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The contribution of this work is the implementation of a new elastic solution method for thick laminated composites and sandwich structures based on a generalized unified formulation using finite elements. A quadrilateral four-node element was developed and evaluated using an in-house finite element program. The C-1 continuity requirements are fulfilled for the transversal displacement field variable. This method is tagged as Caliri's generalized formulation. The results employing the proposed solution method yielded coherent results with deviations as low as 0.05% for a static simply supported symmetric laminate and 0.5% for the modal analyses of a soft core sandwich structure. FINITE ELEMENT USING A GENERALIZED UNIFIED PLATE THEORY 291 the variety of theories developed to solve such complex laminated structures. When carefully analyzed, some of these theories can be grouped into the so-called unified or generalized theories.Li and Liu [9] exploited the concept of superposition and proposed a global-local refined multilayered plate theory. Such theory can be classified as an equivalent single layer (ESL) theory because the number of degrees of freedom was made independent of the number of layers through zig-zag refinements. However, despite its generalization, this formulation lacks unification properties. Unified plate/shell theories can group nearly all existing pate/shell theories into one from which they can be derived.In addition, when using unified formulations, theories with different orders of expansions can be directly compared, because no changes (but the index of the order of the thickness expansion) in the theory or solution method are performed to carry out the comparison. Moreover, other typing and precision errors, which may appear because of the use of different implemented solution methods, can be avoided. In addition, it is a powerful formulation to be implemented as a computer program. The works of Carrera, Demasi and Ferreira et al. [10][11][12][13][14][15][16][17][18][19][20][21][22][23] bring an extensive review and evaluation of the topic. Carrera [10,11] proposed a unification method, which generates a kernel matrix from which infinite axiomatic plate/shell theories can be extracted. Demasi improved Carrera's proposal and generalized it by decoupling the order of the displacement fields in each direction [14][15][16][17][18][19][20][21][22].Considering the aforementioned scope, this paper works on a new finite element solution method to solve thick laminates and sandwich elastic plates. The main contribution of this work is the implementation of a new solution approach, using the FEM, in order to solve unified plate formulations. The novelty of the present work is that the finite element solution is not fully C-0, but it now preserves the C-1 continuity requirements of the transverse displacement field. Carrera [10-13] also showed finite element results, along with closed-form solutions, but they are C-0 solutions. Thus, the present method (tagged as Caliri's generalized formulation -CGF)...

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This paper shows theoretical models (analytical formulations) to predict the mechanical behavior of thick composite tubes and how some parameters can influence this behavior. Thus, firstly, it was developed the analytical formulations for a pressurized tube made of composite material with a single thick ply and only one lamination angle. For this case, the stress distribution and the displacement fields are investigated as function of different lamination angles and reinforcement volume fractions. The results obtained by the theoretical model are physic consistent and coherent with the literature information. After that, the previous formulations are extended in order to predict the mechanical behavior of a thick laminated tube. Both analytical formulations are implemented as a computational tool via Matlab code. The results obtained by the computational tool are compared to the finite element analyses, and the stress distribution is considered coherent. Moreover, the engineering computational tool is used to perform failure analysis, using different types of failure criteria, which identifies the damaged ply and the mode of failure.

Nomenclature C= related to the constitutive matrix in global coordinates and its components F N = functions of the Nth order used in the expansion along the thickness variable H i = two-dimensional Hermite interpolation functions K = related to the global stiffness matrix and its components M = related to the global mass matrix and its components N i = two-dimensional serendipity interpolation function N l = number of plies N τ = order of the complete polynomial chosen for the displacement fields P = vector of normal forces R = vector of moments and twist u = vector of displacement fields u, v = in-plane displacement fields w = transverse displacement field x, y = in-plane coordinate variables z = transverse coordinate variable Γ = plate's in-plane domain δ mn = Kronecker delta ρ = density Ω = plate's three-dimensional domain Subscripts i, j = recurrence indices for node numbering m, n = recurrence indices for Kronecker delta x = first derivative with respect to in-plane x position xy = crossed derivative in respect to in-plane position variables y = first derivative with respect to in-plane y position α = recurrence index related to any expansion term of the three displacement fields α u , α v = related to the order of the in-plane displacement fields polynomial expansion term α w = related to the order of the out-of-plane displacement field polynomial expansion term 1, 2, 3, 4, 5, 6 = local indices used to identify anisotropic material properties Superscripts e = related to the finite element i, j = recurrence indices for node numbering k = current ply r, s = recurrence indices indicating the terms in the displacement polynomial functions T = transpose