Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Effect of reinforcing elements on the deformation and strength of glass-reinforced plastics in tension

The problem of extension of a bar of constant cross section containing reinforcing elements in the form of rods arranged parallel to the generators of the lateral surface is considered. A solution is obtained by the Muskhelishvili method. This solution is analyzed and approximate formulas are proposed for the averaged characteristics of materials (especially glass-reinforced plastics) as a function of the relative contents and mechanical properties of the components. The formulas obtained closely reflect a number of known experimental facts.Mekhanika Polimerov, Vol. 1, No. 2, pp. 104–113, 1965     

Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.     

Asymptotic behaviour of curved rods by the unfolding method

We consider in this work general curved rods with a circular cross‐section of radius δ. Our aim is to study the asymptotic behaviour of such rods as δ→0, in the framework of the linear elasticity according to the unfolding method. It consists in giving some decompositions of the displacements of such rods, and then in passing to the limit in a fixed domain. A first decomposition concerns the elementary displacements of a curved rod which characterize its translations and rotations, and the residual displacements related to the deformation of the cross‐section. The second decomposition concerns the displacements of the middle‐line of the rod. We prove that such a displacement can be written as the sum of an inextensional displacement and of an extensional one. An extensional displacement will modify the length of the middle‐line, while an inextensional displacement will not change this length in a first approximation. We show that the H¹‐norm of an inextensional displacement is of order 1, while that of an extensional displacement is in general, of order δ. A priori estimates are established and convergence results as δ→0, are given for the displacements. We give their unfolded limits, as well as the unfolded limits of the strain and stress tensors. To prove the convergence of the strain tensor, the introduction of elementary and residual displacements appears as essential. By passing to the limit as δ→0 in the linearized system of the elasticity, we obtain on the one hand, a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacements and the limit of the angle of torsion. Copyright © 2004 John Wiley & Sons, Ltd.     

压电梁的多项式解（Ⅱ）——典型问题解析解

从正交各向异性压电介质平面问题,对于材料3个特征根互不相等情况下,以3个拟调和位移函数表达位移、电势、应力和电位移的通解出发,利用调和多项式的显式表达式,结合试凑法,给出了平面压电梁的若干典型问题的解析解,包括悬臂压电梁自由端作用横向集中力和点电荷,悬臂压电梁表面作用线性电势和均布载荷,以及两端简支压电梁作用均布载荷等的解析解．     

Mixed finite element models for free vibrations of thin-walled beams

Simple mixed finite element models are developed for the free vibration analysis of curved thin-walled beams with arbitrary open cross section. The analytical formulation is based on a Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling, and the additional effects of transverse shear deformation and rotary inertia. The fundamental unknowns consist of seven internal forces and seven generalized displacements of the beam. The element characteristic arrays are obtained by using a perturbed Lagrangian-mixed variational principle. Only C⁰ continuity is required for the generalized displacements. The internal forces and the Lagrange multiplier are allowed to be discontinuous at interelement boundaries.

Numerical results are presented to demonstrate the high accuracy and effectiveness of the elements developed. The standard of comparison is taken to be the solutions obtained by using two-dimensional plate/shell models for the beams.     

Distribution of shearing stresses under three-point flexure in beams made of composite materials

Conclusions 1. When short beams made of composite materials with span-to-height ratios within the 4.0–5.0 range are tested in three-point flexure, the profile of shearing stresses over the height of a beam cross section departs appreciably from a parabolic one over approximately 40% of the span length.2. The zones of a perturbed state of stress, which appear in short beams tested in three-point flexure, render this method unusable for determining the interlayer shear strength of composite materials on "ultrashort" specimens.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 56–62, January–February, 1977.     

Optimal Multipliers for Linear Congruential Pseudo-Random Number Generators with Prime Moduli: Parallel Computation and Properties

Two systematic search methods are employed to find multipliers for linear congruential pseudo-random number generation which are optimal with respect to an upper bound for the discrepancy of pairs of successive pseudo-random numbers. The efficiency of these search procedures when executed on parallel systems is assessed by experimental results of a MIMD parallel implementation on a Meiko CS-2 and a workstation cluster. Furthermore the quality of the computed multipliers is evaluated by using the spectral test in dimensions 2–8 and by calculating the actual discrepancy of pairs of the resulting full-period sequences.     

The bending of clamped beams made of materials with low resistance to shear

Oriented glass-reinforced plastics have low resistance to shear. This means that the calculation of beams made of these materials on the basis of the hypothesis of plane cross sections is not possible. We compare several methods of calculating shear and verify them on the basis of experimental data obtained for beams made of oriented glass-reinforced plastics with one or both ends clamped. We also discuss the boundary conditions in the clamped cross section and the effect of axial force in the case of fixed supports.Mekhanika Polimerov, Vol. 3, No. 4, pp. 730–736, 1967     

有限位移理论线弹性动力学二类和三类混合变量的最小势作用量原理和驻值余作用量原理及其应用

两个新的概念,即势作用量的概念和余作用量的概念被引入弹性动力学变分原理中.根据势作用量的概念,最小作用量原理（即Hamilton原理）被改称为最小势作用量原理.根据余作用量的概念,首次提出了驻值余作用量原理.考虑边界条件的变化并应用有限位移理论的功的互等定理,导出了以位移和应力为变分变量的二类混合变量的最小势作用量原理及驻值余作用量原理.应用应变势能密度与应力余能密度的关系式于上述二类混合变量作用量原理,导出了以位移、应力和应变为变分变量的三类混合变量的相关作用量原理.最后,应用拉氏乘子法给出了广义势作用量原理及广义余作用量原理,并且应用大挠度梁二类混合变量最小势作用量原理计算了一悬臂梁的受迫振动.     

Contact Problem for a Pneumatic Tire Interacting with a Rigid Foundation

Based on mixed finite-element approximations, a numerical algorithm is developed for solving a geometrically nonlinear contact problem for a prestressed multilayered Timoshenko-type shell undergoing arbitrarily large displacements and rotations. As unknowns, six displacements of faces of the shell are taken, which allows one to use principally new relationships for components of the Green–Lagrange strain tensor in curvilinear orthogonal coordinates, exactly representing arbitrarily large displacements of the shell as a rigid body. As an example, a tire interacting with a rigid foundation is considered.     

Investigation of the principal dense-winding conditions for wound glass-reinforced plastic pipe

The dense-winding conditions have been found for glass-reinforced plastic pipe wound simultaneously with a large number of strands [conditions (15), (17), and (18)] and relations have been obtained for calculating the winding mechanism kinematics. Kinematic equations have been derived for dense winding by displacement of the strands in the pipe cross section, along the pipe axis, and in both directions at once.Glass-Reinforced Plastics Division of the All-Union Lenin Electrical Engineering Institute, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1057–1064, November–December, 1970.     

Choice of optimal structure of glass laminates with random reinforcement. I

The effect of structural parameters — length, diameter, and distribution of the reinforcing elements — on the mechanical characteristics of glass-reinforced plastics is investigated with reference to the case of glass laminates with randomly distributed, straight, uncut glass fibers in parallel planes. It is shown that the reduced strength of these laminates as compared with unidirectional material is associated with the redistribution of the load between the fibers and the resin and the relative reduction in the number of fibers in the cross section. A formula is proposed for estimating the strength of glass-reinforced plastics with a random distribution of the fibers in parallel planes.All-Union Scientific-Research Institute of Glass-Reinforced Plastics and Glass Fiber, Moscow Region. Moscow Bauman Higher Technical College. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1043–1050, November–December, 1968.     

Finite element construction for simulating delamination of sandwich composite plates and masses

Displacements and transverse normal stresses in sandwich plates and masses have been approximated by the Ambartsumyan iterative approach to constructing mathematical models of the stress-strain state of sandwich structures. A linear distribution of the displacements in the sandwich structure is set up as the first step of the iterative process, while in the subsequent steps the displacement approximations with higher-order polynomials are obtained. The approximation of the compression stresses is based on Hooke's law using the expression of the tangential displacements in the second step and the normal displacements in the third step of the iterative process. Two shear functions are introduced. The finite element is rectangular and has four nodes. The number of degrees of freedom of finite elements is independent of the quantity of the layers that may be orthotropic. The finite element allows us to simulate delamination by a thin low-modulus interlayer. In doing so, the quantity of the layers increases, while the order of the resolving set of equations does not grow. A number of numerical experiments were carried out. It has been shown that the delamination can greatly increase the level of the stresses in the structure. This effect is especially significant for thin structures. The stresses are somewhat lower when taking into account the interlaminar friction.Submitted to the 10th International Conference on Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 251–263, March–April, 1998.     

El concepto de hiperviga como generalización de los modelos unidimensionales en la estática de piezas alargadas

This paper is concerned with the general formulation of linear problems in rod elastostatics, and with the identification of their common formal and structural features, valid for every kinematical hypothesis. The generalized variables (section forces and generalized deformations) defining the 1-dimensional model are introduced in a consistent and natural way, through a convenient factorization of the density of complementary potential energy, for every kinematical constraint which can be expressed as a linear combination of the generalized displacements. The identification of this complementary energy function with the Hamiltonian functional of analytical mechanics allows a systematic procedure to construct the equations which rule this class of problems. In this frame, the main target is to establish the required conditions to write the rod equilibrium equations in a purely statical form (with no interplay of kinematical variables). We primarily conclude that this is possible when the kinematics of the cross-sections is constrained to a rigid body movement. As a consequence, the concept of hyperbeam is proposed in order to define those models with deformation modes beyond rigid body movements of the cross section: in them, section forces and generalized displacements are coupled in the equilibrium equations. This is related to the idea of local static indeterminacy (hyperstaticity) and justifies the new name.     

An Augmented Beam Element for Structural Dynamics Using Unit Deflection Shapes Gained on a 2D Finite Element Mesh of the Cross Section

Wave behaviour of line shaped structures can be analysed if they are simply shaped (beams, rods, etc.) and if the frequency or wavelength is within a range where the underlying theories are still applicable. If the frequencies and wavelengths exceed these limits, the modelling of these line or beam shaped structures is only possible using volume elements which results in considerably high computation time and modelling effort. The idea of this work is to keep the beam like modelling of structures but enhance the solution space such that arbitrary but reasonable deflection shapes of the cross section can be covered. The unit deflection shapes are defined and computed on the basis of a 2D finite element mesh of the beam cross section: warpshapes for primary and secondary torsion and shear forces, Eigenmodes for the problem of a plate in membrane and a plate in bending action, derived warpshapes for in plane shapes and shapes which cover the lateral strains. By this transformation of unknowns the number of system degrees of freedom can be reduced considerably. Expensive pre- and postprocessing steps can be done in parallel which opens the gate for massive performance gains. This method was validated in previous publications of the authors and shall now be extended using unit deflection shapes gained from eigenmodes of an infinite Fourier transformed structure obtained by solving a quadratic eigenvalue problem for the wavenumber k_x. The results of both methods shall be compared. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the theory of layered anisotropic cylindrical shells stiffened with ribs

The deformation, stability and vibration equations for anisotropic cylindrical shells stiffened with individual longitudinal and circumferential ribs are derived without introducing the hypothesis of nondeformable normals. The more general assumption adopted for layered materials (for example, glass-reinforced plastics) of a linear variation of the displacements over the thickness of the shell and the height of the ribs is used; in this case for the points of contact of the shell and the ribs after deformation the common normals form broken lines. The solution of the problem of the stability of a cylindrical shell stiffened with circumferential ribs is examined. For a shell with different, arbitrarily located ribs the problem is reduced to a homogeneous algebraic system of equations equal in number to three times the number of ribs.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 647–654, July–August, 1974.     

Action of a concentrated load on an elastic ring pressed into a circular hole in an anisotropic plate

A solution is obtained for the elastic equilibrium problem for an anisotropic plate with a circular hole into which is pressed a ring of arbitrary cross section symmetrical about the middle surface of the plate. Before deformation the outside radius of the ring differs from the radius of the hole in the plate by the amount of the permissible elastic displacements. A normal concentrated load is applied to the ring. The ring is analyzed in accordance with the theory of thin curved bars. A numerical example is given.I. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 6, pp. 1054–1059, November–December, 1973.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

Piezoelectric beams under small strains but large displacements and rotations

Small strains are consistently incorporated into a model to describe the behavior of piezoelectric beams subjected to large displacements and rotations. While the displacement is assumed to vary in accordance with the Timoshenko assumption, the electric potential has linear variation through each piezoelectric layer thickness. The strong geometric nonlinear effect on the beam electroelastic response is illustrated by static problems of cantilever beams with tip loads and distributed sensing or actuation. The present work seems to be the first to obtain analytical solutions for beams under large displacements and rotations with piezoelectric sensors. One believes that the quality of such solutions can be valuable to validate results predicted by approximate methods.