Physical Nonlinearity Research Articles

Version of the deformation theory of plastic ductility of concrete in a plane stress state

Reinforced concrete as one of the main materials for a wide class of building structures for civil, industrial and transport purposes has a number of specific properties: physical nonlinearity, anisotropy, and crack formation. The behavior of reinforced concrete in the elastoplastic stage before its destruction is more characterized by deformation of concrete. It is shown that the physical nonlinearity of concrete is due to plastic deformations, which are characteristic of various types of stress state. For a triaxial stress state, the system of equations in the mechanics of a deformable solid, it includes two groups of formulas that combine nine equations that include 15 unknowns (three displacements, six strain components, and six stress components). In order for the system to be closed, it is necessary to supplement it with six equations. Such equations are the basic physical relationships that relate the six stress components to the six strain components. The use of linear relationships between stresses and strains introduces the greatest error in the assessment of the stress-strain state (NDS) of structures made of materials with the properties of nonlinearly deformable bodies. In this regard, the more correctly the physical law defining the correlation reflects the material, according to which the material resists various types of deformations, the less error will be allowed in the assessment of the NDS of structures. The article proposes a new approach to the construction of basic physical relationships based on an invariant solution to the problems of mechanics of a deformable solid for concrete in a plane stress state. The correspondence of the proposed dependences to the real stress and deformable state of the material is shown.

Study of the composite action of concrete and reinforcement in the floor in case of emergency impact

The article presents the results of a numerical study of the composite action of concrete and reinforcement in reinforced concrete flooring under emergency loads, taking into account physical and geometric nonlinearities. The study used a nonlinear static method in the LS-DYNA software package, which implements the Continuous Surface Cap Model (CSCM). The simulation model of a monolithic beam floor uses volumetric finite elements for concrete and rod elements for reinforcement. Isofields and graphs of the intensity of stresses and strains in the upper and lower reinforcement, as well as in the concrete at the floor elevation, with an increase in the vertical uniformly distributed load, were built. The load-bearing capacity of the monolithic slab under consideration was determined using the non-collapse criterion. The LS-DYNA software package makes it possible to study the action of load-bearing concrete structures modeling direct concrete reinforcement using reinforcing bars at a significantly nonlinear nature of strain.

Nonlinear Strength Calculation of Shell Structure of Arbitrary Shape Based on FEM

This article discusses the algorithm for calculating the shell structure of arbitrary shape, taking into account the physical nonlinearity of the material used. In determining the parameters of the stress-strain state, a step loading procedure was used. Algorithm for calculating the shell structure of arbitrary shape, taking into account the physical nonlinearity of the used material is discussed in this article. In determining the parameters of the stress-strain state, a step loading procedure was used.

Two-Layer Ferrocement Shells Stress-Strain State Modeling under the Fire Conditions

The article deals with the issues of strength calculation of two-layer ferrocement shells under fire conditions, taking into account the physical nonlinearity of the material. The change in the physicomechanical characteristics of materials under the action of temperature is taken into account. The solution of model problems for shallow shells of double curvature is presented.

Use of generalized finite difference method for calculation of beams from nonlinear elastic material

The introduction into practice of construction of structures made of high-strength steels and other materials having a nonlinear deformation diagram caused the active development of the nonlinear theory of calculation of structures. Replacing Hooke’s law with nonlinear dependencies between stresses and strains leads to so-called physical nonlinearity. For the calculation of such structures, the experimentally obtained dependences between stresses and strains are described using analytical expressions. A number of variants of such approximations have been proposed by various researchers. In this paper, we consider the calculation of beams of symmetrical cross-section made of nonlinear elastic material, for which the dependence between stresses and strains is described by a cubic parabola. This approximation ensures the symmetry of the diagram σ with respect to the tension σ – ɛ compression, and also gives a good match with the experimental curve. The use of the generalized finite difference method for solving the problem allows to reduce the system of nonlinear differential equations to the system of algebraic equations, for the solution of which the method of successive approximations is used. Studies have shown that the proposed method of calculation makes it possible to obtain a fairly accurate solution for a small number of elements. In addition, the presented calculation algorithm is convenient for programming and numerical calculation. As an example of systems that allow calculation using the considered algorithm, beam elements of building structures can act. The calculation of a beam from a nonlinear elastic material on the action of a concentrated force is given.

Nonlinear Deformations of Stiffened Reinforced Concrete Shells

The paper discusses the process of non-linear deformation of shell structures made of reinforced concrete. A mathematical model of deformation in the form of the functional of full potential deformation energy is provided. The model is based on the Kirchhoff–Love hypotheses, and allows accounting for structure reinforcement with stiffeners. An orthogonal network of stiffeners, located from the concave side, is considered as the structure support. Type of load — external, uniformly distributed. The Ritz method is applied to the functional to reduce the variational problem of the functional minimum to a system of nonlinear algebraic equations. Then, for each load value, the problem is solved using iterative methods. Analysis of strength and stability of shallow shells of double curvature and rectangular planform is performed. Values of critical loads, deflection and stress fields are obtained. Curves of deflection depending on load are provided. All results are given in dimensionless parameters. The Mohr–Coulomb criterion was used to analyze concrete strength, and the Lyapunov criterion was used for stability analysis. Influence of the number of stiffeners reinforcing the shell on the resulting stress values is shown. It has been revealed that with account for physical non-linearity of concrete, when the dependence of stresses and deformations is curvilinear, deformations (and deflections as well) of shells increase in comparison with the linear-elastic solution. It has been also found that when nonlinearity is taken into account, redistribution of stresses over the shell field occurs (the maximum stresses shift towards the shell contour).

On the controllability of a creasing singularity in a nonlinear elastic circular sector

The deformation of a circular sector into a full self-contacting circle can be sustained in all homogeneous, isotropic, incompressible materials by surface tractions alone. In this class of nonlinear elastic materials, this works investigates the controllability of such a peculiar mapping having uniform constant strains and a creasing singularity. By performing a perturbative analysis based on small–on–large incremental methods, we determine the critical conditions for the normal traction load to trigger a morphological transition from the circular ground state to an elliptic shape. Such predictions are given for neo-Hookean, Gent and polynomial material models to illustrate how both geometrical and physical nonlinearities concur to this elastic instability.

Determining the influence of physical nonlinearity of soil strength properties on the estimated base resistance

The paper has examined the potential of using nonlinear models of strength in determining the initial critical load on soil, as well as the standardized and estimated base resistance, which makes it possible to reduce labor intensity in the process of determining the strength properties of soils. Based on the analysis and generalization of results from theoretical studies into geomechanical processes using analytical mathematical methods, the formula modifications have been derived that are intended for determining the initial critical load on soil, as well as the standardized and estimated base resistances. We have established interrelation between strength, in particular specific cohesion, and the angle of internal friction, which are included in the strength conditions by Mohr-Coulomb and A. Shashenko, thereby making it possible to improve the procedure for calculating external loads on soil. The dependences of critical loads on base on the mean pressure under the sole of the foundation haven been analyzed in the range of pressure Р =100...500 kPa using the strength conditions by Mohr-Coulomb and A. Shashenko. It has been established that when using generally accepted estimation formulae to determine the critical loads on base, it is required that the pressure range should be taken into consideration at which the properties of soil strength were determined. In this case, using the Shashenko failure criterion to determine critical loads on base makes it possible to properly consider the impact exerted by the mean pressure on them under the sole of the foundation. In contrast to dependences used currently in the Ukrainian, Belarusian, Russian regulatory documents, as well as in other countries’ standards, the resulting formulae make it possible to take into consideration the dependences of soil strength properties on the mean pressure on soil under the sole of the foundation. The results obtained make it possible to improve the reliability of determining the initial critical load on soil, as well as the standardized and estimated base resistances. This is achieved by taking into consideration the nonlinearity of the Mohr limiting circles’ envelope using the strength condition by A. Shashenko

A first order homogenization approach for structural elements: Beam kinematics

AbstractThis contribution deals with a first order homogenization approach for structural elements with the focus on shear deformable beam elements. The homogenization theory itself is based on the well‐known Hill‐Mandel condition. Therefore, the numerical treatment leads to a coupled multiscale approach whose calculation can be performed with the so‐called FE2 scheme. As a result, not only cross‐sectional properties in the linear elastic case can be evaluated but also geometrical and physical nonlinearities can be considered. Since the mesoscale, which will be homogenized, is modelled by 3D brick elements, the considered global structure only needs to be periodic to allow a definition of a representative volume element (RVE). Thus, the procedure is not limited to pure cross‐sectional modelling, but is also able to consider inhomogeneities in the longitudinal direction of the system.As the beam kinematics do not describe the complete deformation gradient or strain tensor, the equilibrium condition introduces a length dependency of the homogenized shear stiffnesses and the stiffnesses which depend on them due to eccentricities. This is due to the fact that the averaged shear deformation of the RVE does not linearly depend on its length multiplied by the beam shear deformation. To overcome this problem, additional constraints are introduced which ensure this linear connection. They are chosen in a way that they do not contribute to the virtual internal work. Therefore, the Hill‐Mandel condition is not affected by them. With these assumptions, the length dependency is eliminated and the averaged shear deformation of the RVE equals the shear deformation of the beam kinematics. As a result, the method is able to reproduce well‐known cross‐sectional values known from the literature and is also capable to evaluate complex nonlinear systems. For the latter, the agreement of the results with those of continuum models is very satisfying.

A longitudinal magnetoelastic wave in a rod with account of the damage of its material

For an electrically conducting rod that performs longitudinal oscillations, a self-consistent system is formulated that includes the equation of rod dynamics, the equation for the variation of the strength of the external magnetic field, and the kinetic equation for the accumulation of damages in the material of the rod. The linearized system and the system of equations, including geometrical and physical elastic nonlinearities, are consecutively considered. It is shown that the waves described by linearized system have dispersion and frequency-dependent damping due to the presence of two types of dissipation, one of which is caused by the damage of the material and the other by a magnetic field. In nonlinear case, an evolution equation is derived that generalizes the Burgers equation, known in nonlinear wave dynamics. Its solutions are found and analyzed.

Floors number influence on the instability parameter of reinforced concrete frame-braced buildings

Abstract This work aims to investigate the floors number influence on the instability parameter limit α1 of reinforced concrete frame-braced buildings; it succeeds another work in this field of knowledge, in which the same question was investigated for wall- and core-braced buildings. Initially, it is showed how the ABNT NBR 6118:2014 (Brazilian code for concrete structures design) defines when a second order analysis is needed. Topics concerning to physical nonlinearity consideration and to the lateral deflection components of frames are also presented. It follows an analytical study that led to the derivation of a method for determining the limit α1 as a function of the floors number and the relation between bending and shear stiffness. Finally, some examples are presented and their results are used for checking the method accuracy.

Optimal Control Synthesis of Semi Active Vehicle Suspension

The paper discusses the problems related with the synthesis of a linear quadratic (LQ) regulator of a semi active vehicle suspension. The synthesis combines the control through output variables with such by the input excitation i.e. classical LQ regulator with Compensator. The physical nonlinearities of the controllable magneto-rheological semi-active damper are taken into an account with the including in the feedback control circuit an inverse damper model. . A reduced suspension model is used to illustrate the synthesis (quarter car model).

Calculation Of Pressure Of Reinforcement Corrosion Products In Reinforced Concrete Element

The existing regulatory framework does not cover the calculation of reinforced concrete elements subjected to corrosion. The calculation of such elements is given in a number of scientific papers. The article deals with a particular problem of calculation-finding the pressure of corrosion products and calculation of detachment of the protective layer. To solve this problem, the polar-symmetric problem was solved and formulas for determining the pressure of corrosion products, stresses and displacements were obtained. The condition of destruction of samples is received based on the theory of strength of G. A. Geniev. The calculations carried out on the obtained dependences have good convergence with the calculations taking into account the physical and geometric nonlinearities, performed in the universal software complex ABAQUS. An engineering method for calculating the separation of the protective layer was developed. The possibility of experimental confirmation of the calculations, as well as the need for additional research in the field of corrosion damages is shown.

Substantiation of Thin-Walled Structures Parameters Using Nonlinear Models and Method of Response Surface Analysis

The article deals with the development of approach for providing of structural strength of complicated thin-walled machine-building structures, which operate in the conditions of geometric and physical nonlinearities using design solution validation. The developed approach is based on the use of mathematical model for stress-strain state taking into account geometric and physical nonlinearities and methods of approximation for constructing functions describing the evaluated characteristics of the object under study. The various factors behind the search for design solutions (including characteristics of strength, rigidity, technological and economic factors), are being added into this function. The developed algorithm for rational parameters search, which takes into account the peculiarities of the response surface shape observed in solving applied problems, is applied to them. In this case, the solution is sought over the whole range of parameter changes. Thus, in the search process, global trends of changes in design decisions are taken into account, and not local ones, as in other approaches. This allows obtain a rational solution that is stable to changes in parameters, which are possible in course of design work and production conditions. Investigations are illustrated on the freight rail-car, tractor cabin frame, carrier personnel hull.

The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells

The axisymmetric propagation of longitudinal waves in an integrally stiffened physically and geometrically nonlinear cylindrical shell is studied. The case is considered when the parameters characterizing nonlinearity, dispersion and thickness–radius ratio are the same order. Using the multiscale asymptotic method, the generalized Schamel equation is derived from the equations of motion of the shell. This equation contains an additional term with the fifth derivative with respect to the spatial coordinate which characterizes the high-frequency dispersion. It is shown that the derived equation does not pass the Painleve test and therefore is not integrated by the inverse scattering transform. The geometric series method using Pade approximants is applied to construct exact solitary-wave solution of the equation. It has been established that for the existence of an exact bounded amplitude solution a “soft” type of physical nonlinearity is necessary. The Schamel–Kawahara equation containing the combined nonlinearity is considered; its exact soliton-like solution is given. Numerical simulation confirmed that the initial perturbation in the form of the exact solution generates a persistently propagating solitary wave.

Identification of physical nonlinearities of a hybrid aeroelastic\u2013pressure balance

This study has presented an improved method for determining physical nonlinearities of weakly nonlinear spring-suspension system and successfully applied to a novel hybrid aeroelastic–pressure balance (HAPB) system used in wind tunnel, which can be used for simultaneously obtaining the unsteady wind pressure and aeroelastic response of a test model. A nonlinear identification method of equivalent linearization approximation was firstly developed on the basis of the averaging method of Krylov–Bogoliubov to model the physical nonlinearity of a weakly nonlinear system. Subsequently, the nonlinear physical frequency and damping were identified using a modified Morlet wavelet transform method and a constant variant method. Using these methods, the physical nonlinear frequency and damping of the HAPB system with a vertical test model were determined and were validated by a time domain method and the Newmark-beta method. Finally, the nonlinear mechanical frequency and damping of the HAPB system with inclined test models were determined in a similar way. This study has not only provided an identification method for determining physical nonlinearities of weakly nonlinear system, but presented the detail for developing a hybrid aeroelastic–pressure balance used in wind tunnel.

Simulation of Performance of CFST Elements Containing Differentiated Profile Tubes Filled with Reinforced Concrete

The approach to calculating CFST elements is considered in which physical non-linearity of materials, geometric non-linearity of the tube and the effect of increasing the strength of the core are taken into account. Finite element models are developed and proposed as the basis for more accurate method of calculating concrete-filled steel elements consisting of differentiated profile tubes filled with reinforced concrete. The technique uses a step iteration algorithm involving analytical dependencies and finite element simulation. The criterion for determining the load bearing capacity of CFST elements was the achievement of the stresses in the tube of the characteristic strength. The possibility of estimating the load bearing capacity of elements by limiting stresses in the core concrete is also implemented. The result of the calculations was obtaining the stress-strain and limiting state of the differentiated profile tubes with CFST elements, and graphic analysis of the regularities of stress redistribution at different stages of performance of columns. In general, with the accepted problem statement we could establish the exact stress-strain state, take into account the elastic-plastic deformations of concrete, its cracking and destruction, and geometric nonlinearity of the tube. The effect of performance of the corrugated sheet as a tube was established.

Model Reduction Methods Applied to A Nonlinear Mechanical System

Modern structures of high flexibility are subject to physical or geometric nonlinearities, and reliable numerical modeling to predict their behavior is essential. The modeling of these systems can be given by the discretization of the problem using the Finite Element Method (FEM), however by using this methodology, it is a very robust model from the computational point of view, making the simulation process difficult. Using reduced models has been an excellent alternative to minimizing this problem. Most model reduction methods are restricted to linear problems, whichmotivated us to maximize the efficiency of these methods considering nonlinear problems. For better accuracy, in this study, adaptations and improvements are suggested in reduction methods such as the Enriched Modal Base (EMB), the System Equivalent Reduction Expansion Process (SEREP), QUASI-SEREP and the Iterated Improved Reduced System (IIRS). The stability of a system is discussed according to the calculation of the Lyapunov exponents and phase space. Numerical simulations showed that the reduced models presented a good performance, according to the commitment of quality and speed of responses (or time saving).

Comparative assessment of the contending force and placement methods for weightless sagging cables

Sagging elastic cables are inherently nonlinear structures due to the presence of self-weight apart from the presence of conventional physical and geometrical nonlinearities. Recently, authors have proposed a force method of weightless sagging elastic cables with linear tension–extension relations and undergoing small elastic displacements to focus explicitly on their characteristic ‘configurational’ nonlinearity. This force method is based upon the rate-type constitutive equations and third-order differential equations of motion. Existing displacement method is reconstructed here for the physically linear weightless sagging elastic cables undergoing finite elastic displacements. In place of nodal forces in force method, deformed state nodal placements are the primary variables in this placement method. Rate-type constitutive equations for two-node weightless planar sagging elastic cables are derived. The objective of the present paper is to compare and contrast these two contending methods in reference to their underlying hypotheses, analytical reach, and quantitative predictions of structural response. Possible equivalence of the proposed third-order and the popular incremental second-order differential equation of motion are explored. Theoretical significance of the present paper is critically evaluated.

Stress-deformed state and form-changing of massive and thin-walled objects

The paper presents the results of the mathematical modeling of the processes of the transformation of massive and thin-walled bodies of rotation under the action of forced displacements and unstoppable temperature fields with large irreversible deformations and variable boundary conditions. As examples, operations of extracting cylindrical glass and hot precipitations of cylindrical work are considered.The processes of processing of metals by stamping are marked by a significant change in the workpieces under the action of poissons, stamps and other tools, often in conditions of uneven heating. In addition, the conditions of interaction with contacting bodies determine the significant influence on the course of change in their stress-strain state. Further improvement of various technological processes of metal processing by embossing to a large extent depends on the completeness and reliability of information on the peculiarities of changing the picture of the VAT of the workpieces in the process of deformation. In this connection, the urgency of the development of research methods for plastic molding, taking into account geometric nonlinearity with large physical-nonlinear deformations, contact interaction and heat transfer conditions on the boundary surfaces, is increasing.In this paper, as output, settlement relationships, and algorithms for solving nonlinear equations systems are taken in the publications of the authors [1, 5, 7].The results of the calculations of technological processes presented in this paper give the basis to conclude that the developed method and implementing its software complex, allow to conduct studies on the modification of shell, massive axisymmetric constructions, taking into account geometric and physical nonlinearity in power and non-stationary temperature loads.