Geometrically non-linear equations in the theory of momentless shells with applications to problems on the non-classical forms of loss of stability of a cylinder

Geometrically non-linear and linearized equations in the theory of momentless shells are set up based on the kinematic relations in [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81]. The use of these equations, unlike in the case of the well-known equations, enables one to avoid the occurrence of spurious bifurcation points in solving real problems. Non-classical problems of the stability of cylindrical shells under an external pressure, axial compression and torsion are considered, which can be formulated on the basis of the derived equations of the theory of momentless shells. Their exact analytical solutions are found and enable one to estimate the quality of the previously obtained relations [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81] and the richness of content of the equations which have been constructed compared with well-known equations in the mechanics of thin shells. It is established that the majority of the new forms of loss of stability of cylindrical shells which are revealed relate to a number of shear forms, the onset of which is possible before the flexural forms which have been well studied up to now, in the case of small values of the shear modulus of a shell material with a very highly pronounced anisotropy in its properties.     

Refined geometrically nonlinear equations of motion for elongated rod-type plate

We derive new refined geometrically nonlinear equations of motion for elongated rod-type plates. They are based on the proposed earlier relationships of geometrically nonlinear theory of elasticity in the case of small deformations and refined S. P. Timoshenko’s shear model. These equations allow to describe the high-frequency torsional oscillation of elongated rod-type plate formed in them when plate performs low-frequency flexural vibrations. By limit transition to the classical model of rod theory we carry out transformation of derived equations to simplified system of equations of lower degree.     

The consistent equations of the theory of plane curvilinear rods for finite displacements and linearized problems of stability

Based on a previously constructed, consistent version of the geometrically non-linear equations of elasticity theory, for small deformations and arbitrary displacements, and a Timoshenko-type model taking into account transverse shear and compressive deformations, one-dimensional equations of an improved theory are derived for plane curvilinear rods of arbitrary type for arbitrary displacements and revolutions and with loading of the rods by follower and non-follower external forces. These equations are used to construct linearized equations of neutral equilibrium that enable all possible classical and non-classical forms of loss of stability (FLS) of rods of orthotropic material to be investigated, ignoring parametric deformation terms in the equations. These linearized equations are used to find accurate analytical solutions of the problem of plane classical flexural-shear and non-classical flexural-torsional FLS of a circular ring under the combined and separate action of a uniform external pressure and a compression in the radial direction by forces applied to both faces.     

Contact formulation of non-linear problems in the mechanics of shells with their end sections connected by a plane curvilinear rod

Starting from the consistent version of the geometrically non-linear equations of the theory of elasticity for small deformations and arbitrary displacements, a Timoshenko-type model that takes account of shear and compression deformations and also an extended variational Lagrange principle, an improved geometrically non-linear theory of static deformation is constructed for reinforced thin-walled structures with shell elements, the end sections of which are connected by a rod. It is based on the introduction into the treatment of contact forces and torques as unknowns on the lines joining the shells to the rods and it enables all classical and non-classical forms of loss of stability in structures of the class considered to be investigated. An analytical solution of the problem of the stability of a rectangular plate, that is under compression in one direction, supported by a hinge along two opposite edges and joined by a hinge with an elastic rod on one of the other two edges, is found using a simplified version of the linearized equations.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

The shell theory including transverse shear deformations and thickness change

A variational method for refining the theory of shells based on power series expansion of displacements has been described. The particular case of a cubic approximation for the tangential displacements and a quadratic approximation for the deflections is considered in detail. A constitutive system of differential equations in the canonical form for the axisymmetrical deformation of cyclindrical shells is derived. As an example, axisymmetrical deformations of a cylindrical shell made of an orthotropic composite material are discussed.Martin Luther Universität Halle-Wittenberg, Fachbereich Werkstoffwissenschaften. Germany. Kharkov State Polytechnical University, Department of Dynamics and Strength of Machines. Ukraine. Published in Mekhanika Kompozimykh Materialov, No. 6, pp. 768–780. November–December, 1997.     

A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges

We prove the existence theorem for solutions of geometrically nonlinear boundary-value problems for elastic shallow isotropic homogeneous shells with free edges under shear model of S. P. Timoshenko. Research method consists in the reduction of the original system of equilibrium equations to a single nonlinear equation for the components of transverse shear deformations. The basis of this method are integral representations for the generalized displacements, containing an arbitrary holomorphic functions, which are determined by the boundary conditions involving the theory of one-dimensional singular integral equations.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

球壳的环向剪切屈曲

通过球壳微元初始屈曲的微分几何分析,推导出一组新的精确的屈曲分支方程,并且应用Galerkin变分法研究铰支球壳承受环向剪切力时的整体稳定性,构造了接近分支点变形状态的屈曲模式,首次求得了从扁球壳到半球壳大范围内的扭转屈曲临界特征值,临界荷载强度和临界应力.     

A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis

This study compares the performance of linear and quadratic tetrahedral elements and hexahedral elements in various structural problems. The problems selected demonstrate different types of behavior, namely, bending, shear, torsional and axial deformations. It was observed that the results obtained with quadratic tetrahedral elements and hexahedral elements were equivalent in terms of both accuracy and CPU time.     

The dynamic process of the inflation of thin elastomeric shells under the action of an excess pressure

A problem of the dynamic process of their deformation is formulated in the momentless approximation for thin shells made of rubber-like elastomers under the action of a time-varying excess hydrostatic pressure. A system of non-linear equations of motion is set up for the case of arbitrary displacements and deformations in which the true deformation of the transverse compression of the shell, corresponding to the use of the modified Kirchhoff–Love model proposed earlier, and the coordinates of the points of the middle surface with respect to a fixed Cartesian system of coordinates, are taken as the required unknown functions. Physical relations connecting the components of the true internal stresses with the elongation factors and the extent of the shear strain are constructed using relations proposed earlier by Chernykh. A finite-difference method is developed for solving the initial-boundary value problem and, on the basis of this, the dynamic process of the inflation of shells of revolution at different rates of pressure increase is investigated and the unstable stages of their deformation are established with a determination of the corresponding limiting (critical) pressure value. After this value has been reached, a further increase in the deformations occurs at decreasing values of the internal pressure.     

斜放四角锥扁网壳的非线性弯曲理论

双层网壳是大型空间结构的主要结构形式,斜放四角锥扁网壳就是其中一种.它主要依靠上、下表层承受载荷,网壳腹部则比较空而且柔.根据斜放四角锥扁网壳的几何和力学特点,在三个基本假定的基础上,把它连续化并等效成一夹层扁壳.先从能量和内力等效的角度来分析它的本构关系,然后运用虚功原理,推导出斜放四角锥扁网壳几何非线性弯曲理论的基本方程.     

Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM.     

The possibility of a theoretical confirmation of the experimental values of the external critical pressure of thin-walled cylindrical shells

The problem of the buckling of elastic, isotropic, thin-walled cylindrical shells with small initial shape defects that are under the action of an external pressure is solved in a geometrically non-linear formulation. Equations that are identical to Marguerre's equations for a shallow cylindrical shell are used in formulating the problem. The solution is constructed by the Rayleigh–Ritz method with the points of the middle surface of the shell approximated by double functional sums over trigonometric and beam functions. The system of non-linear equations obtained is solved by arc-length methods. Cases of the clamped and supported shells when loading with a lateral and uniform hydrostatic pressure are considered. Its deflections from the limit points of the postbuckling branches of its loading trajectory are used as the initial imperfections. An inspection of the different forms of the initial imperfections when they have maximum values of up to 30% of the shell thickness made it possible to obtain practically the whole range of experimentally found critical pressures.     

Forced Vibration of Three-Layered Spherical and Ellipsoidal Shells under Axisymmetric Loads

A variant of vibration theory for three-layered shells of revolution under axisymmetric loads is elaborated by applying independent kinematic and static hypotheses to each layer, with account of transverse normal and shear strains in the core. Based on the Reissner variational principle for dynamic processes, equations of nonlinear vibrations and natural boundary conditions are obtained. The numerical method proposed for solving initial boundary-value problems is based on the use of integrodifferential approach for constructing finite-difference schemes with respect to spatial and time coordinates. Numerical solutions are obtained for dynamic deformations of open three-layered spherical and ellipsoidal shells, over a wide range of geometric and physical parameters of the core, for different types of boundary conditions. A comparative analysis is given for the results of investigating the dynamic behavior of three-layered shells of revolution by the equations proposed and the shell equations of Timoshenko and Kirhhoff-Love type, with the use of unified hypotheses across the heterogeneous structure of shells.     

Asymptotic Expansions and Influence Coefficients for Edge-Loaded Conical Shells

The equations of linear elasticity for rotationally symmetric deformations are expanded using a small parameter related to the thickness to radius of curvature ratio of the shell to obtain the classical thin shell equations of conical shells as a first approximation. These classical equations with variable coefficients permit further asymptotic expansions in the cases of steep as well as shallow cones, yielding systems of equations with constant coefficients. Solutions of these equations are used to compute the influence coefficients relating edge loads and edge displacements.     

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.     

Non-classical forms of loss stability of cylindrical shells joined by a stiffening ring for certain forms of loading

A structure in the form of two coaxial cylindrical shells with different radii, joined by a stiffening ring either rigidly or by hinges, is considered. Starting out from improved equations of general form constructed earlier, a linearized contact problem is formulated that enables all possible classical and non-classical forms of loss of stability to be investigated in the case of axisymmetric forms of loading of the structure. The initial relations of the problem are transformed to an equivalent system of integro-algebraic equations containing integral Volterra-type operators by integrating along the longitudinal coordinate and representing the two-dimensional and one-dimensional required unknowns introduced into the treatment in the form of the sum of trigonometric functions in the circumferential coordinate that, in changing into a perturbed state, allows the possibility of the shell deforming in antiphase forms. A numerical algorithm for constructing solutions of the resulting equations is proposed, based on the method of finite sums, that enables all the boundary conditions of the problem and the conditions for the joining of the shells with the stiffening ring to be satisfied exactly. Retaining and discarding parametric terms in the relations for the shells, the stability of a structure of the class considered is investigated in the case when an external pressure acts on the stiffening ring and, also, in the case of its axial tension during which the stiffening ring is found to be under wrench deformation conditions and, in a shell of larger diameter, subcritical circumferential compressive stresses are formed.     

Nonlinear vibration analysis of tapered Timoshenko beams

The nonlinear flexural vibration analysis of tapered Timoshenko beams is conducted. The equations of motion for tapered Timoshenko beams are established in which the effects of nonlinear transverse deformation, nonlinear curvature as well as nonlinear axial deformation are taken into account. The nonlinear fundamental frequencies of tapered Timoshenko beams with two simply supported or clamped ends are presented.     

On the averaging method for rod equations with quadratic non-linearity

The averaging method is used to approximate solutions of systems of linearly coupled, (quadratic) non-linear dispersive wave equations, which describe extensional–torsional dynamics of a rod. Existence and uniqueness results are established. Error estimates confirm the asymptotic validity of the approximation method on a long time-scale. The linear couplings between the equations imply that resonance can occur inside a single mode of the solution, but energy can also be transferred to other modes.