Elastoplastic Beam Research Articles

A hysteresis model for timber joints with dowel-type fasteners

Predicting the mechanical behaviour and the failure mechanism of timber joints with dowel-type fasteners requires consideration of several factors, including the geometrical and mechanical properties of the metal fastener, the physical properties of timber and the interaction between such elements. This paper proposes a numerical model where a joint is schematized as an elasto-plastic beam in a non-linear medium with a compression-only behaviour. Unlike the differential approach adopted by most of the hysteresis models published in literature, this model predicts the load-displacement response using simple mechanical relationships and basic input parameters. Furthermore, the model is capable of reproducing the effect of the cavity formed around the fastener by timber crushing, and simulates the hysteretic behaviour and the energy dissipation under cyclic conditions. Shear tests are reproduced on nailed steel-to-timber joints in Cross-Laminated Timber and results are compared to the experimental test data obtained on similar single fastener joints. Simulations lead to accurate predictions of both the mechanical behaviour (initial stiffness, maximum load-carrying capacity, global shape of the loading curve and of the hysteresis cycles) and the total energy dissipation observed in the tests.

Simulation of inextensible elasto-plastic beams based on an implicit rate type model

The aim of this paper is to show that it is possible to model the elasto-plastic behavior of an inextensible beam undergoing finite bending by using a novel implicit rate type response relation directly between the bending moment, curvature, and their rates. This is in contrast to conventional approaches that require the integration of elasto-plastic stress–strain relations across the thickness of the beam at each time step. The model proposed here (a) is rate independent and exhibits hysteresis, (b) requires no notion of plastic strain and no integration across the thickness, and (c) does not have a sharp yield point and instead transits smoothly between nominally elastic and inelastic response. The governing equations are solved numerically as a set of first order differential equations with very low order interpolation functions for quasi-static response and are capable of modeling both hardening and softening behavior as well as the formation of plastic hinges, etc. Since there is no yield function, there is no necessity to use a return mapping algorithm that was developed specifically for use in elasto-plasticity with sharp yield functions. This simplifies the numerical algorithm also. The simulations for an elastic perfectly plastic beam are compared with elasto-plastic models in ABAQUSTM, and they show good agreement at a fraction of the computational time. Moreover we have compared simulations of large deformations of an aluminum alloy bar subject to three point bending experiments. The simulations also match the experimental results very well. We demonstrate that it is possible to incorporate frictionless sliding contact (which is needed for the three point bending comparison) in a relatively straightforward manner without the complexity that arises when it is handled in a full three-dimensional (3D) model.

Incremental Analysis of Elastoplastic Beams and Frames Resting on an Elastic Half-Plane

AbstractIn this work, a simple and effective finite element-boundary integral equation (FE-BIE) approach for the static analysis of elastic beams and frames in bilateral frictionless contact with a...

Modeling Load-Displacement Hysteresis Relationship of a Single-Shear Nail Connection

AbstractA finite element based single-shear nail-connection model has been developed based on an existing computer model which idealizes dowel-type wood connection as an elastoplastic beam surround...

Large Elasto-plastic Deflection of Thin Beams with Roller Support Contact

Here, an explicit numerical approach is presented to solve the problem of a three-point bending of a thin elasto-plastic beam undergoing large deflection supported on cylindrical rollers with radius comparable to the deflection. There are three sources of non-linearity in this problem: roller to beam contact, plastic deformations and large deflections. An incremental form of moment- curvature based constitutive law is derived from a uniaxial linearly hardening stress-strain material model. The governing boundary value problem in terms of its slope is transformed into an initial value problem, with the domain up-to the point of contact. The non-linear initial value problem is linearized about the current step and solution for the subsequent step is obtained by employing the classical Runge-Kutta fourth-order method. This solution procedure is repeated for a range of lengths and end angles of beam. Subsequently, a feasible data set is created from the solution space which satisfied the contact configuration condition. It is found that the increase in the radius of roller introduces a stiffening effect in the force response of the structure. A springback analysis is also performed for the beam data from a feasible set which satisfies plastic deformation condition. It is found that springback decreases with the increase in the radius of roller supports.

Exact stress and deformation analysis in elastoplastic Ramberg–Osgood beam

Elastoplastic deformation and stress distributions in a thin or classical Euler–Bernoulli beam are derived by an exact solution methodology. In the formulations phase, the Hencky deformation theory of plasticity is employed. It is assumed that material of Ramberg–Osgood behavior yields according to von Mises criterion. Using the principals of variational calculus and assuming a tri-axial field of stress, governing equations of elastoplastic beam are extracted. Exact solution of resulting nonlinear integro-differential equations are obtained by using the analytical methods of homotopy and Adomian. In the exemplary study of clamped–clamped beam, the fields of deformation are compared with the results of ABAQUS software. Different graphical representations are provided to show the results of analytical solutions and computer simulations. The level of consistency between the computer simulations and solutions of Adomian and homotopy techniques is assessed. At the end, the capability of classical engineering beam theory for the elastoplastic analysis is assessed.

Elastoplastic Analysis of Mechanical Response of Buried Pipelines under Strike-Slip Faults

AbstractA two-dimensional elastoplastic analytical model of stress–strain analysis of the pipeline under strike-slip faults was established based on the plastic flow theory and elastoplastic-beam theory, considering the effects of service loads and the nonlinearities of the pipeline and the pipe–soil interaction. According to the plastic flow theory, the axial, radial, and hoop plastic strains were derived, and the stress–strain relation was established under the plane stress state. The axial stress and strain of the pipeline induced by fault displacements were analyzed based on the elastoplastic beam theory. Additionally, the axial stresses, hoop stresses, and strains induced by the service loads (i.e., the internal pressure and the temperature variation) were also considered. The proposed model was verified by comparing the analytical results and the finite-element results. The results show that the proposed model can predict accurately the mechanical response of the pipeline subjected to strike-slip fa...

Dynamic buckling of an imperfect elastoplastic beam subjected to an impulsive load analyzed using transport equations

In this study, we use transport equations to investigate the dynamic buckling of an imperfect elastoplastic beam, including the effect of axial stress waves, when subjected to an impulsive load. Thus, the governing equations for the elastoplastic behavior of the material are extracted using mass, energy, and momentum transport equations. We assume that the passage of the shock wave through the control volume creates five phases with four boundary discontinuities, i.e., the initial elastic phase, initial plastic phase, fluid phase, secondary plastic phase, and secondary elastic phase. Transport equations are used in integral form as well as non-physical variables to eliminate the discontinuity conditions in the governing equations. These equations are also used for continuum modeling of the elastoplastic behavior of the beam under an impulsive load and a continuous model is presented. Finally, the time histories of stress, strain, and velocity wave propagation along the beam are presented, and the results are validated based on comparisons with known solutions.

An incremental approach for springback analysis of elasto-plastic beam undergoing contact driven large deflection

In this article the solution methodology for a beam on a vee-die undergoing large elasto-plastic deflection along with nonlinear contact development with the die is discussed. A bi-linear stress strain material model is converted into an incremental moment- curvature based constitutive law to ease formulation. The one dimensional governing equation obtained, is highly nonlinear owing to material and geometry and involve boundary condition change. The entire problem is solved in three steps: solving an end loaded cantilever under non-conservative force, followed by choosing the solutions which satisfy the configurational constraint and finally reanalyzing the contact data for springback analysis. The end loaded cantilever problem is solved by an incremental procedure coupled with Runge–Kutta fourth-order explicit initial value solver. Suitable normalization of the pertinent variables of the governing equation paved the way to identify dependence of the responses on a unique non-dimensional parameter. The presented methodology doesn't involve large matrix inversion and so is computationally economic. It may be used in sheet metal manufacturing control facilities to predict springback and reduce the expensive experimental iterations.

Numerical analysis of highly deformable elastoplastic beams

The objective of the present study is to develop a numerical formulation to predict the behavior of highly deformable elastoplastic thin beams. Following the Euler-Bernoulli bending, the axial and shear effects are neglected, and the nonlinear second-order differential equation regarding the angle of rotation is defined based on the specific moment-curvature relationship. Although the formulation can be used for general materials, three constitutive models are employed: linear-elastic, bilinear elastoplastic, and linear-elastic with Swift isotropic hardening. The resultant boundary value problem is solved by means of the fourth-order Runge-Kutta integration procedure and the one-parameter nonlinear shooting method. The performance of the present formulation is investigated via three numerical problems involving finite bending of slender beams composed of elastoplastic materials. For these problems, numerical solutions regarding rotations, displacements and strains for the loading, unloading and reloading phases are provided. Finally, it is shown that the present methodology can also be used to determine the post-buckling behavior of elastoplastic thin beams.

Modelling of the stiffness evolution of truss core structures damaged by plastic buckling

A finite element study based on 1D beam element model is performed in order to investigate the mechanical behavior of an elasto-plastic beam loaded in axial compression over its buckling limit. The mode of loading is related to the damage of truss-cored beams in truss-cored laminates. The analysis takes into account the effects of geometry and material properties. The results of the FEM analysis are used for developing a simple mechanical model based on the basic Euler–Bernoulli beam theory and accounts for the beam compressibility. The model uses phenomenological functions containing parameters related to the basic material and geometrical properties. The presented model is developed in the form of closed solution which does not require complex numerical methods or extensive parametric studies. Predictions of the compressive stiffness degradation of truss-cored composites are made with the proposed model and compared with the results of FEM simulations. The error of the stiffness prediction with respect to the FEM results is within 10% over a 5 fold range of stiffness.

RESIDUAL DISPLACEMENTS‘ PROGRESIVE ANALYSIS OF THE MULTISUPPORTED BEAM / DAUGIAATRAMĖS SIJOS LIEKAMŲJŲ POSLINKIŲ PROGRESINĖ ANALIZĖ

This paper focuses on a shakedown behaviour of the ideally elasto-plastic beams system under variable repeated load. The mathematical models of the analysis problems are created using numerical methods, extremum energy principles and mathematic programming. It is shown that during the shakedown process the residual displacements vary non-monotonically. By solving analysis problem, where the load locus is being progressively expanded, it is possible to determine the upper and lower bounds of residual displacements. Suggested methods are ilustrated by solving multisupported beam example problem. The results are obtained considering principle of the small displacements. Nagrinėjamas idealiai tampriai plastinės lenkiamos strypinės sistemos prisitaikomumo būvis, veikiant kartotinei kintamajai apkrovai. Analizės uždavinių matematiniai modeliai sudaromi, pasitelkus skaitinius metodus, ekstreminius energinius principus ir matematinį programavimą. Parodoma, kad prisitaikant konstrukcijai jos liekamieji poslinkiai gali kisti nemonotoniškai. Išsprendus analizės uždavinį, kuriame progresyviai plečiama apkrovos veikimo sritis, galima nustatyti viršutines ir apatines liekamųjų poslinkių kitimo ribas. Siūloma metodika iliustruota daugiaatramės sijos liekamųjų poslinkių skaičiavimo pavyzdžiu. Rezultatai gauti, esant mažų poslinkių prielaidai.

Cyclic plasticity of trusses

In this paper, the Preisach model of hysteresis, which was already successfully implemented for solving problems of cyclic plasticity of axially loaded bar and cyclic bending of elastoplastic beam, is extended to the structural analysis of trusses subjected to cyclic loading. Although there are numerous well-known models of cyclic plasticity defined, in this paper, it is shown that for uniaxial stress state, hysteresis can be defined, based on the experimental data, in one particular rigorously mathematical form and implemented in finite element equations for trusses subjected to cyclic loading in the plastic domain. Using finite element method, equilibrium equations are obtained and algorithm for numerical solution is defined. Some advantages of this approach are underlined, compared with already existing procedures, such as the Bouc–Wen model implemented in SAP2000 program and generalized plasticity model shown on various numerical examples.

A new phase field model for material fatigue in an oscillating elastoplastic beam

We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.

Nonlinear dynamic analysis of beams with layered cross sections under moving masses

This paper considers the nonlinear dynamic analysis of moving masses applied on beams with layered cross sections. This mechanical system is often used in passenger transportation systems, such as high-velocity trains and magnetic levitation transport vehicles. The structural systems are modeled numerically with the aid of a formulation based on the Finite Elements Method. The main objective of this paper was to present an original numerical approach for a mass-beam system using the concept of positional geometric nonlinearity, which uses nodal positions rather than nodal displacements to describe the finite elements kinematics. By using this methodology, analyses can be performed for moving masses traveling with constant or variable velocity in dimensionless or physical problems. For the analyzed problems, sets of results are presented for the mechanical behavior analyses, such as the moving mass trajectories along the mass-beam coupled system. Moreover, a comparative study is performed on beams with rectangular and rail cross sections. After the passage of the moving masses, the elastoplastic rectangular cross-sectional beam presents a higher stiffness than the elastoplastic rail cross-sectional beam, as a result of the double plasticity that occurs at the extreme surfaces. Another original numerical example presents a beam made of two different materials under the action of a moving mass.

Non-isothermal cyclic fatigue in an oscillating elastoplastic beam with phase transition

We propose a temperature dependent model for material fatigue accumulation, including phase transition in an oscillating elastoplastic beam. The full system of equations consists of the momentum and energy balance equations, equation for the phase transition, and an evolution equation for the fatigue rate. The fatigue rate is supposed to be proportional to the dissipation rate. Due to the phase transition, fatigue rate can also decrease but only in the case of strong melting.

Non-isothermal cyclic fatigue in an oscillating elastoplastic beam

We propose a temperature dependent model for fatigue accumulation in an oscillating elastoplastic beam. The full system consists of the momentum and energy balance equations, and an evolution equation for the fatigue rate. The main modeling hypothesis is that the fatigue accumulation rate is proportional to the dissipation rate. In nontrivial cases, the process develops a thermal singularity in finite time. The main result consists in proving the existence and uniqueness of a strong solution in a time interval depending on the size of the data.

A comparative study of displacement and mixed‐based corotational finite element formulations for elasto‐plastic three‐dimensional beam analysis

The purpose of this paper is to present eight local elasto-plastic beam element formulations incorporated into the corotational framework for two-noded three-dimensional beams. These formulations c ...

Deformation of a three-layer elastoplastic beam on an elastic foundation

We study the thermal force bending of an elastoplastic three-layer beam with a rigid filler; the beam is connected with an elastic foundation. The broken normal hypothesis is adopted to describe the kinematics of a packet nonsymmetric with respect to the thickness. The foundation reaction is described by Winkler’s model. The system of equilibrium equations and its exact solution in displacements are obtained and numerical results for a three-layer metal-polymeric beam are presented.

An introduction to nonlinear problems with random parameters by the Stochastic perturbation‐based FEM

AbstractThe main aim of this paper is to present an algorithm and the solution to the nonlinear problems with random parameters. The methodology is based on the generalized nth order stochastic perturbation method and, on the other hand, on the Finite Element Method adjacent to the physical and geometrical nonlinearities. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is proposed in two different ways – thanks to the straightforward differentiation of the initial incremental equation and, separately, using the modified Response Function Method. This approach is illustrated with the analysis of the simply supported elastoplastic beam loaded centrally with the concentrated force, where the probabilistic moments for the limit load and the reliability indices are determined via the stochastic symbolic computations. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)