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SUMMARYQuasi-conforming formulations of 4-node stress-resultant shell elements are presented. The element formulations use interrelated displacement-rotation interpolations. The formulation also includes drilling degrees of freedom, which improves membrane behavior and allows the modeling of sti ened plates and shells. The proposed treatment for bending provides very good results in the 4-node shell element. The sti ness matrices for the present elements are explicitly expressed and the stresses are taken accurately at the nodal points. Compared to elements using Gauss integration, where the stresses are most accurate at the integration points, the extrapolation procedure needed for post-processing is eliminated in the present shell element. A lot of numerical tests were carried out for the validation of the present 4-node shell element and the results are in good agreement with references.

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The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells.

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The four-node quasi-conforming shell element was extended in the present article to the case of geometrically non-linear behavior of the FGM plates and shells. The high stress occurring in the FGM structures will affect its integrity and the structures is susceptible to failure. Therefore, we focus on the effect of volume fraction of the constituent materials in the mechanical behavior of FGM plates and shells. The material properties are assumed to be varied in the thickness direction according to a sigmoid function in terms of the volume fraction of the constituents. The series solutions of sigmoid FGM (S-FGM) plates, based on the first-order shear deformation theory and Fourier series expansion are provided as the reference solution for the numerical results. In quasi-conforming formulation, the tangent stiffness matrix is explicitly integrated. This makes the element computationally efficient in the non-linear analysis. Several selected examples of non-linear analysis of FGM shells are included in the article for the illustration of possibilities of the presented formulation. It is seen that the present results for the non-linear behavior of FGM plates and shells can provide a useful benchmark to check the accuracy of related numerical solutions.

In this paper, we investigate the natural frequencies and buckling loads of functionally graded material (FGM) plates and shells, using a quasi-conforming shell element that accounts for the transverse shear strains and rotary inertia. The eigenvalues of the FGM plates and shells are calculated by varying the volume fraction of the ceramic and metallic constituents using a sigmoid function, but the Poisson ratios of the FGM plates and shells are assumed to be constant. The expressions for the membrane, bending and shear stiffness of FGM shell elements are more a complicated combination of material properties than a homogeneous element. In order to validate the finite element numerical solutions, the Navier solutions for rectangular plates based on the first order shear deformation theory are also presented. The present numerical solutions for composite and sigmoid FGM (S-FGM) plates and shells are verified by the Navier solutions and various examples of composite and FGM structures. The present results are in good agreement with the Navier theoretical solutions.

The quasi-conforming technique was introduced in the 1980's to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented.The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening.