Stability of a flexible orthotropic plate with free edges

An investigation is made into the postcritical behavior of rectangular flexible orthotropic plates under the action of distributed and concentrated loads. In particular, glass-reinforced plastic plates with different ratios of the moduli of elasticity are studied. The calculation is performed by the method of successive approximations, using finite differences, on a BESM-2M computer. The solutions obtained are compared with experimental data.Mekhanika Polimerov, Vol. 2, No. 4, pp. 585–592, 1966     

Stability of a square plate with an elastic border

We study the problem of stability of a square plate subject to uniform compression in both directions. The elastic border of the plate is characterized by the four stiffness coefficients (corresponding to the number of sides) in relation to the angle of rotation. We obtain an approximate formula for computing the critical load. For the cases when the plate has hinge support along the entire border or is rigidly clamped the results determined by this formula practically coincide with the exact solutions. One table. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 46–50, 1991.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Stress analysis of orthotropic planes weakened by cracks

The stress fields in an orthotropic infinite plane containing Volterra type climb and glide edge dislocations are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surfaces of smooth cracks. The integral equations are of Cauchy singular type and are solved for several different cases of crack configurations and arrangements. The results are used to evaluate modes I and II stress intensity factors for multiple smooth cracks.     

Anti-plane elastodynamic analysis of orthotropic planes weakened by several cracks

Stress analysis is carried out in an orthotropic plane containing a Volterra-type dislocation, the distributed dislocation technique is employed to obtain integral equations for an orthotropic plane weakened by cracks under time-harmonic anti-plane traction. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Bending of a cantilevered glass-reinforced plastic cylindrical shell weakened by a circular opening

The stress concentration is investigated in the neighborhood of a circular opening in a cantilevered glass-reinforced plastic cylindrical shell with a concentrated load at the free end. The problem is solved by the Bubnov method using a Ural-2 computer. The theoretical results have been checked experimentally on glass-reinforced plastic shells.All-Union Correspondence Polytechnic Institute, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 152–157, January–February, 1970.     

Bending of a semi-infinite cantilevered plate weakened by a cut with contacting edges

We solve the problem of the bending of a semi-infinite cantilevered plate containing a cut perpendicular to a clamped edge. Contact of the edges of the cut is taken into account in the two-dimensional formulation on the basis of the model of contacting edges on the face of the plate. We study the effect of the boundary on the distribution of the contact reaction and compute the coefficients of force and moment intensity and determine the breakinge load. We compare the results obtained with the solution of the problem not taking account of the contact of the edges of the cut. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 83–86.     

Mathematical model for crack arrest of an infinite plate weakened by a finite and two semi-infinite cracks

The problem of an unbounded plate weakened by three quasi-static coplanar and collinear straight cracks: two semi-infinite cracks and a finite crack situated symmetrically between two semi-infinite cracks, is investigated. The plate is subjected to uniform unidirectional in-plane tension at infinite boundary. Developed plastic zones are arrested by distributing cohesive yield point stress over their rims. The solution is obtained using complex variable technique. Closed form analytic expressions are derived for load bearing capacity and crack-tip-opening displacement (CTOD). A case study is presented for CTOD and load bearing capacity versus crack length, plastic zone length and inter-crack distance etc. Results are presented graphically and analyzed.     

Cylindrical bending of a thick orthotropic plate

We solve the problem of the stress-strain state of a thick rectangular orthotropic plate in a condition of cylindrical bending under a transverse load. Under this kind of strain the stressed state becomes two-dimensional and depends only on the transverse coordinate and the coordinate along the thickness of the plate. For large ratios of the width of the plate to its thickness we obtain results that are in good agreement with the data of the literature, the difference being at most 0.3%. This difference increases as the ratio decreases. Translated fromMatematichni Metodi to Fiziko-Mekhanichni Polya, Vol. 40, No. 2 1997, pp. 110–116.     

State of stress of flexible plastic shells with an opening

The state of stress of flat flexible shells with an opening is investigated with allowance for the viscoelastic properties of the material. The equilibrium equations and boundary conditions are written in finite-difference form. A nonlinear system of algebraic equations is solved by successive approximations. A method of accelerating the convergence of slowly converging iteration processes is proposed. The effect of the viscoelastic properties of the shell material on its state of stress is investigated with reference to the example of a polymethyl methacrylate shell. The variations of the ring moment and ring forces at the free edge of the shell are plotted for various moments of time, load values, and flatness parameters. It is shown that as soon as the viscosity factor begins to take effect, the state of stress and strain of the shell changes sharply; the concentration of forces and moments increases in the flexible viscoelastic (as compared with the elastic) shell.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1071–1075, November–December, 1973.     

State of stress of flexible viscoelastic plates with an opening

The problem of the state of stress of flexible plates with an opening is investigated with allowance for the viscoelastic properties of the material. The basic equations and the corresponding boundary conditions are formulated. The boundary value problem is solved by numerical integration. The effect of the creep properties of the material on the stress redistribution in the plate is investigated with reference to polymethyl methacrylate. It is shown that the angle of rotation, the ring moment, and the ring forces at the edge of the opening vary with time and as a function of the load.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 4, pp. 708–713, July–August, 1971.     

On bending vibrations of a periodically inhomogeneous orthotropic plate

The matrix technique which has been developed by the author in connection with solving the problem of elastic wave propagation in an unbounded medium with periodic inhomogeneities is applied. The formalism is extended to the case of bending vibrations of inhomogeneous and periodically inhomogeneous orthotropic plates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 179–181, 1989.     

Determination of the stresses in an orthotropic plate with fillets bent by forces normal to the plane of the plate

Methods based on Verkhovskii's hypothesis of nonplanar sections are used to derive equations for the stresses in an orthotropic plate with fillets bent by forces or moments whose plane of action is normal to the plane of the plate. In deriving these equations the depth of the fillets is taken into account.Gor'kii Zhdanov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1112–1115, November–December, 1969.     

Stability of a steady flow guided by flexible walls

Flexible channels conveying fluid may loose stability by divergence or flutter, thus leading to undesirable dynamic behaviour. A lot of investigations have already been done on this subject, each involving structural models expressing a specific type of stiffness. The aim of this contribution is to investigate and compare systematically the influence of various types of structural stiffness on the stability behaviour of the coupled problem. Therefore, a simple, yet general model is built. The fluid is bounded by a rigid and an elastical supported, flexible wall, which exhibits bending and extensional stiffness and considers effects of pre-load. The influence of the various characteristics on the stability of the steady state is discussed and a perturbation approach is used to investigate the influence of small nonlinearities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axisymmetric deformation of flexible orthotropic toroidal shells

Numerical solution of problems for constant-thickness toroidal shells in classical and increased-accuracy formulation is considered for various anisotropies and loads. The numerical approach is based on quasilinearization and discrete orthogonalization methods. The results of numerical examples are presented in graphical form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 70–76, 1988.