The article deals with the transition from corrugated plates and shells to smooth structures of equivalent rigidity. An expression for the potential deformation energy of an infinitely small element of an equivalent smooth shell and formulas that establish a connection between internal forces and generalized deformations of a corrugated structure are given. A review of the formulas for the equivalent stiffness of the corrugated shell during bending, presented in the works of various authors, is carried out. In order to select the dependencies that provide the smallest error when replacing the corrugated shell with a smooth one, a numerical experiment is performed in the LIRA finite element complex. A corrugated plate hinged around the contour under the action of a uniformly distributed projective load is considered. The calculation of the structurally orthotropic construction is performed numerically by the finite difference method. It is also established that the monograph of S.G. Lekhnitsky contains an incorrect formula for the moment of inertia of a sinusoidal corrugation.
Keywords: corrugated structures, plates and shells, finite element method, finite difference method, orthotropy, equivalent stiffness
The article discusses the method for calculating beams with corrugated walls as three-layer structures of equivalent rigidity. The derivation of resolving equations for a one-dimensional finite element of a three-layer beam is given. A hypothesis is introduced that the shelves fully perceive normal stresses, and the wall only works on shear. When obtaining the basic equations, forced deformations are taken into account, which may include creep deformations, temperature deformations, shrinkage deformations, etc. The solution of the test problem for a beam hinged at the ends under the action of a load uniformly distributed over the length is presented. To control the reliability of the results, a finite element analysis was performed in a three-dimensional formulation in the LIRA software package. Shelves of the beam are modeled by flat triangular shell finite elements, and the wall is modeled by rectangular shell FE.
Keywords: corrugated wall beam, three-layer beam, finite element method, equivalent rigidity, stress-strain state
The technique of calculating the metal corrugated structures using the finite element method for an axisymmetric load is considered in the article. One-dimensional finite elements in the form of truncated cones are used. Calculations are performed using the program developed by the authors in the Matlab package. An example of calculation of a ground well rigidly clamped in the base under the action of ground pressure is given. The sinusoidal profile of the corrugation is considered. The graphs of changes in bending moments and ring forces are presented. For a smooth shell of the same thickness, the bending moment in the pinch was 30.3% higher compared to the corrugated, and the maximum value of the ring force was 15.7% higher.
Keywords: metal corrugated structures, cylindrical shell, finite element method, axisymmetric problem, soil well, shell theory, edge effect
Flat bending stability problem of constant rectangular transverse section wooden beam, loaded by a concentrated force in the middle of the span is considered. Differential equation is provided for the cases when force is located not in the center of gravity. The solution of the equation is generated numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the package Matlab. A relationship is obtained between the value of the critical force and the position of the load application point. For this dependence, a linear approximating function is chosen. A comparison of the results obtained by the authors with an analytical solution using the Bessel functions is performed.
Keywords: flat bending stability, secular equation, finite difference method, iteration process
Now elements of natural structures of the world surrounding us form a basis for loan by their architects and designers in their professional activity. It is caused not only by esthetic appeal of natural objects, but also big functionality of their form providing high degree of durability, reliability, adaptation to the changing external conditions, comforts of their use as inhabited and production rooms. In this work the analysis of the VAT of an oviform envelope under the influence of sole weight and intrinsic pressure is provided.
Keywords: intense strained state, finite element method, oviform envelope, sole weight, intrinsic pressure, analysis, efforts, deformation
The general equations of the moment theory of a circular cylindrical shell with creep are considered: static, geometric and physical. We solve the problem of determining the stress-strain state of a shell rigidly clamped in the base when an internal hydrostatic pressure acts on it. The problem has reduced to a linear nonhomogeneous differential equation of the fourth order with respect to deflection. The solution was performed numerically by the finite difference method in the Matlab software package. As a law of connection between creep strains and stresses, the generalized nonlinear Maxwell-Gurevich equation was used. To determine the creep strains, a linear approximation of the first time derivative was used. The shell was made from secondary PVC, and as a result, it was found that in the process of creep in the shell, the circumferential stresses increase by 15%.
Keywords: cylindrical shell; creep; moment theory; polymers; finite difference method
The article proposes the derivation of resolving equations for the bending of triangular finite element of plate with regard to creep. In deriving of the equations we use Lagrange variational principle. The problem is reduced to a system of linear algebraic equations. Creep contributes only to the right side of the system of equations. These equations allow to calculate the plates of arbitrary shape, taking into account the viscoelastic properties of the material. An example of the calculation for a rectangular plate of a secondary polymeric PVC, hinged along the contour and loaded uniformly distributed over the area load is presented. As a law establishing a link between stress and creep deformation we used nonlinear equation of Maxwell-Gurevich. Calculations were performed in Matlab software package. The graphs of change in time of deflection and stresses are presented. Stress during creep vary slightly, a difference between the stresses at the beginning and end of creep process does not exceed 6%. The result of numerical calculation of the maximum deflection value at the end of creep is different from the theoretical on 0.26%.
Keywords: creep, finite element method, bending of plates, polymers, Maxwell-Gurevich equation, long cylindrical rigidity
We investigated the creep of concrete arches based on the following theories: the theory of linear creep by Harutyunyan-Maslov, kinetic theory, the theory of flow, theory of aging, and nonlinear theory of Y. Gurieva. We considered viscoelastic model of the concrete, ie total strain was represented as the sum of elastic strain and creep strain. Solution of the problem was carried out by finite element method. We considered the arch rigidly clamped at the ends and loaded with a uniformly distributed load. Graphs of growth of deflection and stress distribution in the reinforcement and concrete are represented. We obtained the substantial redistribution of stress between the reinforcement and the concrete during creep: in reinforcement stresses increased and in concrete stresses decreased. The strongest redistribution occurs on the theory Y. Gureva.
Keywords: reinforced concrete arch, creep theory of heredity, aging theory, the theory of flow, kinetic theory, finite element method, stress-strain state
The stationary heat conduction problem for radiation-heat shield of the reactor nuclear power plant based on domestic sources of heat was solved. We took into account the dependence of the thermal conductivity of the concrete on the temperature, which leads to nonlinear problem. The solution is performed using the finite element method in combination with the method of successive approximations. We used the simplex triangular elements. The problem was solved in axisymmetric formulation. As a design scheme we used a rigidly clamped at the base thick-walled cylinder. It was found that the inclusion of thermal conductivity depending on the temperature leads to a slight (2.5%) increase of temperature in core of the construction.
Keywords: thermal conductivity, finite element method, the steady-state temperature field, radiation-heat shield, thick-walled cylinders
The article presents basic equations for reinforced concrete elements that are experiencing bending moment and axial force, taking into account the creep of concrete. The stress-strain state of reinforced concrete statically determinate three–hinged arch is investigated on the basis of these equations. Also for this task we gained the resolving equations of finite element method and compared the numerical-analytical calculation with the numerical performed using the finite element method to the arch loaded with a uniformly distributed load and having the shape of a circular arc. The calculations used viscoelastic model, according to which the total deformation is the sum of concrete elastic deformation and creep. We consider a rectangular cross section with symmetrical reinforcement. It is shown that because of creep stress redistribution between concrete and reinforcement arises.
Keywords: finite element method, the creep of concrete, viscoelasticity, reinforced concrete arch, the stress–strain state.
The article provides information about method of sustainability calculation of compressed polymer rods taking into account nonlinear creep . As the law of the relationship between stress and strain is used nonlinear equation of Maxwell-Gurevich. Derived from the analysis of the resolving equations with time tends to infinity, we obtain an expression for a long critical force in the case of constant rigidity of the rod.
Keywords: nonlinear creep, rod, stability, Maxwell-Gurevich, finite difference method, long critical force, relaxation viscosity, viscoelasticity, high elasticity modulus.
The technique of determining the stress-strain state of the polymer thick cylindrical shells in flat tension conditions with effects of temperature and creep deformation. As the law of the relationship between stress and strain is used nonlinear equation of Maxwell-Gurevich. Solution is performed numerically by finite element method.
Keywords: nonlinear creep, cylinder, the Maxwell-Gurevich equation, finite element method, relaxation viscosity, viscoelasticity, high elasticity modulus, plane stress and temperature.
We obtained the solution of the optimization problem for the step-parallel beam with constant width of the cross-section. The problem was solved by minimizing the strain energy while the volume of beam is constant. The solution was made for the case of simple support at the ends and evenly distributed along the length load. We introduced the parameter α, which is the ratio of height of the average part to a height of side parts. The optimal value of α was found. At this value the stiffness of the beam is maximum at constant volume. The obtained beam of maximum stiffness is not equal strength.
Keywords: optimization, speed-prismatic beam, the strain energy, variable stiffness, minimum
We solved the problem of optimizing the gable beam by minimizing the strain energy at constant volume. The problem reduced to an integral equation for determining the optimum angle of the beam. This equation was solved numerically by the method of bisection. Integral was calculated using the method of trapezoids. Solution was made in software package Matlab. The optimum angle depends on the width of the cross section b, and the volume of the beam. It was found that with increasing of width of the cross section the optimum tilt angle decreases. Also the volume increases by increasing the width of the beam.
Keywords: optimization, gable beam, strain energy, variable rigidity, minimum volume, method of bisection
The phenomenon of buckling under the creep of concrete arches was investigated. Solution of the problem carried out by means of the finite element method. To analyze the stability we used Newton-Raphson method. It has been established that there is a long-term critical load, beyond which the growth of the deflection has not fading character. As the equation of the relationship between the creep deformation and strains we used viscoelastoplastic hereditary model of aging concrete. To determine the creep strain we used a linear approximation with respect to time. It was found that the long critical load for considered arch was in 1.44 times lower than the instant critical load.
Keywords: reinforced concrete arch, stability, creep, geometric nonlinearity, finite element method, Newton-Raphson method