Physical Nonlinearity Research Articles

The influence of the reinforced concrete deformability in the design of slender columns

This work aims to evaluate the influence of the main parameters related to the deformability of reinforced concrete columns in the design of pin-ended rectangular slender columns, under combined bending and axial force. For this, the objective is to analyse the influence of the elasticity modulus of the concrete and the type of coarse aggregate on the deformability of slender columns and consequent dimensioning, and to propose an adaptation of the parabolic law, inserting the influences of the coarse aggregate and the initial tangent modulus of elasticity in this equation, being able to use it to evaluate the local second order effects. Concrete characteristic strengths in the range 20–50 MPa (Group I) and 55–90 MPa (Group II) and CA-50 steel (fyk=500MPa) and their respective constitutive laws are considered. Geometric non-linearity is defined from the exact solution of the second-order differential equation for two simultaneous loading cases: geometric imperfection of the column axis and equal moments applied at its extremities. Physical non-linearity is defined from the moment–curvature diagram of the cross-section under the action of the compression force. The constitutive laws used in this diagram are: the bilinear law with horizontal threshold for steel, and for concrete in compression, the parabola-rectangle diagram on the dimensioning of the cross-section at the ultimate limit state under combined bending and axial force, and the law given in MC-90, MC-2010 [13], EC2-2010 [17] and in the Brazilian Bridge Standard ABNT NBR 7187 [20]. It is also examined the parabola of degree n, corrected for the local physical non-linearity consideration (i.e. of the column’s length). These laws explicitly or implicitly include the modulus of elasticity of concrete and the type of aggregate in it, through the factor αe defined in ABNT NBR 6118 [14]. The concrete strength in tension is disregarded in this work, that is, the tension-stiffening effect for the cracked column is taken equal to zero, which is on the safe side in case of slender columns, as its cracking moment increases with increasing axial compression force. Besides, this strength depends also on the height of the cross-section, frequently considered in prestressed beams. Another important parameter, besides the concrete characteristic strength, is the geometric ratio of the total reinforcement. Thus, it is possible to determine more precisely and securely the second order effect of the slender column, and the consequent variation in reinforcement in relation to the basic case in which αe=1. The results showed that the influence of the type of aggregate can be considered irrelevant for the determination of reinforcement of columns with low slenderness. However, for columns with high slenderness and high strength concrete, the type of aggregate had a significant influence on the design of these columns, where the column with aggregate αe=0,7 required twice the reinforcement compared to the column with aggregate αe=1. In addition, a parabolic law adapted to analyse the second order effects of slender columns is proposed with the replacement of fckby∝0∙αe∙fcd0.

Elastic-viscoplastic analysis of plate bending with Reissner's theory by the Boundary Element Method

The development of a formulation of the Boundary Element Method (BEM) for application to elastic-viscoplastic analysis of plate bending is presented in this work. Reissner's theory is used, valid for thin and thick plates. Initially, Reissner's plate bending theory is presented, including the consideration of physical nonlinearity, and then, integral equations applied to Reissner's plates are shown. The theory for considering viscoplasticity is presented, as well as the procedures for its use with the BEM. In order to solve the equations that govern the problem, an incremental procedure is used and von Mises’ and Tresca's yield criteria are considered. This process offers an alternative method of solution for elasto-plastic problems, when steady-state condition is reached. Numerical examples are presented and results are compared with available solutions.

Finite element modeling of the joint action of flow slide and protective structure

Introduction. In the context of the problem of plane deformation, a finite-element model of a natural landslide slope is developed. It allows for the joint work of a flow slide and a protective engineering structure. The Drucker-Prager model is used to take into account the physical nonlinearity of the slope layer material. To activate the kinematic instability, a viscoelastic interlayer is introduced into the design scheme, along which the landslide layer slides. Materials and Methods. Numerical experiments were performed using the ANSYS Mechanical software package, which implements the finite element method in the form of the displacement method. Slope discretization is performed on the basis of PLANE42 flat four-node finite elements. To simulate the displacement of the landslide layer relative to the fixed base, the combined viscoelastic elements COMBIN14 were used. Results. A physically nonlinear model of a natural landslide slope consisting of a base, a landslide layer, and a viscoelastic interlayer, is formalized. An engineering technique for analyzing the stress-strain state of the “slopeprotective structure” system has been developed, taking into account the kinematic instability of the landslide layer. A series of computational experiments was carried out. Discussion and Conclusion. Based on the calculations performed, it is shown that the proposed method enables to specify the force action of the landslide layer on the protective structure and, thereby, to increase the reliability of the risk assessment when activating the landslide process.

Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach

This paper applies a robust generalised proportional integral (GPI) controller to address the problems of stabilisation and position tracking in voltage-controlled magnetic levitation systems, with consideration of the system’s physical parameters, non-linearities and exogenous disturbance signals. The controller has been developed using as a basis a model of the tangent linearised system around an arbitrary unstable equilibrium point. Since the approximate linearised system is differentially flat, it is therefore controllable. This flatness gives the resulting linearised system a relevant cascade characteristic, thus allowing simplification of the control scheme design. The performance of the proposed GPI controller has been analysed by means of numerical simulations and compared with two controllers: (i) a standard proportional integral derivative (PID) control, and (ii) a previously designed exact feedforward-GPI controller. Simulation results show that the proposed GPI control has a better dynamic response than the other two controllers, along with a better performance in terms of the integral squared tracking error (ISE), the integral absolute tracking error (IAE), and the integral time absolute tracking error (ITAE). Finally, experimental results have been included to illustrate the effectiveness of the proposed controller in terms of position stabilisation and tracking performance when appreciable non-linearities and uncertainties exist in the underlying system. Comparative graphs and metrics have shown a superior performance of the proposed GPI scheme to control the magnetic levitation platform.

НЕСУЩАЯ СПОСОБНОСТЬ ФЛАНЦЕВОГО СОЕДИНЕНИЯ БАЛОК. ПОКРЫТИЕ ОДНОЭТАЖНОГО ПРОМЫШЛЕННОГО ЗДАНИЯ (ОПЗ) С УЧЕТОМ ЕГО ГРИБОВИДНОСТИ

The article deals with issues related to the analysis of steel frame structures. The subject of the study is the construction sector. The object of research is flanged joints of beams of coverings of one-storey industrial buildings. In order to achieve these goals, the following methods are used: analysis, synthesis, description, generalization and comparison. According to the results of the work, the features of forming the quality of interaction between flanged joints and high-strength bolts in the form of a nodal connection of coating beams are determined. Additionally, the analysis of features during the design work and production of flanged connections used for the implementation of popular design solutions for steel cross frames of single-storey buildings is carried out. The main directions of analysis of the state of steel structures in the presence of signs of deformation (mushroom shape) of the flange connection are disclosed. The obtained data helps to optimize the composition of structural reinforcement elements, as well as reduce the level of labor intensity in the production and installation of flange joints of single-storey buildings’ coating beams. In the course of the study, an experiment is conducted in which the principle of symmetry is used. The unidirectional connection in and between nodes is modeled using the elements considered. In this case, it is necessary to mention the phenomenon of the adhesive effect of the nodes during compression and the lack of adhesion during tension. The model takes into account the physical nonlinearity of the material properties, the behavior of which is set by bilinear dependencies under load

Piercing soft solids: A mechanical theory for needle insertion

In this paper we investigate the mechanical problem of piercing a soft solid body with a needle. This phenomenon is controlled by the critical condition of needle insertion. Needle insertion involves physical and geometrical nonlinearities and a complex failure mechanism. To overcome the complexity of the problem, we describe needle insertion as a sharp transition between two needle-specimen configurations, namely ‘indentation’ and ‘penetration’. The sharp configurational change emerges from a mechanical instability and follows the principle of energy minimum. We describe the needle-specimen system in terms of the force applied to the back of the needle and the axial displacement of the needle tip toward the material. At small needle displacements, the energetically favoured configuration is indentation. Conversely, when the needle displaces beyond a critical threshold, it penetrates the specimen by rupturing its surface. This creates a new energetically favoured configuration: penetration. Our analysis considers a cylindrical needle with a spherical tip, neglects friction and adhesion between the needle and the material, and assumes quasi-static conditions. Despite the mathematical simplicity of our analysis, our theoretical predictions on the needle insertion force have been validated against experiments with surprising accuracy. Our method provides an effective predictive tool, which can be extended to account for different indenter geometry and material behavior.

Convergence of the finite element method and the semi-analytical finite element method for prismatic bodies with variable physical and geometric parameters

In this paper, a numerical study of the convergence of solutions obtained on the basis of the developed approach [1, 3, 4, 5] is carried out. A wide range of test problems for bodies with smoothly and abruptly varying physical and geometric characteristics in elastic and elastic-plastic formulation are considered. The approach developed within the framework of the semi-analytical method to study the stress-strain state of inhomogeneous curvilinear prismatic bodies, taking into account physical and geometric nonlinearity, requires substantiation of its effectiveness in relation to the traditional FEM and confirmation of the reliability of the results obtained on its basis.
 The main indicators that allow comparing the SAFEM and FEM include the rate of convergence of solutions with an increase in the number of unknowns and the amount of charges associated with solving linear and nonlinear equations. For the considered class of problems, the convergence is determined by such factors as the nature of the change along Z3’ of the geometric and mechanical parameters of the object. The uneven distribution of mechanical characteristics is associated with the presence of the initial heterogeneity of the material, the development of plastic deformations, and the dependence of material properties on temperature. The same factors also affect the convergence of the iterative process, since the conditionality of the SAFEM matrix depends on them. In order to determine the area of effective application of the SAFEM, a wide range of test cases are considered.
 In all cases, the semi-analytic finite element method is not inferior in approximation accuracy, and in some problems it is 1.5-2 times superior to the traditional method of scheduling elements. finite element method.

Experimental study of nonlinearity of the mechanical properties of layered woven CFRP

The article presents the results of an experimental study of the nonlinearity of the mechanical properties of layered composite materials made by vacuum infusion based on carbon fabrics and an epoxy resin. The analysis of existing approaches to the modeling of composite materials considering their physical nonlinearity is carried out. A method is proposed for taking into account the degradation of the properties of a composite material, based on the fact that the damage parameters are replaced by the functional dependence of elastic constants on deformations, based on experimental data. The results of the application of this method are given for the test data analysis of two types of layered woven carbon-fiber-reinforced-plastics with a cross-hatching structure.

Питання міцності і стійкості неоднорідних конструкцій ракетно-космічної техніки при урахуванні пластичності і повзучості

Shell structures provide a compromise between strength and mass, which motivates their use in rocket and space hardware (RSH). High and long-term loads cause plastic and creep deformations in structural elements. RSH structures feature inhomogeneity: design inhomogeneity (polythickness, the presence of reinforcements, openings, etc.) and technological inhomogeneity (defects produced in manufacturing, operation, storage. and transportation, defects produced by unforeseen thermomechanical effects, etc.). These factors, which characterize structural inhomogeneity, are stress and strain concentrators and may be responsible for an early failure of structural elements and inadmissible shape imperfections. In inhomogeneous structures, different parts thereof are deformed by a program of their own and exhibit a different stress and strain level. In accounting for a physical nonlinearity, which is governed by plastic and creep deformations, the following approach to the determination of the stress and strain field is efficient: the calculation is divided into stages, and at each stage parameters that characterize the plastic and creep deformations developed are introduced: additional loads in the equilibrium equations or boundary conditions, additional deformations, or variable elasticity parameters (the modulus of elasticity and Poisson’s ratio). Successive approximation schemes are constructed: at each stage, an elasticity problem is solved with the introduction of the above parameters. Special consideration is given to the determination of the launch vehicle and launch complex life. This is due to damages caused by alternate high-intensity thermomechanical loads. The basic approach relies on the theory of low- and high-cycle fatigue. The plasticity and the creep of a material are the basic factors in the consideration of the above problems. This paper considers various aspects of the solution of RSH strength and stability problems with account for the effect of plastic and creep deformations.

Energetic effects analysis of shear waves and their second harmonics in a waveguide with non-ideal component contact

In this article the energetic and kinematic effects that occur in the elastic shear wave and its second harmonics propagation are investigated. The waveguide consists of anisotropic elastic crystal layer of cubic system m3m class enclosed between crystal halfspaces of same anisotropy class. A slipping contact is assumed in the contact zone of waveguide parts. The research is based on a model of general geometrical end physical nonlinearity in dynamic deformation processes. It allows to use elastic potential with the quadratic and cubic deformation components and the deformations with nonlinear terms. The approach of nonlinear elastic wave characteristics expansion into rows of a small parameter is used. Due to this approach at the first stage it's necessary to solve the problem of finding the components of the localized shear wave displacement vector (the problem of the first approximation). In the second stage, using the obtained results of the first approximation problem, the representation of the components of the displacement vector for the second harmonics of the localized elastic wave is solved in analytical form (the problem of the second approximation). By using the obtained kinematic results, the energy effects can be evaluated in the form of a vector of the average for the period of power flow. Specific results of the study of the amplitude-frequency and energy characteristics of the localized shear type elastic waves in the considered waveguide structure were obtained using computer algebra methods.The calculations of cinematic and energetic characteristics (that in contrast to linear SH harmonic are P-SV type waves) have been carried out for NaCl layer and germanium halfspaces waveguide.

КОЛЕБАНИЯ И УСТОЙЧИВОСТЬ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНОЙ ВЕСОМОЙ БУРИЛЬНОЙ КОЛОННЫ ДЛЯ УГЛУБЛЕНИЯ НЕФТЕГАЗОВЫХ СКВАЖИН

Heavy weight drill pipe (HWDP) in wells are hollow, weighty rods with stepwise changing physical properties (for example, stiffness), and each link of the string can deform according to geometrically nonlinear laws. They are the most critical part in the drilling process, transmitting power from the drilling rig to the rock failing tool, and are in hydrodynamic and contact interaction with the borehole walls, and are always curved. This occurs due to the curvature of the well itself, and under the action of its own weight, contact forces, as well as centrifugal forces in the case of rotation of the pipe. In this case, the curvature of the HWDP axis can be significantly influenced by the geometric nonlinearity of the deformation of its pipes. A review of this issue revealed a number of poorly studied problems, which include accounting for both phy- sically and geometrically nonlinear problems, accompanied by various types of complications (loss of stability HDWP, pipe breaks, etc.), as well as other processes in the elements of a dynamic drilling system (DDS). In this paper, based on the use of modern methods for studying dynamic processes in mechanical systems, a method is proposed for studying longitudinal oscillations of a geometrically nonlinear HWDP of its stability under torsion, taking into account the physical nonlinearity in the process of its deformation. The dependences characte- rizing this process are found.

Artificial boundary conditions for nonlinear time fractional Burgers’ equation on unbounded domains

This paper is concerned with the design of artificial boundary conditions (ABCs) for nonlinear time fractional Burgers’ equation on unbounded domains, that describes the propagation of many kinds of waves in physics. It is a major challenge to design appropriate ABCs to obtain the numerical solution and simulate the real physical phenomena, owing to the unboundedness of the physical domain, nonlocality and nonlinearity. The ABCs on the introduced artificial boundaries are constructed by applying the ideas of operator splitting approach and artificial boundary method to obtain a reduced initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. The boundedness and convergence are rigorously analyzed. Numerical simulations are presented to validate accuracy and feasibility of our proposed method.

Free-space optical neural network based on thermal atomic nonlinearity

As artificial neural networks (ANNs) continue to make strides in wide-ranging and diverse fields of technology, the search for more efficient hardware implementations beyond conventional electronics is gaining traction. In particular, optical implementations potentially offer extraordinary gains in terms of speed and reduced energy consumption due to the intrinsic parallelism of free-space optics. At the same time, a physical nonlinearity—a crucial ingredient of an ANN—is not easy to realize in free-space optics, which restricts the potential of this platform. This problem is further exacerbated by the need to also perform the nonlinear activation in parallel for each data point to preserve the benefit of linear free-space optics. Here, we present a free-space optical ANN with diffraction-based linear weight summation and nonlinear activation enabled by the saturable absorption of thermal atoms. We demonstrate, via both simulation and experiment, image classification of handwritten digits using only a single layer and observed 6% improvement in classification accuracy due to the optical nonlinearity compared to a linear model. Our platform preserves the massive parallelism of free-space optics even with physical nonlinearity, and thus opens the way for novel designs and wider deployment of optical ANNs.

USAGE OF CONSTRUCTION-ORIENTED SOFTWARE SCAD FOR ANALYSIS OF WORK OF MACHINE-BUILDING STRUCTURES

Purpose. In the case of analysis of work of the machine-building structures, the direct usage of construction-oriented software developments is impossible, since ideology and methodology for solving various tasks in construction and machine-building are different. Therefore, in the conducting of practical calculations, there is a need for a certain adjustment of the approaches put in the program complexes and their adaptation to the engineering industry. The presentation of the author's experience of the construction-oriented software SCAD usage for Windows for analyzing the work of various machine-building structures, their components and assemblies is the immediate purpose of the publication. Methodology. During a long period of time the author was engaged in analyzing the work of building, mainly thin-walled, steel structures using the Finite Element Method based on the SCAD for Windows software package. At the same time, a considerable number of machine-building structures were considered, including railroad rolling stock units. Most of these tasks grew into a scientific and research problem that needed to be thoroughly researched and analyzed before giving design recommendations. Findings. The publication presents more than a dozen different tasks, typical for the machine-building industry, which the author had to deal with. Static and quasi-static problems, the problem of motion in time, the contact problem, the problem of the cracks deve-lopment, the physical and geometric non-linearity are among them. Accordingly, for each of these problems the main challenges, features and practical techniques developed during the work are presented, as well as the constructed finite element models are presented as an illustration. Originality. The experience of construction-oriented software product usage on the basis of the Finite Element Method for analyzing of the work of machine-building structures is generalized. A number of practical methods and approaches to the solution of various problems in the machine-building sphere are set forth. Practical value. The application of the given information, methods and approaches allows not only to solve concrete practical problems of the machine-building sphere, but also to obtain correct and practically acceptable solutions.

Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem{−Δu+λu=g(u)inRN,N≥3,u∈H1(RN),∫RN|u|2dx=ρ>0, where ρ is prescribed and (λ,u)∈R×H1(RN) is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball in L2(RN) of radius ρ is demonstrated, which allows to provide general growth assumptions imposed on g. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at 0.

Подходы к уточнению расчетных моделей для оценки напряженно-деформированного состояния конструкции на основе анализа данных систем мониторинга

Object and purpose of research. This paper discusses numerical simulation possibilities in terms of stress-strain monitoring for marine engineering structures. This approach can simulate the behavior of strain gauges for both elastic and plastic material behavior. Materials and methods. FEM-based simulation of strain gauge operation process taking into account geometric and physical non-linearity. Main results. Development of refined FE models for sensor installation area of stress-strain monitoring system. Numerical simulation of uniaxial and triaxial strain gauge operation. Time histories of strain gauge readings for linear and non-linear behavior of material. Sensitivity analysis of strain gauges in terms of various strain types. Update of strain gauge arrangement for the best description of structural strains. Conclusion. These results demonstrate and confirm a strong potential of numerical models in development of stress-strain monitoring systems for engineering structures. Simulating strain gauge operation, these models make it possible to determine global strained state of given structure as per strain gauging data for some of its areas.

Alternative analytical and finite element models for unbonded flexible pipes under axisymmetric loads

The calculation of stress components in tensile armour wires of flexible pipes subjected to combined tension and bending is challenging given the intrinsic geometrical and physical nonlinearities. Mechanical loading simulation in finite element models yields heterogeneous curvature distribution, directly impacting stress evaluation. Addition of two auxiliary thermal layers internally and externally subjected to radial and longitudinal thermal expansions simulates equivalent axisymmetric loading, achieving relatively uniform curvature distribution. Calculation of equivalent thermal loads and expansion coefficients involves time consuming trial and error processes by running several finite element models. Therefore, a theoretical formulation that takes into consideration thermal sheaths with orthotropic thermal expansion coefficients and respective thermal loads is developed. A three-dimensional finite element model is also developed for direct application of axisymmetric mechanical or equivalent thermal loadings. A 6″ flexible pipe is employed in a case study to compare the finite element model results within themselves and with the analytical mechanical and thermal models. The analytical results are shown to be rigorously identical. The finite element models present small differences when compared to the analytical results allowing combined application of thermal and bending loading that results in constant curvature away from the boundaries.

Measurement of unsteady aerodynamic force on a galloping prism in a turbulent flow: A hybrid aeroelastic-pressure balance

This paper proposes a novel hybrid aeroelastic-pressure balance (HAPB) technique for the measurement of unsteady aerodynamic force on a galloping prism. HAPB wind tunnel tests were performed to simultaneously observe the unsteady aerodynamic force and galloping response of a test model. The amplitude-dependent nonlinear damping and stiffness of the HAPB system that include the non-wind-induced aerodynamic force that is caused by the interaction between the oscillating model and its surrounding ‘still’ air were identified using a wavelet method. The non-wind-induced aerodynamic force was determined by a forced vibration technique. Subsequently, the galloping response was calculated by substituting the unsteady aerodynamic force and physical nonlinearities into the governing equation of motion. The results show that (1) the proposed HAPB technique is effective in obtaining unsteady crosswind forces; (2) the unsteady self-excited force with excluding the non-wind-induced aerodynamic force can accurately predict the galloping response of the test model whereas the classical quasi-steady theory fails for the prediction. This study has not only provided a HAPB used in wind tunnel, but also addressed shortcomings of the classical quasi-steady theory in predicting galloping instabilities of slender prisms.

A‐stable linear two‐step time integration methods with consistent starting and their equivalent single‐step methods in structural dynamics analysis

AbstractA spectral consistent starting procedure is proposed for the first‐order‐type A‐stable linear two‐step (LTS) time integration methods in structural dynamics analysis. The accuracy analysis for the LTS methods in structural dynamics is presented based on the first‐order model, which enables the algorithms in first‐order transient systems to be extended to structural dynamics under the umbrella of a consistent framework. It is indicated that the approximation of the loads plays an essential role in the equivalence between the LTS methods and some single‐step methods. An algorithmic equivalence feature, which is stricter than the spectral equivalence, is revealed between the two‐leg generalized‐α methods in first‐order systems and the A‐stable LTS methods. The measures of the error constant in the LTS methods and the ultimate spectral radius are extended to optimize a class of single‐step methods of a given convergence order. The spectral consistent starting procedure for the A‐stable LTS methods is developed by utilizing the algorithmic equivalence of the single‐step methods, which results in an optimal A‐stable LTS (OALTS) method possessing a consistent spectral radius with controllable numerical dissipation in the starting step and the steps thereafter. Comparing with the equivalent single‐step methods, the present OALTS method does not need the auxiliary variables, and its displacement, velocity and acceleration can achieve second‐order accuracy simultaneously. The performance of the present OALTS method is verified by numerical examples including physical damping, external loads, and/or nonlinearity.

TORSIONAL VIBRATIONS OF A CIRCULAR ELASTIC ROD TAKING INTO ACOUNT PHYSICAL NONLINEARITY

. In this work, on the basis of the equations of torsional vibrations of a circular rod derived by the author, taking into account physical nonlinearity, the problem of torsional vibrations of a rod is numerically solved. A comparative analysis of the results obtained for nonlinear and linear cases is carried out. To solve the problem, a numerical finite difference method is applied. The approximation of the oscillation equation and boundary conditions leads to a system of algebraic equations, the solution of which is not mathematically difficult. On the basis of the obtained numerical data of the problem, graphs of the dependences of the torsional displacement and stresses on time were constructed. The main conclusions made on the basis of the constructed graphs of displacement and stresses are presented.