis of fatigue delamination propagation in composite materials

A novel approach is proposed for the use of cohesive elements in the analysis of delamination propagation in composite materials under high-cycle fatigue loading. The method is applicable to delamination propagation within the Paris-law regime and is suitable for the analysis of three-dimensional structures typical of aerospace applications. The major advantages of the proposed formulation are its complete independence of the cohesive zone length – which is a geometry-dependent parameter – and its relative insensitivity to mesh refinement. This is only possible via the introduction of three nonlocal algorithms, which provide (i) automated three-dimensional tracking of delamination fronts, (ii) an estimation of direction of crack propagation and (iii) accurate and mesh-insensitive integration of strain energy release rate. All calculations are updated at every increment of an explicit time-integration finite element solution, which models the envelopes of forces and displacements with an assumption of underlying constant cyclic loading. The method was implemented as a user-defined subroutine in the commercial finite element software LS-Dyna and supports the analysis of complex three-dimensional models. Results are presented for benchmark cases such as specimens with central cut plies and centrally-loaded circular plates. Accurate predictions of delamination growth rates are observed for different mesh topologies in agreement with the Paris-laws of the material.     

Validation of the tangent formulation for the solution of the non-linear Eshelby inclusion problem

An extension of the Eshelby problem for non-linear viscous materials is considered. An ellipsoidal heterogeneity is embedded in an infinite matrix. The material properties are assumed to be uniform within the ellipsoid and in the matrix. The problem of determining the average strain rate in the ellipsoid in terms of the overall applied strain rate is solved in an approximate way. The method is based on the non-incremental tangent formulation of the non-linear matrix behavior [Acta Metall. 35 (1987) 2983]. In the present work this approximate solution is verified with a good agreement by comparing to finite element calculations for various inclusion shapes and loading conditions.     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

Elastic–plastic FE analysis of a notched shaft under multiaxial nonproportional synchronous cyclic loading

An elastic–plastic finite element analysis is presented for a notched shaft subjected to multiaxial nonproportional synchronous cyclic tension/torsion loading. The elastic–plastic material property is described by the von Mises yield criterion and the kinematic hardening rule of Prager/Ziegler. The finite element program system ABAQUS is used to solve the boundary value problem. Special emphasis is given to explore the effects of the stress amplitude, the mean-stress, and the mutual interactions on the local stress–strain responses at the notch root.     

Stabilized finite element method for simulation of freeway traffic

The modeling of traffic flow is a key tool to simulate and predict the behavior of traffic systems. Macroscopic traffic simulation models are based on advection dominated coupled non-linear partial differential equations. The solution of such advection dominated equations with the method of finite elements is leading to the development of stabilization techniques. The choice of suitable stabilization parameters is often application-dependent. A stabilized finite element procedure on the basis of a Galerkin/least-square approximation is presented for systems of transient advection-dominated equations. A general rule for computing suitable element stabilization parameters is outlined which uses the spectral radius of the differential operators and the specific element expansion. The application of this approximation to a macroscopic traffic model shows the applicability of this approach. Simulation results of typical phenomena of jam formation in freeway traffic are presented.     

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.     

Advancements in subsea riser analysis using quasi-rotations and the Newton–Raphson method

Beam structures undergoing finite deflections and rotations in space have extensive application in the subsea industry particularly for the analysis of holistic systems with larger numbers of mooring and riser components. In using the finite element analysis approach, there is an increasing requirement for large element sizes which preserve accuracy with regard to the coupling of axial, bending and torsion response.The authors outline a method for improving the current state of practice for the analysis of riser systems. The approach draws on the convected coordinates method, Euler–Bernoulli beam theory, the principle of virtual work and the finite element method. Two quasi-rotation measures are developed including a quasi-material rotation definition for rotational deformation relative to the convected axis of a beam and a quasi-space rotation definition to deal with the path dependent nature of rotations in three dimensions.The novel aspect of this work is to relate the rate of change of the quasi-material rotation vector along the beam axis to a linear transformation of the beam axis rate-of-rotation vector through utilising the convected coordinates axes system. In this way, incremental values of quasi-material rotation are directly linked to incremental values of nodal quasi-space rotation and a global Newton–Raphson solution technique for interconnecting beam elements is straightforward to assemble.Furthermore, this leads to accurate definitions of coupled axial, bending and torque response for beams with significant deflection. The approach has particular advantages in the analysis of subsea riser sections. Also, the accuracy of the solution is preserved for a fewer number of elements compared to alternative solutions for computationally sensitive load cases with highly non-linear loading regimes.     

An FE analysis of reinforced subgrade under automobile loading

An FE analysis procedure was presented to predict the behavior of soil-geogrid interaction under automobile loading. The dynamic interactions between the transverse bars, the longitudinal ribs and the soil were simulated by a system consisting of nonlinear springs, dashpots and masses, to study the deformation properties of the reinforced soil. The equivalent stiffness and damping ratios could be determined with the shaking table. The dynamic responses of a reinforce subgrade were analyzed with the 3D finite element approach. This approach is programmed and applied to analyze the soil-geogrid interaction under dynamic loading. The comparative analysis of the response of the reinforced subgrade and that of the subgrade without reinforcement shows that the geogrid placed at the bottom of the base layer may effectively reduce the accumulative plastic deformation due to the cyclic automobile loading.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

Plane strain extension of a strip made of a material with stress state type dependent properties and weakened by cuts with circular base

The limit properties of many heterogeneous materials such as grounds, concrete, ceramics, cast-iron alloys, and various heat-resistant and powder materials, as well as the properties of many composite materials, depend on the loading conditions. Neglecting the effects exhibited by such materials may result in nonconservative limit load analysis for some types of loading and possibly in an overly increased end product weight by failing to take into account stronger material properties for other types of loading. This paper presents a possible approach to modeling the behavior of such materials under plastic deformation, which is demonstrated for the sample problem on the extension of a strip weakened by cuts with circular base. An analytic solution on the basis of a rigid-plastic model of the material and a numerical solution by the finite elementmethod with elastic strains and small strengthening taken into account are presented.     

Cyclic plasticity modeling with the distribution of non-linear relaxations approach

Motivated by the distribution of non-linear relaxation (DNLR) approach, a phenomenological model is proposed in order to describe the cyclic plasticity behavior of metals under proportional and non-proportional loading paths with strain-controlled conditions. Such a model is based on the generalization of the Gibbs's relationship outside the equilibrium of uniform system and the use of the fluctuation theory to analyze the material dissipation due to its internal reorganization. The non-linear cyclic stress–strain behavior of metals notably under complex loading is of particular interest in this study. Since the hardening effects are described appropriately and implicitly by the model, thus, a host of inelastic behavior of metals under uniaxial and multiaxial cyclic loading paths are successfully predicted such as, Bauschinger, strain memory effects as well as additional hardening. After calibrating the model parameters for two metallic materials, the model has demonstrated obviously its ability to describe the cyclic elastic-inelastic behavior of the nickel base alloy Waspaloy and the stainless steel 316L. The model is then implemented in a commercial finite element code simulating the cyclic stress–strain response of a thin-walled tube specimen. The numerical responses are in good agreement with experimental results.     

双轴反复荷载作用下钢筋混凝土空间框架结构滞回全过程分析

实际地震中结构所承受的地震作用是多维的。结构的恢复力特性是反映其抗震性能的一个重要属性。对地震作用下钢筋混凝土柱、梁塑性铰区的计算模型进行了分析,给出了塑性铰区等效长度的简便计算公式;建立了“有限纤维”空间线性梁单元模型并推导出了其刚度矩阵;使用“弥散法”来考虑梁锚固钢筋在结点区的粘结滑移对结构整体变形的影响。最后使用空间杆系模型对一承受双向反复荷载的框架结构进行了计算分析。分析表明所建立的计算模型是精确有效的。