Method Of Elastic Solutions Research Articles

Elastic deformation analysis of rotational wood-dowel welding joint system based on the variational method

Abstract This study aimed to investigate the elastic deformation of rotational wood-dowel welding joints by using the variational method, as there was no efficient tool to access the deformation behavior of welding joints perpendicular to axis orientation so far. Based on the variational method, the displacement control equation of axially-loaded dowel-welded wood joints was derived. By incorporating the boundary conditions and using an iterative approach, the parameters required in the control equation were determined. The analytical expression ϕ(r) for the deformation function of the wood substrate under elastic conditions was proposed, thereby determining the displacement and axial force of the single wood-dowel. The elasticity modulus was the primary influence parameter on the function ϕ(r). However, the welding depth had a more critical impact, and reasonable combination of parameters could lead to a better interaction between the wood dowel and substrate. The comparison between calculated results and test trials collected from the previous studies indicated that the elastic solution method could accurately estimate the pull-out ultimate bearing capacity and deformation characteristics of the welding joints. This work can provide unique perspective to understand the technology of rotational wood-dowel welding joints, and a good reference suggestion for the manufacture of multi-dowel welding joints.

Nonlinear Deformation Model for Analysis of Temperature Effects on Reinforced Concrete Beam Elements

The results of the study are aimed at developing a nonlinear strain model for reinforced concrete beam elements in the general case of force actions and temperature effects for different durations. As a rule, temperature effects on structures involve temperature drops, causing heterogeneity of mechanical and rheological properties of concrete and reinforcement. The article has approximating expressions needed to take into account the effects of temperature and heating time on the values of temperature-induced strain and mechanical and rheological properties of heavy concretes with C30–C60 strength classes. The properties of concrete and rebars are heterogeneous from top to bottom and across the width of the cross section. A physically nonlinear problem is solved using the method of elastic solutions combined with the method of stepwise increases in temperature and force loading. The cooling of a heated reinforced concrete element to normal temperature is considered a short-term effect. Strength criteria of portions of a concrete cross section, crack closure conditions, and the ability of cracked sections to take loads in compression amid a change in the sign of stresses are determined. The stress–strain state (SSS) analysis of reinforced concrete beams, made according to the proposed method, is compared with the experimental studies using (i) values of thermal bending moments in statically indeterminate structures, (ii) cracking forces, and (iii) values of deformations (elongations and curvatures) of the elements in the longitudinal axis. Good agreement between the calculated and experimental values of controllable criteria confirms the reliability of physical relationships (i) developed for heterogeneous reinforced concrete beam elements and (ii) applied to the complex cases of temperature and force effects.

Nonlinear Calculation of the Reinforced Concrete Road Pavement Slabs of Highways on the Elastic Basis by the Zhemochkin Method

A rectangular reinforced concrete slab is considered taking into account its physical nonlinearity on a linearly elastic homogeneous base under the action of a vertical external load. The anisotropy and heterogeneity of the slab are due to the properties of reinforced concrete, as well as the formation of cracks from the action of an arbitrary load during operation. The nonlinear problem was solved by the Zhemochkin method using the iterative algorithm of the Ilyushin elastic solution method. The Ritz method (determining the deflections of a slab with a pinched normal) and the Boussinesq solution (determining the displacements of points on the surface of an elastic half-space) were used to determine the coefficients of the resolving equations of the Zhemochkin method. At the first iteration, the slab was calculated as linearly elastic, orthotropic, and homogeneous; at subsequent iterations, it was calculated as linearly elastic, anisotropic, and inhomogeneous at each Zhemochkin site. The deflections of the middle surface of the slab from a unit force were determined as a series according to the first five particular Clebsch solutions. Experimental and numerical researches have been carried out. The latter – with the help of the MATHEMATICA computer program. The results obtained showed that the proposed calculation method allows one to accurately describe the distribution of settlements and reactive stresses under the slab. Verification of the methodology for static nonlinear calculation of a rectangular reinforced concrete slab, taking into account its physical nonlinearity, was carried out by comparing the results of calculations of maximum settlement and average pressures under the slab, obtained using the proposed methodology, and the results obtained using the layer-by-layer summation method and modern software systems Lira and PLAXIS 3D.

Application of variable-height beams with wooden sub-rafter elements in the roof structures of industrial buildings

The investigation pertains to the coating utilized in single-story industrial buildings. Frame constructions with spans of 24, 30, and 36 meters are examined, employing wood-based elements as rafter structures. The reinforced concrete rafter structures exhibit a pitch of 2-3 meters and are configured in the shape of an I-beam. The truss structures along their length are subdivided into seven sections, featuring variable lengths, flange widths, rib thicknesses, and cross-section heights. Deflection calculations consider the nonlinearity of concrete and reinforcement deformations, adhering to prevailing building codes. The elastic solutions method is employed in conjunction with the finite difference method. The proposed coating designs are distinguished by their ease of manufacturing, transportation, and element installation. The wood-composite rafter structure boasts a lower mass compared to reinforced concrete elements, facilitating installation with a lightweight crane and overall diminishing the coating's weight without compromising its structural integrity. Several beam characteristics for spans of 24, 30, and 36 meters include respective mid-span heights of 1.2 meters, 1.4 meters, and 1.5 meters; volumes of 8.23 cubic meters, 9.25 cubic meters, and 10.6 cubic meters; and weights of 19.8 tons, 22.2 tons, and 25.4 tons. The proposed solution allows for the integration of bending moment and stiffness diagrams for the rafter beam configuration.

The non-axisymmetric loading of an elastoplastic three-layer plate in its plane

The statement of the boundary value problem on the deformation of a circular three-layer plate in its plane under the action of a non-axisymmetric load is herein presented. The materials of thin carrier layers obey the hypotheses of the theory of small elastoplastic deformations. The relatively thick filler is physically non-linearly elastic. A system of non-linear differential equilibrium equations in partial derivatives is obtained. A general technique for solving the problem in displacements based on the Fourier method and Ilyushin’s method of elastic solutions is proposed. The case of an external cosine load is considered. An iterative solution of a boundary value problem for a physically non-linear plate is obtained. The corresponding solution of the elastic problem is written out in the final form. The obtained solution is numerically tested.

Повторное знакопеременное нагружение упругопластической трехслойной пластины в температурном поле

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

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.

Determining the stress concentration change with time in a viscoelastic orthotropic solid

The procedure for solving the plane problem of the linear theory of viscoelasticity by the finite element method is described. Based on the virtual work principle and the assumption of the constancy of the strain rate at small intervals of time, the matrix form of the equilibrium equations of the finite-element approximation of a body is written. The solution procedure is described for the constitutive relations in the Boltzmann—Volterra integral form. This integral is transformed into an incremental form on a time mesh, at each interval of which the problem is solved by the finite element method with unknown increments of displacements. The numerical procedure is constructed by ununiformly dividing the time interval, at which the study is conducted. In this case, the stiffness matrix requires recalculation at each time step. The relaxation functions of the moduli of a viscoelastic orthotropic material are described in the form of the Proni—Dirichlet series. The solution to the problem of determining the change over time of the stress concentration in a body with a round hole in a viscoelastic orthotropic plate is presented. To construct a numerical solution, the three moduli of orthotropic material were written using one exponent with the same relaxation time. For these initial data, an analytic expression for the viscoelastic components of the stiffness matrix of an orthotropic plate under plain stress conditions is constructed. Numerical examples are presented for several ratios of the hole radius to the size of the plate. These results are compared with the solution obtained for an infinite plate by inverse transformation by a numerical method of the well-known analytic elastic solution.

Estimation of the earth dam strength with inelastic soil properties

The study aims to develop the models, methods, and algorithms for assessing the earth’s structure strength with inelastic-plastic properties of soil and to evaluate the strength of earth dams. For the mathematical statement of the problem, the principle of virtual displacements, the finite element method, and the elastic solution method were used. To assess the safety factor at each point in the structure, the Coulomb – Mohr theory of strength was used. The stress-strain state (SSS) of specifically designed earth dams with a height of 195 m was studied under various static effects, with inelastic elastic-plastic properties of soil. The isolines of equal values distribution of safety factor for a dam body, as a whole, were obtained, which allow stating the presence, size, and location of the zones in which the strength condition is met.

Развитие методов расчета стержневых элементов стальных конструкций при многопараметрическом загружении

In order to reduce the volume of calculation operations, a reverse course of the study of the strength and stability of the rod elements of steel structures in the general case of loading is proposed. For a given limit state in the cross section of the elastic rod element corresponding to the ultimate strain , using the elastic solution method and the «Section» algorithm, the real ultimate state in strength is established. The results obtained in this way are more reliable, since the development of plastic deformations (and for thin-walled elements - section reductions), are determined by the combined action of efforts. The stability of rod elements is known to be characterized by a violation of the equilibrium deformed state. Therefore, following the solutions of the strength problems, at the same loads, a number of stress states are considered in the most loaded section with various values ( ),according to which the «deformation» forces and the corresponding working cross-sectional area and the rod stiffness are determined. Using the latter, the inverse analytical solution of deformation problems determines the largest general loading parameter corresponding to the buckling coefficient.

Deformation of a Three-Layer Rod with a Compressible Core in a Neutron Flow*

We studied the deformation of a three-layer elastoplastic rod with a compressible core in a neutron flow. To describe the kinematics of the sandwich asymmetric across the thickness, the broken line hypothesis are adopted: Bernoulli’s hypothesis for the thin face layers and Tymoshenko’s hypothesis with linear approximation of displacements over the thickness for the compressible core. The resistance of the core in the tangential direction is taken into account. The constitutive stress–strain relations correspond to the theory of small elastoplastic deformations. The system of differential equilibrium equations is derived using the variational method. The kinematic boundary conditions of the rod is free support of the ends by fixed rigid supports. The boundary-value problem is solved by determining four functions of deflections and longitudinal displacements of the midsurfaces of the face layers. The analytical solution is obtained on the basis of the method of elastic solutions. Its numerical analysis is carried out in the case of an evenly distributed load.

FINITE DIFFERENCE METHOD FOR SOLVING THE ELASTOPLASTIC PROBLEMS OF ANISOTROPIC BODIES

Usually, for the numerical solution of elastoplastic boundary value problems based on deformation theory of plasticity is used an elastic solutions method proposed by A.A. Ilyushin. In literatures the method of elastic solutions relative to boundary value problems of the theory of plastic flow is called the method of initial stresses or the method of initial deformations. In this paper, to solve the boundary value problems of the deformation theory of plasticity of transversally isotropic bodies, it is used relatively “simple” finite-difference method considered in combination with the iterative method, that is, the elastic solution method. The essence of the method is to construct symmetric finite-difference equations, separately for internal and boundary nodes of the area under consideration, and to solve them with respect to “central” or boundary node displacements and the organization of the iterative process. Elastoplastic problems are solved numerically for isotropic and transversely isotropic parallelepipeds under various boundary and boundary conditions. The obtained results are consistent with the known solutions, which shows the validity of the applied methodology. It is explored the spreading of the zone of plasticity and the effect of anisotropy on their distribution.

Time-dependent creep analysis of a functionally graded simple blade using first-order shear deformation theory

ABSTRACT Thermo-elastic and history of stresses and displacements of a rotating functionally graded simple blade is studied using method of successive approximation and a variable thickness cantilever beam model. The rotating blade geometry and loading are mathematically defined so that one can define his own blade profile and loading using any particular function. The blade is subjected to a transverse distributed force, an inertia body force due to rotation and a distributed temperature field due to a thermal gradient between the tip and the root. The creep behaviour is modelled by Sherby’s constitutive model. All mechanical and thermal properties except Poisson’s ratio are assumed to be longitudinally variable based on the volume fraction of reinforcement. The governing differential equations of the problem are obtained using principle of virtual work as well as first order shear deformation theory. Coefficients of these differential equations are variable so by division method, these are solved and the distribution of stresses and displacements are presented for three different volume content of reinforcement. By using Mendelson’s method of successive elastic solution, history of stresses and displacements are obtained. It has been found from history of stresses and displacements that they converge to their steady state condition almost after 50,000 hours.

Deformation of an Elastoplastic Three-Layer Circular Plate in a Temperature Field

A boundary-value problem on axisymmetric deformation of a circular elastoplastic plate, asymmetric across its thickness, in a temperature field is formulated. The physical equations of state used correspond to the theory of small elasticoplastic strains. The effect of a temperature field on the stress-strain state of the plate is considered. The intensity of thermal flow is constant. To describe the kinematics of the plate package, the hypotheses of a broken line are accepted: in the thin load-bearing layers, the Kirchhoff hypotheses are valid; in the rigid core, incompressible across it thickness, the Tymoshenko hypothesis is valid. The work of core in the tangential direction is taken into account. The equilibrium equations are derived by the variational method. An analytical solution of the boundary-value problem is obtained by the method of elastic solutions. A numerical analysis of the stress-strain state of a plate with a hinged contour is carried out.

Time-dependent creep analysis of a functionally graded beam with trapezoidal cross section using first-order shear deformation theory

Time-dependent creep analysis of a rotating functionally graded cantilever beam with trapezoidal longitudinal cross section subjected to thermal and inertia loading is investigated using first-order shear deformation theory (FSDT). The model described in this paper is a simple simulation of a turbine blade working under creep condition. The material is a metal based composite reinforced by a ceramic where the creep properties of which has been described by the Sherby\' s constitutive model. All mechanical and thermal properties except Poisson\'s ratio are assumed to be variable longitudinally based on the volume fraction of constituent. The principle of virtual work as well as first order shear deformation theory is used to derive governing equations. Longitudinal distribution of displacements and stresses are investigated for various volume fractions of reinforcement. Method of successive elastic solution is employed to obtain history of stresses and creep deformations. It is found that stresses and displacements approach their steady state values after 40000 hours. The results presented in this paper can be used for selection of appropriate longitudinal distribution of reinforcement to achieve the desired stresses and displacements.

Calculation of thin isotropic shells beyond the elastic limit by the method of elastic solutions

The paper focuses on the model of calculation of thin isotropic shells beyond the elastic limit. The determination of the stress-strain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method. In the present work the building of the solution based on the equilibrium equations and geometric relations of linear theory of thin shells in curved coordinate system α and β, and the relations between deformations and forces based on the Hirchhoff-Lave hypothesis and the small elastic-plastic deformations theory are presented. Internal forces tensor is presented in the form of its expansion to the elasticity tensor and the additional terms tensor expressed the physical nonlinearity of the problem. The functions expressed the physical nonlinearity of the material are determined. The relations that allow to determine the range of elastic-plastic deformations on the surface of the present shell and their changing in shell thickness are presented. The examples of the calculation demonstrate the convergence of elastic-plastic deformations method and the range of elastic-plastic deformations in thickness in the spherical shell. Spherical shells with the angle of half-life regarding 90 degree vertical symmetry axis under the action of equally distributed ring loads are observed.

Один вариант метода упругих решений в задаче об определении положений равновесия растягиваемой стержневой системы с разупрочняющимися элементами

Один вариант метода упругих решений в задаче об определении положений равновесия растягиваемой стержневой системы с разупрочняющимися элементами

Analytical model of stress field in submerged arc welding butt joint with thorough penetration

Analytical model of temporary and residual stresses for butt welding with thorough penetration was described assuming planar section hypothesis and using integral equations of stress equilibrium of the bar and simple Hooke’s law. In solution the effect of phase transformations (structure changes and structural strains) has been taken into account. Phase transformations during heating are limited by temperature values at the beginning and at the end of austenitic transformation, depending on chemical composition of steel while the progress of phase transformations during cooling is determined on the basis of TTT-welding diagram. Temperature values at the beginning and at the end of transformation are conditioned by the speed of heating. Kinetics of diffusional transformation is described basing on Johnson-Mehl-Avrami-Kolmogorov equation, while martensitic transformation, basing on Koistinen-Marburger equation. Stresses in elasto-plastic state are determined by iteration, using elastic solutions method with changeable longitudinal modulus of elasticity, conditioned by stress-strain curve. Computations of stress field have been conducted for one-side butt welded of two steel flats made from S235 steel. It has enabled a clear interpretation of influence of temperature field and phase transformation on stresses caused by welding using Submerged Arc Welding (SAW) method.

Plastic deformations in the ring spherical shells

In the present work the results of the study of plastic deformations distribution in the thickness in ring spherical shells are presented. Resolving differential equations system is based on the Hirchhoff-Lave hypothesis, linear thin shells theory and small elastic-plastic deformations theory. The studying of the development area of plastic deformations in shells thickness are performed with using the results of the elastic solutions method. The basic relations of elastic solutions method that allow to determine the distribution areas of plastic deformations in shells thickness and along the generatrix are presented. The diagram of intense stress dependence from the strain intensity with linear hardening is received. The numerical solution is performed by orthogonal run method. Long and short spherical shells under the operation of three evenly distributed ring loads are observed. The shells have a tough jamming along the contour at the bottom and at the top. Dependency between tension intensity and deformations intensity is accepted for the case of a material linear hardening. Area of plastic deformations in shells thickness for three kinds of ring spherical shells are shown. The results for the loads differed by the value in twice are presented.

The influence of temperature differences for the analysis of thin orthotropic cylindrical shell

A problem of elasticity of cylindrical shell made of orthotropic different resistant material is considered in this article. The problem is uncoupled, i.e. it consists of two separate problems: of structural mechanics and of thermodynamics. The article contains research of a shell within the framework of shallow shells theory. The solution is based on the technical hypotheses of Kirchhoff-Love. The relations of Treschev A.A. are used in the capacity of physical dependencies. The main relations are used in the form of the elastic solutions method of Ilushin A.A. The process of heat transfer is described by classic equation of heat conductivity. The article introduces a system of solving equations in combined form supplemented by initial conditions and boundary conditions.