This paper presents the formulation of exact stiffness matrices applied in linear generalized beam theory (GBT) under constantand/or linear loading distribution in the longitudinal direction. Also, the assortment of the correct exact stiffness matrixand the corresponding shape function are presented based on main transversal deformation mode, which can be dividedinto: (1) dominant distortion mode; (2) dominant torsion mode; (3) and critical distortion– torsion mode. Special attention isgiven to the hyperbolic– trigonometric shape functions, which are organized in a system of vector in function of longitudinaldirection and a coefficient matrix obtained from the completeness requirement. This approach has the benefit of compactingthe terms of the stiffness matrix and systematizing the boundary conditions of an ELEMENT by applying the completenesscoefficient matrix as a transformation matrix. As a result, in linear analysis, a single ELEMENT can represent the stress anddisplacement fields. Moreover, due to the higher-order continuous derivatives properties of hyperbolic– trigonometric shapefunctions, the generalized internal shear is obtained without the typical discontinuity of Hermitian shape functions. A fulland detailed example, applied in a thin-walled circular hollow cross section, provides not only an illustration of the presentedapproach, but also a quick introduction point in GBT.

The need for compatibility between degrees of freedom of various ELEMENTs is a major problem encountered in practice during the modeling of complex structures; the problem is generally solved by an additional rotational degree of freedom [1-3]. This present paper investigates possible improvements to the performances of strain based cylindrical shell FINITE ELEMENT [4] by introducing an additional rotational degree of freedom. The resulting ELEMENT has 24 degrees of freedom, six essential external degrees of freedom at each of the four nodes and thus, avoiding the difficulties associated with internal degrees of freedom (the three translations and three rotations) and the displacement functions of the developed ELEMENT satisfy the exact representation of the rigid body motion and constant strains (in so far as this allowed by compatibility equations). Numerical experiments analysis have been conducted to assess accuracy and reliability of the present ELEMENT, this resulting ELEMENT with the added degree of freedom is found to be numerically more efficient in practical problems than the corresponding Ashwell ELEMENT [4].

Introduction: Mandibular first molar is the most important tooth with complicated morphology. In FINITE ELEMENT (FE) studies, investigators usually prefer to model anterior teeth with a simple and single straight root, it makes the results deviate from the actual case. The most complicated and time-consuming step in FE studies is modeling of the desired tooth, thus this study was performed to establish a FINITE ELEMENT method (FEM) of reconstructing a mandibular first molar with the greatest precision.Materials and Methods: An extracted mandibular first molar was digitized, and then radiographed from different aspects to achieve its outer and inner morphology. The solid model of tooth and root canals were constructed according to this data as well as the anatomy of mandibular first molar described in the literature. Result: A three-dimensional model of mandibular first molar was created, giving special consideration to shape and root canal system dimensions.Conclusion: This model may constitute a basis for investigating the effect of different clinical situations on mandibular first molars in vitro, especially on its root canal system. The method described here seems feasible and reasonably precise foundation for investigations.

In this paper, the static bending, free vibration, and dynamic response of functionally graded piezoelectric beams have been carried out by FINITE ELEMENT method under different sets of mechanical, thermal, and electrical loadings. The beam with functionally graded piezoelectric material (FGPM) is assumed to be graded across the thickness with a simple power law distribution in terms of the volume fractions of the constituents. The electric potential is assumed linear across the FGPM beam thickness. The temperature field is assumed to be of uniform distribution over the beam surface and through the beam thickness. The governing equations are obtained using potential energy and Hamilton's principle based on the Euler-Bernoulli beam theory that includes thermo-piezoelectric effects. The FINITE ELEMENT model is derived based on the constitutive equation of piezoelectric material accounting for coupling between the elasticity and the electric effect by two nodes Hermitian beam ELEMENT. The present FINITE ELEMENT is modelled with displacement components and electric potential as nodal degrees of freedom. The temperature field is calculated by post-computation through the constitutive equation.Results are presented for two-constituent FGPM beam under different mechanical boundary conditions. Numerical results include the influence of the different power law indexes, the effect of mechanical, thermal, and electrical loadings and the type of in-plane boundary conditions on the deflection, stress, natural frequencies, and dynamic response. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.

In this paper an efficient method is developed for the formation of null bases of FINITE ELEMENT models comprised of four-node quadrilateral plane stress and plane strain ELEMENTs corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs to the FINITE ELEMENT models and using an ELEMENT with new equilibrium tractions and stress field for the formation of localized self-equilibrating stress systems. The efficiency of the present method is illustrated through some examples.