Lagrange Variational Principle Research Articles

Multi-criteria parametric optimization of the displacement and weight of a shell of minimal surface on a circular contour consisting of two inclined ellipses under thermal and power loading with consideration of geometric nonlinearity

Thin shells are well suited to optimal design problems, and the finite element method and gradient descent method make it possible to solve inverse problems in structural and applied mechanics. The calculation process takes into account geometric nonlinearity. In modern numerical studies, a formulation taking into account geometric nonlinearity in the finite element method is used, namely, a stepwise loading procedure. It becomes necessary to take into account the relations between displacement vectors and their derivatives and strain increments. These relations help to determine the stiffness matrix of the finite element at each loading step, which leads to a qualitative study of real displacements in the structure. The geometrically nonlinear calculation of a shell of minimum surface on a circular contour consisting of two inclined ellipses in a multi-criteria parametric optimization is performed by the finite element method. The finite element method is a universal variational method that is focused on solving the most complex problems of elasticity theory and applied and structural mechanics using separate calculation complexes. Stiffness matrix - for the whole body is formed on the basis of the finite element stiffness matrix. The system of solving equations of the finite element method is formed using the Lagrange's variational principle, according to which the total potential energy P of a finite element model of a body in a state of stability and equilibrium has a minimum value. As part of the sensitivity analysis, gradients of the design variables of the structure, displacements in the form of partial derivatives of these characteristics along the design variables, and shell thickness are calculated. The sensitivity information serves as the basis for building an optimal design algorithm using the gradient descent method of the objective function. Standard multi-criteria parametric optimization allows for an average of 10% steel savings, with geometric nonlinearity increasing to 20%. At this study site, 24.4% of sheet steel was saved, which is a significant relative saving. This methodology of the authors shows high results for innovative design and production of steel thin shells in Ukraine and around the world, and also makes it possible to apply several types of optimization to one research object.

Building of a Mathematical Model for Solving the Elastic–Gas-Dynamic Task of the Gas Lubrication Theory for Petal Bearings

Petal gas bearings are widely used in industrial turbine installations. However, there are practically no mathematical models that take into account the nonlinear dependence of the elastic element deformations of the gas-dynamic bearing on the pressure. It is very difficult to obtain a converged solution with a given accuracy in an acceptable time. On this basis, a mathematical model of the motion of a gas-dynamic bearing with a petals package was proposed. The model takes into account the pressure dependence of the elastic element deflections of the bearing, as well as the bearing design features and its elastic elements. A feature of solving the presented task was the joint solution of two subtasks: gas lubrication and deformations of the elastic elements of the bearing. The problem was that the contact areas of the elastic elements are not known in advance. The search for deflections or the formulation of the elasticity problem was based on the Lagrange variational principle. Petals are in the shape of thin cylindrical shells. The solution was achieved by minimizing the potential energy of the system of deformed petal shells using the first-order gradient method. The solution to the gas-dynamics problem was achieved by applying an explicit finite-difference approximation in time. To implement the proposed model, a numerical algorithm was developed to solve the related tasks of dynamics and elasticity by combining the methods of variational calculus, optimization, and an explicit finite-difference time scheme. The order of accuracy of convergence of the solution corresponded to 10−5. The study demonstrated that the load-carrying capability of the bearing increased by 2–4 times with an increase in the number of petals. The results of experimental studies allowed us to estimate the interval for the ascent speed of the rotor, which was used as the initial conditions for numerical modeling.

Optimal design of the forced oscillation frequencies of the minimum surface shell on a circular contour consisting of two inclined ellipses under thermal and power loading

The problem of choosing the optimal design solution is a multi-purpose complex task, the solution of which requires taking into account a wide range of requirements for different stages of implementation. There are several levels of solving optimal design problems. The first level deals with general design models, including the main structural elements, taking into account the physical and geometric features of their behavior. This level can be called the "global" level of optimization. Due to the fact that the automated calculation of structures is impossible without the use of finite element models, and at the next "detailed" level, the parameters that make up this model (coordinates of nodes, physical and geometric properties of individual elements, etc.)
 The choice of a finite element model for optimal design is directly related to the complexity (the possibility of computer implementation) of the optimal design problem. To reduce the computational procedures associated with solving the problem of numerical analysis, an approach based on the approximation of system parameters (displacements, forces, frequencies, waveforms, etc.) can be used.
 The problem of forced oscillations for shells of minimal surfaces is solved by the finite element method using the Lagrange's variational principle. The potential and kinetic deformation energies of the shell of a minimal surface are defined in matrix form as the sum of the potential and kinetic energies of each finite element.
 The optimization algorithm for single-criterion parametric optimization is performed as follows: the objective function is the weight of the shell of the minimum surface on a circular contour consisting of two inclined ellipses, the design variables are the thickness of the shell, and the constraints are represented by the first forced vibration frequency. The results of changing the objective function are a decrease in the weight of the shell by 3050 kg of C240 steel, which is a percentage equivalent of 13.4% without loss of strength and stability of the shell of the minimum surface on a circular contour consisting of two inclined ellipses, which proves the effectiveness of the authors' methodology.

Variational Estimation Methods for Sturm–Liouville Problems

In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated through the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are unitary presented in final part of the paper, which ends with applications.

Investigation of the stress-strain state of a hollow cylinder with a coating based on the gradient model of thermoelasticity

The study of the stress-strain state of a thermoelastic hollow cylinder with a homogeneous coating is carried out taking into account the scale effects. Aifantis' one-parameter gradient model is used to account for scale effects. Equilibrium equations and boundary conditions for a composite hollow thermoelastic cylinder are obtained on the basis of the Lagrange variational principle. In comparison with the classical formulation of the problem, additional boundary conditions and conjugation conditions are set for moment stresses and displacement gradients. The dimensionlessness of the task of thermoelasticity has been carried out. Solving the problem of uncoupled thermoelasticity begins with finding the radial temperature distribution of a layered cylinder on the basis of solving the problem of heat conduction in the classical formulation. The solution of the problem in displacements is presented as a sum of solutions in the classical formulation of the problem and additional boundary layer terms found on the basis of the asymptotic properties of the modified Bessel functions. Simplified analytical expressions are obtained for finding radial displacements, radial and circumferential Cauchy stresses, nonzero components of the tensor of moment and total stresses. On specific examples, calculations of the radial distribution of displacements and stresses of a composite cylinder in the case of both mechanical and thermal loading are carried out. The limits of applicability of the asymptotic solution of the problem are investigated. The difference between the radial distribution of displacements and stresses found on the basis of solutions to the problem in the classical formulation and in the gradient formulation is shown. It was found that the Cauchy radial stresses experience a jump at the boundary of the cylinder and the coating, which is explained by the continuity of radial displacements and their first derivatives. The components of the moment stress tensor either take on peak values or experience a jump at the interface. The moment stresses are proportional to the square of the gradient parameter, at small values of which they have values that are much less than the values of the total stresses. With an increase in the dimensionless scale parameter, the values of radial displacements and total circumferential stresses decrease, but moment stresses increase.

Finite element modeling of creep bending of polymer plates

Thin rigid plates are considered in the article and the derivation of the equations of bending of a quadrangular finite element of a plate is given that takes creep into account creep. The equations are derived using the Lagrange variational principle. The relationships between the integral characteristics of stresses (linear bending and torque moments) and the curvatures of the middle surface are taken as physical relationships. Given as an example, the problem of analyzing a rectangular polymer plate made of secondary PVC, hinged along the contour and loaded with a load uniformly distributed over the area, is reduced to a system of linear algebraic equations. Maxwell-Gurevich equation was choosen as the equation of state between stresses and creep strains. Graphs of stress and deflection changes over time are presented. The stresses in the creep process change insignificantly, the difference between the stresses at the beginning and at the end of the creep process is less than 5%.

Axial tension of nonlinear elastic composite shells of zero Gaussian curvature with a rectangular hole

The formulation of physically nonlinear problems for composite shells of zero Gaussian curvature weakened by a rectangular hole under the action of axial loading is given. The initial equations are the equations of the theory of non-sloping shells, in which the Kirchhoff–Love hypotheses take place. Geometric relationships are written in vector form, and physical relationships are based on the deformation theory of plasticity for anisotropic materials. The system of resolving equations is obtained from the Lagrange variational principle. A technique has been developed for the numerical solution of two-dimensional physically nonlinear problems for orthotropic composite shells of this type, based on the use of the method of additional stresses and the method of finite elements. A variant of the finite element method is proposed, the peculiarity of which lies in the vector approximation of the sought values and the discrete execution of the geometric part of the Kirchhoff–Love hypotheses (at the nodes of finite elements). Using the developed technique, the nonlinear elastic state of an organoplastic conical shell with a rectangular hole, which at the ends is reinforced with frames and loaded with uniformly distributed tensile forces, has been investigated.

Stability analysis of wooden arches with account for nonlinear creep

Introduction. The paper deals with the calculation of wooden arches taking into account the nonlinear relationship between stresses and instantaneous deformations, as well as creep and geometric nonlinearity, are considered. The analysis is based on the integral equation of the viscoelastoplastic hereditary aging model, originally proposed by A.G. Tamrazyan [1] to describe the nonlinear creep of concrete. Materials and Methods. The creep measure is taken in accordance with the work of I.E. Prokopovich and V.A. Zedgenidze [2] as a sum of exponential functions. The transition from the integral form of the creep law to the differential form is shown. The relationship between stresses and instantaneous deformations for wood under compression is determined from the Gerstner formula, and elastic work is assumed under tension. The solution is carried out using the finite element method in combination with the Newton-Raphson method and the Euler method according to the scheme that involves a stepwise increase in the load with correction of the stiffness matrix taking into account the change in the coordinates of the nodes with the sequential calculation of additional displacements of the nodes, which are due to the residual forces. The proposed approach for increasing the accuracy of determination of creep deformations at each step provides using the fourth-order Runge-Kutta method instead of the Euler method. Results. Based on the Lagrange variational principle, expressions are obtained for the stiffness matrix and the vector of additional dummy loads due to creep. The method developed by the authors is implemented in the form of a program in the MATLAB environment. Calculation examples are given for parabolic arches simply supported at the ends without an intermediate hinge and with an intermediate hinge in the middle of the span under the action of a uniformly distributed load. The results obtained are compared in the viscoelastic and viscoelastic formulation. The reliability of the results is validated through the calculation in the elastic formulation in the ANSYS software package. Discussion and Conclusions. For the arches considered, it is found that even with a load close to the instant critical, the growth of time travel is limited. Thus, the nature of their work under creep conditions differs drastically from the nature of the deformation of compressed rods.

The Stress State in the Boundary Region of a Conical Shell according to a Refined Theory

In this article, on the basis of a refined theory, we study the stress state of a conical shell of the “boundary layer” type. In comparison with the classical theory, the sought displacements of the shell are approximated by polynomials along the coordinate normal to the median surface with a degree two units higher. Based on the equations of the three-dimensional elasticity theory and the Lagrange variational principle, a system of differential equations of equilibrium in displacements with variable coefficients is obtained. The solution of the formulated boundary value problem is carried out by using the methods of finite differences and matrix sweep. The results can be used when assessing the strength and durability of shell structures.

Refined Nonlinear Deformation Model of Sandwich Plates with Composite Facings and Transversal-Soft Core

Following up on the results obtained earlier, a refined nonlinear model of static deformation of sandwich plates with transversal-soft core and facings with low stiffness of transverse shear and transverse compression is constructed for the case of cylindrical bending. It is based on the use of linear approximations in thickness for deflections of the external layers, cubic approximation in thickness for tangential displacements, and simplified three-dimensional equations of elasticity theory that can be integrated along the transverse coordinate with introduction of two unknown functions representing constant transverse tangential stresses in thickness for transversal-soft core. The kinematic relations for the facings are constructed in a geometrically nonlinear quadratic approximation. They allow, taking into account physical nonlinear behavior of the material under transverse shear conditions, the description of non-classical transverse-shear forms of stability loss (FSL) in both compression and bending conditions. Based on the generalized Lagrange variational principle, one-dimensional nonlinear equations of equilibrium and conjugation of the facings with the core by tangential displacements are constructed to describe the static deformation process with high rates of variability of the parameters of the stress-strain state (SSS).

The energy method for calculating the cantilever strip bending flat form stability taking into account its own weight development

An effective version of the energy method is recommended when calculating the rectangular cantilever strips for stability of a flat bending shape taking into account its own weight. The essence of this method’s variant is to use the Lagrange variational principle instead of the condition for the potential strain energy equality and the external forces work. The proposed approach makes it possible to perform the calculations’ machine implementation and take into account an arbitrary number of the series members. The solution to the problem for the cantilever beam is presented taking into account its own weight and the concentrated force action.

МЕТОД ЭНЕРГИИ В РАСЧЕТЕ НА УСТОЙЧИВОСТЬ ПЛОСКОЙ ФОРМЫ ИЗГИБА КОНСОЛЬНОЙ ПОЛОСЫ С УЧЕТОМ СОБСТВЕННОГО ВЕСА

the problem of bending a strip by a force applied at the end is not of practical interest. Such method of loading and securing the ends is interesting only because it is most convenient to implement it on experience, which makes it possible to verify the Prandtl theory. When conducting experiments, there is a need for two corrections: it is necessary to evaluate the influence of the self-weight of the strip and the effect of increasing or decreasing the point of force application. As we are talking about small corrections, it is quite enough to use only the first approximation for calculations. An effective version of the energy method is recommended. It is used to calculate rectangular cantilever strips for stability of a flat bending shape, taking into account its own weight. The essence of this variant of the method is to use the Lagrange variational principle instead of the condition for equality of the potential strain energy and the work of external forces. The proposed approach allows us to perform machine implementation of calculations and take into account an arbitrary number of members of the series. The presented solution of the problem for the cantilever beam takes into account its own weight and the action of concentrated force.

Mathematical Models and Methods for Calculating the Stress-Strain State of Aircraft Panels from Composite Materials Taking into Account the Production Technology

The relations of the mathematical model for studying the stress-strain state of structurally anisotropic panels made of composite materials are given. The mathematical model of the reinforcing element is clarified under conditions of one-sided contact with the skin. The influence of panels manufacturing technology is taken into account: residual temperature stresses and prestressing of reinforcing fibers. Based on the Lagrange variational principle, an eighth-order resolving equation and natural boundary conditions are obtained. A class of exact solutions of boundary value problems is considered. In the MATLAB operating environment, a software package has been developed. The results of the numerical implementation of the solution and the results of field experiments are presented.

Combined effects of surface energy and couple stress on the nonlinear bending of FG-CNTR nanobeams

This paper investigates the size-dependent nonlinear bending of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams. Chen–Yao’s surface elasticity and modified couple stress theories are adopted to describe surface effects and couple stress effects, respectively. These nanobeams, in which the carbon nanotube (CNT)-reinforced phases are assumed to be distributed in a gradient along the thickness, are subjected to a uniform pressure and rest on a nonlinear elastic foundation. In accordance with the Euler–Lagrange variational principle, the governing equations and boundary conditions for the FG-CNTR nanobeams, which involve geometric nonlinearity due to the von Kármán strain relations, are obtained. Then, with the assistance of the two-step perturbation technique, the load-deflection relationship is determined for nanobeams subjected to simply supported (SS) and clamped–clamped (CC) boundary conditions. Finally, the impacts of various factors, including surface properties, characteristic material length, elastic foundation, geometric factors, layout type and volume fraction of CNTs, on the mechanical behaviors of CNT-based nanobeams are examined. The numerical results reveal that the combination of surface effects and couple stress helps to enhanceq the stiffness of the nanobeams. Furthermore, the size-dependent nonlinear bending of the FG-CNTR nanobeam is markedly affected by the content and layout type of the reinforcements.

Nonlinear Bending Analysis of Functionally Graded CNT-Reinforced Shallow Arches Placed on Elastic Foundations

This research explores the nonlinear bending behaviors of functionally graded carbon nanotube-reinforced (FG-CNTR) shallow arches with unmovable simply supported ends and clamped–clamped ends; these arches are subjected to a uniform radial pressure and rest on a nonlinear elastic foundation. The temperature-dependent material properties of the arches are considered. Within the framework of Reddy shear deformation theory possessing von Karman nonlinearity, the motion equations and boundary conditions for the FG-CNTR arches are determined by the Euler–Lagrange variational principle. Then, a two-step perturbation technique is adopted to determine the load–deflection relationship analytically. To verify the validity of the developed model and related perturbation solutions, a numerical investigation is conducted for shallow arches with five distribution patterns of carbon nanotube (CNT) reinforcements uniaxially aligned in the axial direction. Finally, the influences of various factors, including the elastic foundation, layout type, and volume fraction of CNTs and geometric factors, on the nonlinear behaviors of FG-CNTR shallow arches are examined. The obtained results show that the load–deflection curves exhibit less snap-through instability as the CNT volume fraction increases. The transverse shear stress versus the thickness of FG-CNTR shallow arches is markedly affected by the layout type and content of reinforcements.

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. A notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. A closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.

The Stressed State of the “Boundary Layer” Type in Cylindrical Shells Investigated according to a Nonclassical Theory

Two variants of a refined theory for calculating the stress–strain state in the boundary zones of cylindrical shells are presented. The relevant mathematical models are based on equations of the three-dimensional elasticity theory and the increase over the thickness of the shell in the orders of the polynomials that approximate the sought displacements. The Lagrange variational principle is applied to the value of the shell’s total energy functional defined more exactly with respect to the classical Kirchhoff–Love theory. The formulated boundary problems allow determination with different degrees of accuracy of additional stressed states of the “boundary layer” type. The calculated results obtained in this work are compared with the results obtained according to the classical theory. It has been established that the above stresses make a significant contribution to the total stressed state of the shell and should be considered when designing and testing machine structures for strength and longevity.

Formulation of Problems in the Bernoulli—Euler Theory of Anisotropic Inhomogeneous Beams

A procedure of reducing the three-dimensional problem of elasticity theory for a rectilinear beam made of an anisotropic iuhomogeueous material to a one-dimensional problem on the beam axis is studied. The beam is in equilibrium under the action of volume and surface forces. The internal force equations are derived on the basis of equilibrium conditions for the beam from its end to any cross section. The internal force factors are related to the characteristics of the strained axis under the prior assumptions on the distribution of displacements over the cross section of the beam. To regulate these assumptions, the displacements of the beam’s points are expanded in two-dimensional Taylor series with respect to the transverse coordinates. Some physical hypotheses on the behavior of the cross section under deformation are used. The well-known hypotheses of Bernoulli—Euler, Timoslienko, and Reissner are considered in detail. A closed system of equations is proposed for the theory of anisotropic iuhomogeueous beams on the basis of the Bernoulli—Euler hypothesis. The boundary conditions are formulated from the Lagrange variational principle. A number of particular cases are discussed.

Investigation of three-layer rectangular panel bending by variation method

The task of transverse bending of a rectangular three-layer panel with the lightweight filler is considered. The differential equation system of transverse bending of the panel is generated within the scope of Lagrange variational principle. An explicit solution of the differential equation system in double trigonometric Fourier series is obtained. It is shown that the maximum deflection of the three-layer panel can be compared with the three-layer beam deflection, with a multiple panel width excess over its length.

Theory of Moderately Large Deflections of Sandwich Shells Having a Transversely Soft Core and Reinforced Along Their Contour

Variants of sandwich structural elements in the form of plates and shells with a transversely soft core are analyzed. Their outer, load-carrying layers are reinforced along their outer contour with elastic bars to ensure the transfer of loads to the layers during their interaction with other structural elements. For such structures, at small strains and moderate displacements, a refined geometrically nonlinear a theory is constructed that allows one to describe their subcritical deformation and reveal all possible buckling modes (cophasal, antiphasal, mixed flexural, mixed shear-flexural, and arbitrary modes including all the listed ones) of the load-carrying layers and the reinforcing elements (flexural, shear-flexural, and pure shear ones at various subcritical stressstrain states). This theory is based on considering the interaction forces of outer layers with the core and of the layers and core with the reinforcing bars as unknowns. To derive the basic equations of static equilibrium, boundary conditions for the shell and stiffening bars, the conditions of kinematic conjunction of outer layers with the core and of the outer layers and core with the reinforcing bars, and the generalized Lagrange variational principle proposed earlier are utilized. The theory suggested differs from all known variants by a high degree of accuracy and maningfulness at a minimum number of unknown two-dimensional functions for the shells, one-dimensional functions for the reinforcing bars, and one and two-dimensional contact forces of interaction between structural elements.