Stability of composite shells with an elastic core

The stability in axial compression of orthotropic cylindrical shells with an elastic core is investigated with allowance for the following factors: transverse shear strains in the shell material, precritical state of stress of the core, and the moments due to the surface forces exerted on the shell by the core. The numerical results obtained are compared with the results of approximate methods in which the above-mentioned factors are disregarded.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 544–547, May–June, 1976.     

A refined model for three-point bending of sandwich panels 2. Transverse stresses

According to the relationships derived in [1], transverse normal and tangential stresses in a sandwich panel have been analyzed. Asymptotic formulas for the stress concentration area in the vicinity of point forces are derived. Analytical estimates of a normal stress at the central and end sections of the panel are deduced. The Saint-Venant effect of the degeneration of a panel of finite length into an infinite strip is studied. For the estimation of the concentration of the transverse tangential stress, the possibility of a superposition of the solution of the slippage problem of the face layers and the classical solution allowing for shear is substantiated. It is shown that the local Reissner-type effects are specified by reducing the concentration of the tangential stress in the face layers along the longitudinal coordinate and transition to the steady tangential stress state in the filler layer. The concentration coefficients of the tangential stress are derived as functions of the dimensional parameters of the panel section.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 66–93, January–February, 1998.     

Stresses in plastics reinforced with anisotropic fibers and subjected to transverse normal loading

The question of the stress distribution in plastics reinforced with anisotropic fibers and subjected to transverse normal loading is considered. The stresses in the components are determined by the methods of the theory of elasticity using stress functions. The theoretical relations obtained are used to construct diagrams showing the distribution of the tangential, radial, and shear stresses in the composite and the isoclines of the concentration coefficient for a carbon-reinforced plastic. The results obtained for the carbon-reinforced plastic are compared with the analogous results for a glass-reinforced plastic.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 244–252, March–April, 1973.     

A refined model for three-point bending of sandwich panels. 1. Deflections and bending stresses

Design formulas for the flexural characteristics of sandwich panels under three-point loading by point forces, taking into account local effects, have been derived. Transverse deformation of the normal in the modified model is deduced in terms of the difference between deflections of face layers. It is considered that the rotation of the normal depends also on shear of the filler. The deflections, local curvatures, and bending stresses, dependent on the face-layer thicknesses and transverse characteristics of the filler, are studied. The danger of initial failure caused by the local moment stresses at the central panel section is shown. Comparative estimates refining the conventional designs are established.Institute of Polymer Mechanics, Latvian Academy of Sciences, Riga LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 747–767, November–December, 1997.     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

Bending features of a sandwich panel with asymmetric structure under local loads

The bending characteristics of a composite panel with asymmetric layered structure under local surface loads are obtained. A refined version of the applied theory is developed using the analytical solution of the bending problem of a sandwich plate with arbitrary asymmetric structure under a point load. Local effects are investigated within the limits of a discrete model allowing for the specific character of elastic properties of a soft filler. The advantages of the solution are expressions of bending characteristics — layer curvatures, displacements, and stresses — in a closed form. It is shown that these characteristics can vary several times depending on the asymmetry parameters of the structure. Degeneration peculiarities of the solution, stemming from the slipping of layers or, otherwise, their rigid linking by the Kirchoff—Love hypothesis, as well as from account of the transverse shear and compression of the normal, are examined in line with the degeneration of geometric and physical parameters of the discrete model adopted. The results obtained are illustrated by curves and surfaces for the characteristics studied.Submitted for the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 717–742, November–December, 1999.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of the sign of the shear stresses on the shear strength of reinforced plastics

The effect of the sign of the shear stresses on the shear strength has been investigated for a glass-reinforced plastic of the SVAM type with various ratios of the longitudinal and transverse reinforcement in shear in the plane of reinforcement in axes turned through 45° relative to the direction of reinforcement. More than 80 tubularspecimens were tested. The shear strengths T ₄₅ ⁺ and T ₄₅ ^– corresponding to shear stresses of different signs can be found from uniaxial tests in tension and compression in the direction of the reinforcement.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 262–268, March–April, 1971.     

Torsion of laminated cylindrical shells with adhesive interlayers

Based on refined equations of the Timoshenko-type shell theory, the contact stresses in torsion of a two-layer cylindrical shell with an adhesive interlayer are numerically studied. The effect of the geometric and physical-mechanical parameters of the load-carrying layers and adhesive interlayer of the shell on the distribution of the interlaminar tangential stress is analyzed. The results are presented graphically.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 501–506, July–August, 1998.     

Effect of transverse shear on the stability of an orthotropic cylindrical shell with an elastic core in axial compression

The effect of transverse shear on the stability of a composite shell in axial compression has been investigated. The core is treated as an isotropic elastic cylinder bonded to the inner surface of the shell. The effect of the tangential shearing forces between the shell and the core is taken into account.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 267–273, March–April, 1973.     

Hydrodynamic analogy for direct shear force and bending moment evaluation in a plate

A brand new interpretation of the plate bending equations is given using hydrodynamic analogy. It permits one to determine directly the shear forces and bending moments of a plate without the need of finding deflections. In engineering design of a plate it is more important to know shear forces and bending moments than the deflections. The existing numerical methods of solution of plate problems consist in determining deflections; then shear forces and bending moments are obtained by differentiating the deflection three and two times which produces great loss of accuracy. The hydrodynamic analogy method has the advantage over other numerical methods because the shear forces and bending moments are obtained directly, without the need of finding deflections and because they are obtained with better accuracy. The hydrodynamic analogy can be applied to a plate of arbitrary shape, with arbitrary boundary conditions under an arbitrary loading.     

Analysis of a 3‐node mixed‐shear‐projected triangular element method for Reissner–Mindlin plates

This article analyses an existing 3‐node hybrid triangular element, called MiSP3, for Reissner–Mindlin plates which behaves robustly in numerical benchmark tests (Ayad, Dhatt, and Batoz, Int J Numer Method Eng 42 (1998), 1149–1179). Based on Hellinger‐Reissner variational principle and the mixed shear interpolation/projection technique of MITC family, the MiSP3 element uses continuous piecewise linear polynomials for the approximations of displacements and a piecewise‐independent equilibrium mode for the approximations of bending moments/shear stresses. Due to local elimination of the parameters of moments/stresses, the element is almost of the same computational cost as the conforming linear triangular displacement element. We derive uniform stability and convergence results with respect to the plate thickness. The main tools of our analysis are the self‐equilibrium relation of the moments/stresses approximations, the properties of the mixed shear interpolation and the discrete Helmholtz decomposition of the shear stress approximation. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 241–258, 2017     

Optimization of cylindrical composite shells with regard to their critical mode of imperfections

This study deals with the optimum design of composite shells under external pressure with material strength and loss of stability according to the critical mode of imperfections taken as the failure criterion. The problem of optimum design is solved and the critical mode is obtained by nonlinear optimum programming for which the geometric and initial imperfection parameters are treated as variables. Numerical results are obtained for a cylindrical composite shell supported freely at its ends. The effect of shear forces between layers on the load-carrying capacity of the shell is also investigated.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 613–620, September–October, 1998.     

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.     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.Presented at the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Timoshenko Institute of Mechanics, Ukranian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 465–472, July–August, 2000.     

Mathematical justification of a one-dimensional model for general elastic shallow arches

We present a bending model for a shallow arch, namely the type of curved rod where the curvature is of the order of the diameter of the cross section. The model is deduced in a rigorous mathematical way from classical tridimensional linear elasticity theory via asymptotic techniques, by taking the limit on a suitable re-scaled formulation of that problem as the diameter of the cross section tends to zero. This model is valid for general cases of applied forces and material, and it allows us to calculate displacements, axial stresses, bending moments and shear forces. The equations present a more general form than in the classical Bernoulli–Navier bending theory for straight slender rods, so that flexures and extensions are proved to be coupled in the most general case. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Resolvents of the theory of orthotropic shells with low shear stiffness in force-moments and stress functions

The problem of the theory of orthotropic layered shells with low shear stiffness is reformulated in force-moments and stress functions on the basis of new strain continuity relations constructed with allowance for transverse shears.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 950–952, September–October, 1970.     

Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates

This paper presents a low order stabilized hybrid quadrilateral finite element method for ReissnerMindlin plates based on Hellinger-Reissner variational principle,which includes variables of displacements,shear stresses and bending moments.The approach uses continuous piecewise isoparametric bilinear interpolations for the approximations of the transverse displacement and rotation.The stabilization achieved by adding a stabilization term of least-squares to the original hybrid scheme,allows independent approximations of the stresses and moments.The stress approximation adopts a piecewise independent 4-parameter mode satisfying an accuracy-enhanced condition.The approximation of moments employs a piecewise-independent 5-parameter mode.This method can be viewed as a stabilized version of the hybrid finite element scheme proposed in [Carstensen C,Xie X,Yu G,et al.A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates.Comput Methods Appl Mech Engrg,2011,200:1161-1175],where the approximations of stresses and moments are required to satisfy an equilibrium criterion.A priori error analysis shows that the method is uniform with respect to the plate thickness t.Numerical experiments confirm the theoretical results.