Physical Nonlinearity Research Articles

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.

Nonlinear dynamics of heterogeneous shells Part 1. Statics and dynamics of heterogeneous variable stiffness shells

The increasing complexity of the constructive forms and shell elements structure leads to the need to develop both the theory and methods for solving static and dynamic problems for non-homogeneous (heterogeneous) shells.By the shell heterogeneity, we mean heterogeneity in a broad sense: these are inclusions in the shell body of the different rigidity elements and, as a special case, these are holes; the material inhomogeneity caused by a change in the stress–strain state under the influence of both static and dynamic loads; stiffeners; and by taking into account the physical nonlinearity and different modulus of the shell material.This work is devoted to the mathematical model creation of the statics and dynamics for non-homogeneous shells in the above sense and consists of two parts.In the first part, a mathematical model of statics and dynamics for rectangular shells described by the Kirchhoff–Love kinematic model is constructed. Geometric nonlinearity is taken into account on the basis of T. von Kármán’s geometric model, physical nonlinearity — according to the deformation theory of plasticity, based on the elasticity variable parameters method. Stiffness heterogeneity is taken into account using the Heaviside function. The original equations were obtained from Hamilton’s variational principle. A numerical experiment is performed using the Faedo–Galerkin method in higher approximations. The convergence of this method is investigated.In the second part, the nonlinear dynamics of axisymmetric elastic variable thickness shells is investigated. In contrast to the first part of this work, where the Faedo–Galerkin method in higher approximations was used to reduce partial differential equations to the Cauchy problem, in this part the Ritz method in higher approximations is employed. For differential operators used in two parts of this work, according to the research of S.G. Mikhlin, the Ritz and Faedo–Galerkin methods are equivalent. The convergence of the applied numerical method is investigated. An approach is proposed for the zones study of oscillation types for variable thickness shells using dynamic modes maps.

Деформационный расчет и устойчивость стержневых элементов стальных конструкций с несимметричным сечением

To develop practical recommendations for calculating the stability of rod elements of steel structures with an asymmetric cross-section, which is formed as a result of uneven corrosion damage, strengthening or reduction of the cross-section, there is proposed a numerical and analytical solution of the deformation problem, taking into account physical and geometric (according to the spatially deformed scheme) nonlinearities. It allows reducing the calculation time by several orders of magnitude as compared to the existing numerical solutions. The analytical solution of the deformation problem of elastic rods compressed-bent in two planes is based on the qualitative proximity of bending forms of deformation under central compression, namely, sinusoids with corresponding forms obtained by non-deformational calculation, on a square parabola. The latter was replaced by a sinusoid, which allowed obtaining a general solution. The manifestations of physical nonlinearity are compensated by additional loading of the elastic rod with a fictitious force with biaxial eccentricities, which is assessed using the «section» algorithm in the most loaded section, taking into account its spatial displacements. As a result, there was obtained a system of numerical-analytical equilibrium equations in dimensionless parameters, the solutions of which, as the load increases, allow assessing the coefficient of loss of spatial stability.

Учет физической нелинейности при расчете плоской железобетонной рамы с использованием вычислительного комплекса SCAD

The aim of the paper is to analyze the results of the determination of forces and motions in a flat slab under static load, obtained considering the physically nonlinear operation of reinforced concrete. The deformation theories used for composite materials were compared. A linear analysis was carried out, along with a nonlinear automated analysis of forces and motions in a slab in the SCAD Office software (version 21.1.9.7) based on various deformation theories. An automatic calculation of a slab was conducted in accordance with the guidelines of the SP 52-103-2007 “Concrete monolithic building structures”, according to which the nonlinear operation of reinforced concrete structural elements should be considered by introducing reducing coefficients. A comparative analysis of the obtained forces and motions in a flat reinforced concrete slab was carried out. An inconsistency was found between the values calculated based on the theory of physical nonlinearity and those calculated according to the recommendations of SP 52-103-2007. In comparison with the nonlinear analysis based on SP 52-103-2007, the maximum error comprised 55.8%. The obtained values of motions, according to the recommendations of SP 52-103-2007, were also inconsistent. The largest discrepancy in the results of the nonlinear analysis was 49.4%, while the largest discrepancy in the linear analysis was 69%. Thus, the study showed inconsistency between the results obtained using the guidelines and the results of linear and non-linear analysis. The difference in the results demands additional verification when using non-linear analysis in calculations for strength and stiffness as a means of clarifying the stress-strain state of the studied structure.

Meshless approach based on MLS with additional constraints for large deformation analysis

In the present work, we are interested to develop a meshless approach, based on the strong form MLS approximation with additional constraints, to solve the nonlinear elastic and elsto-plastic problems for regular and irregular distribution of points. We adopt a plastic behavior law based on the total deformation theory, which is very convenient when the physical nonlinearity is more important than the effect of irreversible process and the loading history. In plasticity, one encounters discontinuities of rigidity where the application of asymptotic developments seems difficult or impossible. To apply the Taylor series expansion, regularization methods have been adapted. The strong form MLS approximation allows us to avoid the inconvenient of the numerical integration, while the asymptotic developments help us to reduce the computation cost observed in the incremental law of plasticity and the iterative methods. For irregular points distribution, we can get an ill-posed least squares problems due to a singular moment matrix of MLS approximation. To avoid this difficulty, we propose a modified MLS approximation by introducing additional constraints which allows to increase the error functional used in the derivation of the shape functions.

ON AUTOMATIC SEGMENTATION METHOD USING IN SOLVING NONLINEAR INITIAL BOUNDARY VALUE PROBLEMS OF MECHANICS OF SOFT-SHELL STRUCTURES

An approach to improving nonlinear initial boundary value problems of thin-walled structures solution algorithm is represented in the work. The algorithm is based on using line method and parameter differentiation method and is oriented to analysis of composite structures behavior at arbitrary physical and geometrical nonlinearity. Obviously, the less restrictions the calculation algorithm imposes on problem statement, the more features of its realization must be taken into account when developing corresponding software module. The drawback of universality of the algorithm represented is that its direct implementation in some cases can lead either to wrong results, or to loss of calculation stability. To avoid these problems we suggest automatic segmentation method using as a step of the algorithm developed. The essence of the method lies in testing fulfillment of the condition of normalized integral matrices of initial and conjugate differential equation systems orthogonality at the step of problem preprocessing. In the points of integration interval where orthogonality condition is not fulfilled segmentation of the interval is carried out. For testing this approach we select nonlinear problems of dynamic inflation of an infinite cylinder of Mooney-Rivlin material and a sphere of Neo-Hookean material by suddenly applied pressure. It is shown that automatic segmentation method allows improving period and amplitude calculation accuracy, as well as increasing iteration process convergence rate. A problem of dynamic inflation of a hinged hemisphere of Neo-Hookean material by suddenly applied pressure is considered. On the basis of its solution results we suggest the approach to determining parameters of calculation algorithm which allow obtaining correct solution.

ДИНАМИКА БЕЗБАЛОЧНЫХ ЖЕЛЕЗОБЕТОННЫХ КАРКАСОВ СООРУЖЕНИЙ ПРИ ПОВРЕЖДЕНИЯХ ПЛИТ ПРОДАВЛИВАНИЕМ

An approach to the assessment of dynamic effects for reinforced concrete non-beam frames of structures under combined emergency actions is considered. Primary impacts imply an excess of operational loads, which causes damage to the slab-column junction from punching to the onset of a secondary impacts. Secondary impacts include considering the removal of any column from the design model over a finite time. The analysis of dynamic effects based on two approaches: the first is based on the energy method of G.A. Geniev and assumes a quasi-static assessment of the stress-strain state of the damaged system; the second is the analysis of the transient dynamic process taking into account physical and geometric nonlinearity. An approximate approach to modeling the damage of a reinforced concrete slab from punching is proposed and further prospects for its improvement are considered. The degree of danger at emergency impacts for structures pre-damaged by punching, as well as the influence of these damages on the survivability of frames under these effects, has been established. An example of assessing survivability on a model of an underground parking frame in the presence of damage from punching for one of the column-slab joints is considered.

Digital modeling of the mechanics of mobile road pavements made of high-molecular low-pressure polyethylene for application in the Arctic

The paper addresses the possibility of using temporary road pavements in the form of slabs made of high-molecular low-pressure polyethylene. The load applied to temporary road pavements was simulated by using wheeled and tracked vehicles, which corresponds to the real operating conditions, including in the oil and gas industry. When modeling the loading of temporary road pavements, the physical and geometric nonlinearity of the problem was taken into account. The results of the numerical modeling of pavement behavior show that the arrangement of temporary roads and sites in the Arctic, using polymer materials, is a promising and feasible direction.

Исследование прочности и устойчивости трубобетонных элементов конструкций обратным численно-аналитическим методом

In order to reduce the volume of calculations, the research of the strength and stability of concrete-filled steel tube structural members is carried out by the inverse numerical-analytical method. The limit forces are numerically assessed, according to the elastic limit state in the rod`s cross section, corresponding to the maximum permissible deformation of the most stressed fiber beyond the elastic limit. In stability problems, the limit forces are taken as deformation ones in the most loaded section, and, being combined with fictitious forces compensating for physical nonlinearity, they allow, using the inverse analytical deformation calculation of the elastic rod, to determine its the actual loading on the supports. Taking into account the practical application of this method, the solutions are constructed in dimensionless parameters, using the steel members with the calculated section parameters taken as mechanical characteristics. Comparison analysis of the obtained numeric calculation results of stability with the data obtained during the experimental studies of foreign researchers has shown a good consistency.

Stress-strain state of concrete filled steel tubular columns of annular cross-section taking into account physical nonlinearity

The article presents the derivation of the resolving equations for calculating concrete filled steel tubular columns with an annular cross-section under central compression, taking into account physical nonlinearity. The solution is performed numerically by the finite difference method. The deformation theory of plasticity of concrete by G.A. Geniev is used as the law of deformation

Optimal configuration of RC frames considering ultimate and serviceability limit state constraints

Abstract Most current structural design codes are based on the concept of limit states, that is, when a structure fails to meet one of its purposes, it is said that it has reached its limit state. In the design of reinforced concrete structures, the Ultimate Limit State (ULS) and the Serviceability Limit State (SLS) must be checked. Therefore, this paper presents an optimization scheme for reinforced concrete plane frames, in which the objective is to minimize the cost of structures for three cases of constraints: the first is related to ULS and SLS; the second refers only to the ULS; and the third is related only to the SLS. Computational routines for checking limit states of beams and columns are implemented in MATLAB, following the requirements of the Brazilian code. Structural analyses are performed by using the MASTAN2 software, taking into account geometric nonlinearities and a simplified physical nonlinearity method. The objective function considers the cost of concrete, reinforcement and formwork, and the optimization problems are solved by genetic algorithms. Two numerical examples of frames are presented. Regarding the optimal characteristics related to each type of limit state, it is noted that the beams and columns tend to have larger and more reinforced cross sections in the case of the ULS. Even so, optimal structures related to the ULS often do not satisfy SLS and vice versa, which indicates that the optimal characteristics related to each limit state may be different. In addition, it is observed that the SLS is less restrictive than ULS.

ОЦЕНКА ВЛИЯНИЯ ОТПОРА ГРУНТА И ФИЗИЧЕСКОЙ НЕЛИНЕЙНОСТИ ЖЕЛЕЗОБЕТОНА ПРИ РАСЧЕТЕ ПОДЗЕМНЫХ СБОРНЫХ ЖЕЛЕЗОБЕТОННЫХ ТРУБОПРОВОДОВ ДЛЯ ВОДЫ

One of the most common types of hydraulic structures is reinforced concrete pipelines used in various areas of water management. They are used in water supply and disposal systems as supply and discharge pipelines of hydroelectric power plants and pumping stations, in the system of land reclamation. Due to the prevalence of these structures and their large lengths, design issues play an important role. Currently, when calculating reinforced concrete pipelines, both monolithic and prefabricated, the pipe ring of a single width is considered as a linear-elastic body under the conditions of a plane-deformed state. At the same time, it is considered that reinforced concrete pipelines have a sufficiently high rigidity, so the impact of soil resistance due to their deformation is very small and it is not taken into account when designing. In addition,in reinforced concrete, from which pipes are made, even at low loads, nonlinear deformations occur associated with the development of its creep. Non-linear operation of reinforced concrete also occurs in the case of cracks in the pipeline shell, and in its most stressed zones cracks may be formed and stresses may appear in the reinforcement. These factors lead to a change in the rigidity of the cross sections of the pipeline shell and redistribution. The article considers how the presence of expanded joints affects the stress state of the underground pipeline shell,how justified is the assumption that the ground resistance has a small effect on the operation of the prefabricated water supply pipeline, and also there is examines the influence of the physical nonlinearity of reinforced concrete on the results of its calculation in the spatial setting.I

Simplified micro-modelling of masonry cross vaults: stereotomy and interface issues

The importance and the relevance of masonry cross vaults is evident in churches, palaces and cloisters of almost all European countries. However, despite their relevance and long-lasting history, which prove an accurate design with respect to gravitational loads, their seismic behaviour is still poorly understood. In this regard, the present work deals with the finite element micro-modelling analysis of masonry cross vaults subjected to horizontal loads. Assuming the blocks rigid-infinitely resistant, the interface represents the only source of physical nonlinearities. Consequently, block interlocking assumes a capital importance, here discussed in the light of the ancient rules of thumb. From the computational point of view, the mechanical properties of the interface are assessed by means of a sensitivity analysis. Comparisons between numerical and experimental results available in literature are presented in terms of ultimate strength capacity and failure mechanisms.

Nonlinear analysis of a crossing-beams system on an elastic with usage of the Mathematica software

In this paper, the authors consider the method of calculating an infinite system of cross beams on an elastic base by the variationaldifference method. The system of cross beams on an elastic base is most often modeled as shallow strip foundations for buildings of various functional purposes. The variation-difference method is one of the numerical and analytical methods for calculating building structures, it is based on the variational principles of the Ritz-Timoshenko method and on the minimum of the total potential energy of the entire system according to the Lagrange principle, and is also close to the real operating conditions of the foundation – base. A single-layer isotropic artificial base was used as an elastic base in the work, as an elastic layer limited in thickness. The algorithm of nonlinear calculation is based on the use of the iterative method of elastic solutions. The physical nonlinearity of the material of reinforced concrete beams is taken into account through the asymptotic dependence “Moment-curvature”. Numerical approbation of the results of elastic and nonlinear calculations of the system of cross beams on an elastic base was carried out using the MATHEMATICA software package.

Use of the hyperelastic model for plastic materials by example of the three-bar truss

Particular features and traits of a model of large deformations applied for static analysis and design of metal structures with plasticity are considered. Foundations of the deformation theory of plasticity relevant for incompressible materials are formally generalized for the model of hyperelastic body described by equation of state in the Finger form. The behavior of the model is analyzed for the case of uniaxial tension. The collected results are used to explore a symmetrical three-bar truss made of two materials. Relations describing behavior of the truss under conditions of the geometric and physical nonlinearities are obtained. Specifics arising in the analysis and design of the truss using standard structural alloys are analyzed.

NONLENEAR DEFORMATION OF GRANULAR COMPOSITES

The model of nonlinear deformation of a layered material with physically nonlinear layers is proposed. The laminate is considered a two-component material with random layers. The basis is the stochastic differential equations of the physically nonlinear theory of elasticity L.P. Khoroshun. The solution to the problem of the stress-strain state and effective properties of the composite material is constructed by the averaging method. An algorithm for determining the effective deformable properties of a layered material with physically nonlinear layers has been developed. The solution of nonlinear equations taking into account their physical nonlinearity is constructed by an iterative method. The law of the relationship between macrostresses and macrostrains in a layered material and the dependence of average strains and stresses in its layers on macrostrains has been established. Curves of material deformation are plotted for different values of the volumetric content of its filler. The dependence of the effective deformative properties of the laminated material on the volumetric content of the filler has been studied. The effect of nonlinearity of layers on the deformation of a layered composite material is investigated. It was found that the nonlinearity of the layers significantly affects the effective deformative properties and the stress-strain state of laminated materials.

Frequency and Power Dependence of Passive Intermodulation in Connectors With Contact-Material Nonlinear Modeling

Passive intermodulation (PIM) is generally regarded as an unavoidable interference in radio frequency (RF) circuit and system, which exhibits dynamic characteristics with the change of signal frequencies and powers. In this letter, a prediction model of frequency and power dependence of PIM is theoretically proposed and experimentally verified. Physical nonlinearities of truncated distortion and tunneling effect are introduced to provide explanations for the contact surface-induced PIM. Considering nickel is the nonlinear material source in connector coating, the impact of signal frequency on PIM is investigated by calculating the current density in the magnetic area coupled with skin effect theory. Through the combination of two PIM generation mechanisms, a novel equation-based equivalent circuit model is provided to simulate the third-order intermodulation (IM3) powers in connector resulting from different signal frequencies and powers. The frequency and power dependence characteristics are observed from both the theoretical predictions and measurements.

Extension of incremental sequentially linear analysis to geometrical non-linearity with indirect displacement control

The most widely used method for simulating the non-linear behaviour of concrete and masonry structures is the Newton–Raphson method with arc-length control (N-R method). However, this method may fail to produce converged results because of softening, negative tangent stiffness, bifurcations or snap-back. Sometimes, convergence can be obtained by controlling degrees of freedom in the failure process zone or by applying sequentially linear analysis (SLA). However, the location of the failure is often not known a priori and geometrical non-linearity needs to be included.Recently, incremental sequentially linear analysis (ISLA) has been proposed, which is based on a combination of the N-R method and SLA. The solution search path follows damage cycles sequentially with secant stiffness corresponding to local damage increments, which traces both damage history (explicit) and displacement history (implicit). The objective of this paper is to demonstrate that ISLA can be applied to problems that behave geometrically nonlinear in addition to physically nonlinear.In this paper, we introduce a method that combines ISLA with indirect displacement control. This method stabilises localised damage process areas and avoids the global unloading caused by geometrical and physical non-linearity. The method uses one or more control points, which are positioned independently of the failure process zones.Two masonry walls were tested and analysed. The load was perpendicular to their planes and evenly distributed. The walls were supported on two or four edges. Stable post-peak results were computed for large geometrical non-linear displacements, and localised crack propagation was computed robustly and correctly.

Stress-strain state cylinder-plate-cable-stayed roof buildings (structures) with various forms of external support contour

Relevance. A new wave-like combined (complex) coating design for large-span buildings - a cylinder-plate-cable-stayed roof, combining three types of structures: a cylindrical shell of zero Gaussian curvature, plate and cable-stayed (hanging) roofs are presented. This combination of structures and materials for roof large areas was not chosen by chance. The cable works in tension with its entire cross section only, and the cylindrical shell and plate work in two (longitudinal and transverse) planes. In combination with external influence, they create the necessary design strength, taking into account, at the same time, a rational choice of materials (steel and reinforced concrete). New architectural and constructive solutions of a large-span building with a cylinder-plate-cable-stayed roof are proposed taking into account the different geometric shapes of the external support contour in a form of a semicircle, semiellipse, etc. The aim of the work is to analyze the influence of the external support contour (semicircle, semiellipse, and other forms) on the spatial work of internal forces in the combined cylinder-plate-cable-stayed roof of a large-span building. Methods. Results of static numerical analysis of spatial models of large-span buildings with different types of external supporting contour in the centralized roofs were made in the FEMAP with NX NAS-TRAN software package. This complex belongs to the CAE - class which implements the finite element method (FEM) and allows on the basis of the physical and geometric nonlinearity of the deformation of structures. Results. Calculation study carried out comparative numerical analysis of the stress-strain state of a complex cylinder-plate-cable-stayed roof with different outlines of the support contours on the effect of vertical loads. This determines their rational choice given due consideration to the total cost and useful area of the building. The results of calculating large-span buildings with cylinder-plate-cable-stayed roofs for horizontal (wind) load, on the basis of determination of aerodynamic coefficients, are supposed to be published in the next article.

An Analysis of the Reliability of a Bus Safety Structure on Carrying Out the Numerical and Experimental Tests

In the paper, the reliability of a spatial tubular structure of a bus safety frame formed of different steel profiles is discussed. A methodology for the bus safety structure modeling is presented herein by applying numerical methods that enable us to simulate virtually a test for assessing bus rollover crashworthiness according to the United Nations Economic Commission for Europe (UNECE) Regulation No. 66, and also to assess and ensure the reliability and safety of the structure under operating conditions. The simulation has been performed by applying the mixed method of kinematical analysis and finite elements. In the course of the calculations, physical and geometrical non-linearity of materials was assessed. In addition, an experimental rollover test according to UNECE Regulation No. 66 was performed in this work, striving to verify the provided methodology for modeling by applying numerical methods. For the experiment, an identical safety structure and a rollover stand (identical to the one used in modeling) were used. The rollover test was shot by a Phantom v711 high-speed camera. In the paper, the results of kinematical and dynamic analysis from applying the finite element method and the ones of the experimental test, as well as their comparisons, are provided. It is assessed whether the developed safety structure model is reliable and suitable for use.