New variational-asymptotic formulations for interlaminar stress analysis in laminated plates

A new approach for the three-dimensional stress analyses in composite laminates is presented. The method of composite expansions along with Hellinger-Reissner variational formulation is employed to derive the interior and edge layer problems for high order approximations. A new method of finite element analysis for solving the edge layer problem is developed based on the modified mixed-hybrid method. Comparison of the present solution with an assumed displacement finite element solution is made to demonstrate the accuracy and efficiency of the present approach.

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Zusammenfassung Eine neue Methode zur dreidimensionalen Spannungsanalyse von Verbundlaminaten wird dargelegt. Die Methode der gewünschten Entwicklungen zusammen mit dem Hellinger-Reissner'schen Variationsprinzip führt zur Herleitung des inneren Problems und des Grenzschichtproblems für höhere Näherungen. Eine neue Methode wird entwickelt, welche mit Hilfe von finiten Elementen, vom gemischt hybriden Verfahren ausgehend, das Grenzschichtproblem löst. Der Vergleich, der hier vorgeschlagenen Lösung mit einer klassischen, auf Verschiebungselementen beruhenden Lösung zeigt die Genauigkeits- und Effizienzvorteile der hier dargelegten neuen Methode.

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This research was supported by Technion V. P. R. fund (grant No. 030-717), New York Metropolitan research fund.     

Postbuckling characteristics of angle-ply laminated truncated circular conical shells

The postbuckling characteristics of the angle-ply laminated composite conical shells subjected to the torsion, the external pressure, the axial compression, and the thermal loading considering uniform temperature change are studied using the semi-analytical finite element approach. The finite element formulation is based on the first-order shear deformation theory and the field consistency principle. The variations in the stiffness coefficients along the meridional direction due to the changes in the ply-angle and the ply-thickness of the filament wound conical shells are incorporated in the finite element formulation. The nonlinear governing equations are solved using the Newton–Raphson iteration procedure coupled with the displacement control method to trace the prebuckling followed by the postbuckling equilibrium path. The presence of asymmetric perturbation in the form of a small magnitude load spatially proportional to the linear buckling mode shape is considered to initiate the bifurcation of the shell deformation. The influence of semi-cone angle, ply-angle and number of circumferential waves on the prebuckling/postbuckling response of the anti-symmetric angle-ply laminated circular conical shells is investigated.     

Yield condition for circular cylindrical shells made of a fiber-reinforced composite

A yield condition is obtained for circular cylindrical shells made of a definite class of fiber-reinforced composite material whose components possess plastic properties. It is shown that, in the plane of generalized stresses — the axial bending moment and the circumferential force (when the axial force is absent) — the yield curve consists of two linear and four curvilinear sections. By approximating the curvilinear sections, we get a piecewise linear yield condition described by a hexagon in the plane indicated. The nonlinear equations and the corresponding piecewise linear equations of the yield condition for particular cases are given in the form of tables. In solving specific boundary-value problems, we consider a circular cylindrical shell simply supported at its ends and loaded with a uniform internal pressure, for which the load-carrying capacity is determined in relation to the mechanical properties of composite components and some characteristic geometrical parameters. The results of numerical calculations are represented in the form of graphs. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 655–666, September–October, 2006.     

An accurate method to predict the stress concentration in composite laminates with a circular hole under tensile loading

The paper presents a theoretical-numerical hybrid method for determining the stresses distribution in composite laminates containing a circular hole and subjected to uniaxial tensile loading. The method is based upon an appropriate corrective function allowing a simple and rapid evaluation of stress distributions in a generic plate of finite width with a hole based on the theoretical stresses distribution in an infinite plate with the same hole geometry and material. In order to verify the accuracy of the method proposed, various numerical and experimental tests have been performed by considering different laminate lay-ups; in particular, the experimental results have shown that a combined use of the method proposed and the well-know point-stress criterion leads to reliable strength predictions for GFRP or CFRP laminates with a circular hole. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 531–570, July–August, 2007.     

Explicit through-thickness integration schemes for geometric nonlinear analysis of laminated composite shells

Degenerated shell elements were found to be attractive in solving homogeneous shell problems. Direct extension of the same to layered shells becomes computationally inefficient as, in the computation of element matrices, 3-D numerical integration in each layer and summation over the layers have to be carried out. In order to make the formulation efficient, explicit through-thickness schemes have been devised for linear problems. The present paper deals with the extension of the same to geometric nonlinear problems with options of small and large rotations. The explicit through-thickness integration becomes possible due to the assumption on the variation of inverse Jacobian through the thickness. Depending on the assumptions, three different schemes under large and small rotation cases have been presented and their relative numerical accuracy and computational efficiency have been evaluated. It has been observed that there is no sacrifice on the numerical accuracy due to the assumptions leading to the explicit through-thickness integration, but at the same time, there is considerable saving in the computational time. The computational efficiency improves as the number of layers in the laminate increases. The small rotation formulation with the assumption of linear variation of Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient, as applied to geometric nonlinear layered shell problems.     

Analysis of free damped vibrations of laminated composite cylindrical shells

Damped free vibrations of multilayered composite cylindrical shells are investigated. Vibration and damping analysis of cylindrical shells is performed by using the first-order shear deformation theory (FOSDT). Based on other researchers' works, two damping models are developed, i.e., the energy method (EM), and the method of complex eigenvalues (MCE). Several numerical examples of the damped free vibration problem of laminated composite cylindrical shells have been solved and comparison has been made with the results of other authors.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 646–659, September–October, 1995.     

Free vibrations of composite laminated doubly-curved shells and panels of revolution with general elastic restraints

A unified numerical analysis model is presented to solve the free vibration of composite laminated doubly-curved shells and panels of revolution with general elastic restraints by using the Fourier–Ritz method. The first-order shear deformation theory is adopted to conduct the analysis. The admissible function is acquired by using a modified Fourier series approach in which several auxiliary functions are added to a standard cosine Fourier series to eliminate all potential discontinuities of the displacement function and its derivatives at the edges. Furthermore, the general elastic restraint and kinematic compatibility and physical compatibility conditions are imitated by the boundary and coupling spring technique respectively when the composite laminated doubly-curved panels degenerate to the complete shells of revolution. Then, the desired results are solved by the variational operation. Large quantities of numerical examples are calculated about the free vibration of cross-ply and angle-ply composite laminated doubly-curved panels and shells with different geometric and material parameters. Through the sufficient conclusions obtained from the comparison, it can be seen that highly accurate solutions can be yielded with a little computational effort. To understand the influence of different boundary conditions, lamination schemes, material and geometrical parameters on the vibration characteristics, a series of parametric studies are carried out. Lastly, results for vibration of the composite laminated doubly-curved panels and shells subject to various kinds of boundary conditions and with different geometrical and material parameters are also presented firstly, which can provide the benchmark data for other studies conducted in the future.     

State of stress around a circular opening in orthotropic spherical shells with linearly varying thickness

The state of stress of a thin spherical shell of orthotropic material of linearly varying thickness weakened by a circular opening at the pole is investigated. A solution in Bessel functions is constructed for the differential equations describing the state of stress of the shell. Graphs are presented for the dependence of the stress concentration coefficients at the edge of the opening and the uniformity of the state of stress on the coefficient characterizing the variation of the thickness. It is shown that by a suitable choice of this coefficient it is possible to obtain a shell in which the stresses at the edge of the opening and on the equator are equal.     

On the theories used for the wave propagation in laminated composite thin elastic shells

In this paper, the problem of propagation of free harmonic waves in cross-ply laminated thin elastic shells is considered. For this problem, a theoretical unification of the most commonly used, in physical and engineering applications, thin shell theories which take into consideration transverse shear deformation effects is presented. In more detail, the problem is formulated in such a way that by using some tracers, which have the form of Kronecker's deltas, the obtained stress-strain relations, constitutive equations and equations of motion produce, as special cases, the corresponding relations and equations of the transverse shear deformable analogs of Donnell's, Love's and Sanders' theories. Using an eigenvalue form solution of the equations of motion, a comparison of corresponding numerical results obtained on the basis of all of the afore-mentioned theories is made. Comparisons with corresponding results obtained on the basis of the classical thin shell theories of Donnell, Love, Sanders and Flugge are also made.

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Resumé On étudie un problème de propagation d'ondes harmoniques libres dans une fine coque cylindrique, élastique, composée de lamelles croisées. On présente une unification des théories les plus courantes que les ingénieurs et physiciens appliquent aux problèmes de coques minces sous considération de la déformation due au cisaillement transverse. En détails, le problème est formulé de telle façon qu'en utilisant des opérateurs de trace (sous forme du symbole de Kronecker) les relations obtenues: contrainte-déformation, équations constitutives et équations de mouvement donnent comme cas spéciaux les relations correspondantes et les équations des problèmes analogues (déformation de coques minces par cisaillement transverse) des théories de Donnell, Love et Sanders. En utilisant une solution aux valeurs propres des équations de mouvement, on compare les résultats numériques obtenus grâce aux théories mentionées ci-dessus aux résultats correspondants sur la base des théories classiques de Donnell, Love, Sander et Flugge.

     

Limiting equilibrium of a plane with a circular hole

The elastoplastic state of a compressible isotropic plane with a circular hole is studied by the method of a small parameter. An unknown boundary separating the domain of limiting equilibrium and the elastic domain is determined. We construct the complex Kolosov-Muskhelishvili functions that describe the elastic state of a plane and compare these with the solutions of Galin's problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 980–981. July, 1993.     

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.     

A two-dimensional layerwise-differential quadrature static analysis of thick laminated composite circular arches

In this paper, a high accuracy and rapid convergence hybrid approach is developed for two-dimensional static analyses of circular arches with different boundary conditions. The method essentially consists of a layerwise technique in the thickness direction in conjunction with differential quadrature method (DQM) in the axial direction. Hence, the high accuracy and fast convergence of DQM with generality of layerwise formulations for modeling the transverse deformations of arbitrary laminated composite thick arches are combined. This results in superior accuracy with fewer degrees of freedom than conventional finite element method (FEM) or finite difference method (FDM). The convergence behavior of the method is shown and to verify its accuracy, the results are compared with those obtained based on the first order shear deformation Reissner–Naghdi type shell theory and also higher order shear deformation theory. The effects of opening angles, ply angle, boundary conditions, and thickness-to-length ratio on the stress and displacement components are studied.