Number Of Mathematical Layers Research Articles

Free-edge stress fields in cylindrically curved cross-ply laminated shells

This paper discusses a semi-analytical approach to the determination of three-dimensional stress fields in cylindrically curved orthotropic symmetrically or unsymmetrically laminated composite shells under bending load under consideration of free-edge effects. The analysis method incorporates a plane-strain analysis in the innermost regions of the shell (the so-called inner solution) in conjunction with a free-edge solution which enables the accurate determination of the interlaminar stress field at the free shell edges. The free-edge solution employs a discretization of the laminated shell into a number of mathematical layers with respect to the thickness direction wherein in each layer interface approximate and a priori unknown displacement functions are introduced. Lagrangian shape functions are used to interpolate between the displacement functions in the interfaces, and various orders of shape functions are investigated in this paper. The differential equations and boundary conditions that describe the current stress problem are derived from the principle of minimum elastic potential of the laminated shell. The resultant differential equations allow for an exact solution for the displacement functions whereas the boundary conditions at the free shell edges are satisfied in an integral sense. The results of the developed analysis method are verified by comparison with detailed finite element computations, and it is found that the semi-analytical approach works with comparable accuracy, however only at a fraction of the required computational effort.

3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles

The exact three-dimensional (3D) shell model proposed in the present paper is able to perform the thermal stress analysis of simply-supported Functionally Graded Material (FGM) spherical and cylindrical shells, cylinders and plates. The model is based on the 3D equilibrium equations for spherical shells developed using an orthogonal mixed curvilinear coordinate system. The use of this reference system allows the investigation of cylindrical shells, cylinders and plates as particular cases of spherical shells by means of simple considerations on the radii of curvature. The 3D shell model uses a layer-wise approach and the exponential matrix method to calculate the general and the particular solutions through the thickness direction z. The system of second order differential equations in z is not homogeneous because of the thermal terms which are externally defined. The system is reduced to a group of first order differential equations in z simply redoubling the number of variables. The solution is in closed form in the in-plane directions α and β because of the hypotheses of simply-supported boundary conditions, harmonic forms for displacement and temperature fields, and isotropic behavior in the in-plane directions for functionally graded materials. In order to define the equivalent thermal load, the temperature profile through the thickness is separately defined by means of three possible ways. Using the hypothesis of temperature amplitudes imposed at the top and bottom external surfaces in steady-state conditions, the temperature profile can be: imposed as linear through the entire thickness direction, calculated by solving the 1D version of the Fourier heat conduction equation, or calculated by solving the 3D version of the Fourier heat conduction equation. The effects of different temperature profiles on the displacement and stress analyses of FGM plates and shells are here remarked. The first order differential equation system in z has not constant coefficients because of the presence of radii of curvature for shells and through-the-thickness variable elastic and thermal coefficients for the FGM layers. An appropriate number of mathematical layers is introduced to calculate the curvature influence for shells and the elastic and thermal material coefficients for FGM layers. Therefore, the system can be considered as differential equations with constant coefficients. The proposed results allow the evaluation of thickness ratio, geometry, lamination scheme, thickness material law and temperature profile effects in the related thermal stress analysis of single-layered and sandwich FGM plates, cylinders, spherical and cylindrical shells.

Free-edge stress fields in cylindrically curved symmetric and unsymmetric cross-ply laminates under bending load

In this paper, an analysis method for the determination of displacement, strain and stress fields in cylindrically curved cross-ply laminates under bending load is presented. The considered cross-ply laminates may be either symmetrically or unsymmetrically laminated and are clamped at one end while the other end is loaded by an evenly distributed bending moment. The analysis method employs a layerwise plane strain approach in the inner regions of the laminate in which the stresses in each layer are represented by adequate formulations for Airy’s stress function. In the regions of the free laminate edges where significant three-dimensional and possibly singular interlaminar stress fields are to be expected, the plane-strain approach is upgraded by a layerwise displacement-based formulation wherein the physical laminate layers are discretized into a number of mathematical layers with respect to the thickness direction. The governing differential equations for the unknown additional displacement functions with respect to the width coordinate in the form of the Euler-Lagrange equations stemming from the underlying variational statement can be solved exactly and eventually lead to an eigenvalue problem that needs to be solved numerically. Usage of continuity conditions between the individual laminate layers and formulation of adequate boundary conditions at the free edges in an integral sense then lead to complete representations for displacements, strains and stresses at every location in the considered laminate. While the analysis approach relies on a discretization of the laminate into a number of mathematical layers with respect to the thickness direction and further requires a numerical solution of a quadratic eigenvalue problem, it provides closed-form analytical solutions concerning the width direction and can thus be classified as being a semi-analytical solution. The presented analysis method is compared to the results of comparative finite element simulations and is shown to be in good agreement, however with only a fraction of the computational effort that is required for according finite element simulations.

A statically compatible layerwise stress model for the analysis of multilayered plates

This paper proposes a new layerwise model for multilayered plates. The model, called SCLS1 for Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction, complies exactly with the 3D equilibrium equations and with the free-edge boundary conditions. As in the LS1 model initially proposed by Naciri et al. (1998), the laminated plate is considered as a superposition of Reissner plates coupled by interfacial stresses which are considered as generalized stresses. However, the divergences of the interlaminar transverse shears are introduced as additional generalized stresses in the SCLS1 model. Also, a refined version of the new model is obtained by introducing several mathematical layers per physical layer. Contrary to the LS1 model which is derived by means of the Hellinger–Reissner principle, the new model is derived by means of the minimum of the complementary energy principle. This ensures the convergence of the refined SCLS1 solution to the exact 3D solution as the number of mathematical layers per physical layer increases. The new model has been implemented in a new version of the in-house finite element code MPFEAP . Several comparisons are made between LS1, SCLS1 and full 3D FE models in order to assess the performances of the new model which shows itself very effective.

Stress concentration phenomena in the vicinity of composite plate-doubler junctions by a layerwise analysis approach

This paper presents an analysis method for the determination of displacements, strains and stresses in a laminated composite plate subjected to tensile load that is reinforced by a doubler. The analysis approach consists of two parts. Firstly, a ‘global’ solution that is based on Classical Laminated Plate Theory (CLPT) is introduced. Secondly, a ‘local’ model is derived that enables the assessment of the three-dimensional stress state near the plate-doubler junction. The local solution employs a discretization of the physical layers into a number of mathematical layers which necessitates the numerical treatment of a quadratic eigenvalue problem. Consequently, the current approach can be classified as being a semi-analytical layerwise analysis method. Results are generated for several different plate-doubler configurations, and it is found that the current analysis model delivers excellent results when compared to finite element simulations, however with only a fraction of the computational time and effort that are needed for the FEM analyses.

In-plane Free Vibration Analysis of Laminated Thick Circular Deep Arches

In order to accurately determine the natural frequencies of laminated thick circular deep arches, a solution procedure based on the two-dimensional theory of elasticity is presented. The layerwise finite element method is used to discretize the equations of motion and the related boundary conditions through the thickness of the arch. The formulations are general so that the effects of the variation of beam curvature across the cross-section, the transverse shear and normal stresses and rotary inertia can easily be implemented. Exact solutions are obtained for antisymmetric cross-ply laminated thick circular deep arches with simply supported boundary conditions. The convergence of the method is shown against the number of mathematical layers used for modeling the arches. The effects of different parameters such as opening angle, thickness-to-length and orthotropy ratios on the natural frequencies of the arches are outlined. Due to high accuracy of the method, the results can be used as benchmarks for future research.