A solid shell element for the shell structured eyeball with application to radial keratotomy

The present dissertation describes the research work carried out towards: I. developing of analytical and numerical models for accurate and efficient analysis of laminates and sandwiches under different loading conditions (static analyses, impulsive loading and low velocity impacts); II. developing of techniques to improve structural performances of composites structures by tuning their energy absorption mechanism; A higher order theory (AD-ZZ) including zig-zag effects, transverse shear continuity, variable transverse displacement and layerwise representation is presented for analysis of laminates and sandwiches. This theory is cast in such a way that the unknowns in the model will involve only five variables like equivalent single layer models: the three displacement components and the two shear rotations of the normals on the reference surface. It is advantageous to formulate the theory in this way, as it provides the opportunity to refine its through-the-thickness representation by adding computational layers in the thickness direction without increasing the number of unknowns. After extensive evaluation of the accuracy obtained by the above mentioned theory for a variety of structures, the AD-ZZ model is enriched by adding a set of continuity functions in order to treat laminates and structures with in-plane discontinuities (e.g. bonded joints, two material wedge) under a unified approach without adding new unknowns. Speaking of techniques to improve structural performances, two approaches are presented: variable stiffness composites and stitching. The former tool works on the strain energy of the structure computing fibre distributions that minimize unwanted energy contributions and maximize the wanted ones. The latter prescribes the insertions of transverse reinforcements in sandwich structures. The homogenized mechanical properties of this structure, required by the AD-ZZ model to perform the analysis, are evaluated through virtual material tests using a 3D FE model. The results obtained applying these techniques demonstrate that both are effective in reducing transverse stresses at critical interfaces and, in certain cases, in improving bending stiffness. New C0 displacement-based and stress-based finite elements for analysis of laminated and sandwich plates based on the AD-ZZ model are developed. To achieve this goal, a new technique that converts derivatives of functional degrees of freedom contained in the AD-ZZ is presented. Both the elements satisfy all the requirements of computationally efficient finite element models for analysis of multilayered structures, namely 1) the number of degrees of freedom is independent from the number of layers; 2) no shear correction factors or penalty number are added in the formulation. Consistent shear fields are obtained for the present finite element formulations. Numerical results demonstrate that these new elements are robust, accurate and computationally efficient for analysis of multilayered structures. In order to study structures subjected to impact loading a new simulation procedure is developed. It is based on the AD-ZZ model and considers the crushing behaviour of soft-like media without performing each time a detailed 3D finite element analysis. Numerical results show that the response of composites undergoing low velocity impact is very accurately predicted with low computational effort. Overall, the present procedure holds great promise for analysis of laminated and sandwich structures undergoing low velocity impact

Free vibration analysis of skewed open circular cylindrical shells

This paper presents a zeroth-order (non-gradient based) methodology for topology optimization of continuum structures. The solid and void elements in the design domain are represented by binary design variables. As in the SIMP method, design variables are assumed to represent the non-dimensional density of the individual finite elements and their contribution to structural stiffness in the model. The optimization procedure begins by specifying a desired volume fraction over a discretized domain and associated boundary conditions. Based on a randomized distribution of solid and void elements, a finite element analysis is performed to identify the strain energy of individual elements. While keeping the volume fraction fixed, the solid elements with low strain energy are converted into void elements whereas the void elements with high strain energy are converted into solid elements. The strain energy threshold for element exchange is specified at the beginning and can be adjusted in the course of optimization. Through subsequent repetition of analysis and exchange steps, the optimal topology corresponding to minimum compliance emerges with no gray elements to filter out. The general characteristics of the Element Exchange Method are demonstrated through solutions of several benchmark problems.

Buildings as well as other structures may be idealized as an assembly of elements connected at joints or nodal points. These elements can be unidirectional such as beams or rod elements, two dimensional like plates and shell elements, and three dimensional such as solid elements. The structure may be modeled as a shear building when the horizontal diaphragms at the floor levels of a multistory building are assumed to be rigid. In such a model, it is assumed that: (1) the total mass of the structure is concentrated at the levels of the floors, (2) the horizontal diaphragms at the floor levels are plane rigid, and (3) the deformation of the structure is independent of the axial force present in the columns. These assumptions transform the problem from a system with an infinite number of degrees of freedom (due to the distributed mass) to a system that has only as many degrees of freedom as it has lumped masses at the floor levels.

SUMMARY A study is made of the effect of mesh distortion on the accuracy of transverse shear stresses and their first-order and second-order sensitivity coefficients in multilayered composite panels subjected to mechanical and thermal loads. The panels are discretized by using a two-field degenerate solid element, with the fundamental unknowns consisting of both displacement and strain components, and the displacement components having a linear variation throughout the thickness of the laminate. A two-step computational procedure is used for evaluating the transverse shear stresses. In the first step, the in-plane stresses in the different layers are calculated at the numerical quadrature points for each element. In the second step, the transverse shear stresses are evaluated by using piecewise integration, in the thickness direction, of the three-dimensional equilibrium equations. The same procedure is used for evaluating the sensitivity coefficients of transverse shear stresses. Numerical results are presented showing no noticeable degradation in the accuracy of the in-plane stresses and their sensitivity coefficients with mesh distortion. However, such degradation is observed for the transverse shear stresses and their sensitivity coefficients. The standard of comparison is taken to be the exact solution of the three-dimensional thermoelasticity equations of the panel.

Relevance. The use of the finite element method for determining the stressstrain state of thin-walled elements of engineering structures predetermines their discretization into separate finite elements. Splitting irregular parts of the structure is impossible without the use of triangular areas. The triangular elements of shell structures are joint in displacements and in their derivatives only at the nodal points. Therefore, ways to improve the compatibility conditions at the boundaries of triangular elements are relevant. Aims of research. The aim of the work is to improve the compatibility conditions at the boundaries of adjacent triangular elements based on equating the derivatives of normal displacements in the middle of the boundary sides. Methods. In order to improve the compatibility conditions at the boundaries of triangular elements in this work, the Lagrange functional is used with the condition of ensuring equality in the middle of the sides of adjacent elements derived from normal displacements in the directions of perpendiculars tangent to the middle surface of the shell. Results. Using the example of analysing an elliptical shell, the efficiency of using a joint triangular finite element is shown, whose stiffness matrix is formed in accordance with the algorithm outlined in this article.

This is the first of a two part paper on three‐dimensional finite elements with rotational degrees of freedom (DOF). Part I introduces an 8‐node solid hexahedron element having three translational and three rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 20‐node hexahedron element. The new element produces a much smaller effective band width of the global system equations than does the 20‐node hexahedron element having midside nodes.A small penalty stiffness is introduced to augment the usual element stiffness so that no spurious zero energy modes are present. The new element passes the patch test and demonstrates greatly improved performance over elements of identical shape but having only translational DOF at the corner nodes.

In this paper, the solid tetrahedron element with drilling degrees of freedom is presented. This element has only four nodes. Each node has three translational components and three drilling ones. By using this element, the high level of analytical accuracy can be obtained. On the other hand, the free mesh method creates tetrahedron elements locally around every nodal point. Consequently, the element with intermediate nodes is not available to the free mesh method. However, the proposed solid tetrahedron element with drilling degrees of freedom is especially effective to the free mesh method, because it has no intermediate nodes. In this paper, the feature of accuracy of this element is shown through several numerical examples, and the effectiveness of the free mesh method is also described.

When a thin plate is machined by EDM on one side, a certain curvature results owing to tensile residual stress generated on the surface of the workpiece. If the machined plate is tempered, some change in curvature of the thin plate appears to occur due to reduction in residual stress and changes in the degenerated layers.This study was conducted to examine changes in curvature of a machined thin plate under various tempering conditions, in relation to those in the properties (phase changes) of the machined surface, using an X-ray diffractometerReduction in residual stress was concluded to occur due to stress relaxation, phase changes (transformation of γ Fe to α Fe) and the precipitation of carbides (Fe3C, M7C3). Change in curvature of thin plate was essentially the same as that noted by the X-ray stress measuring method. As a result, there occurs considerable plastic deformations (slip, twinning and lattice defects) of the EDM surface, owing to thermal effects.

In this study, an efficient finite element model with two degrees of freedom per node is developed for buckling analysis of axially functionally graded (AFG) tapered Timoshenko beams resting on Winkler elastic foundation. For this, the shape functions are exactly acquired through solving the system of equilibrium equations of the Timoshenko beam employing the power series expansions of displacement components. The element stiffness matrix is then formulated by applying the developed shape functions to the total potential energy along the element axis. It is demonstrated that the resulting shape functions, in comparison with Hermitian cubic interpolation functions, are proportional to the mechanical features of the beam element, including the geometrical properties, material characteristics, as well as the critical axial load. An exhaustive numerical example is implemented to clarify the efficiency and simplicity of the proposed mathematical methodology. Furthermore, the effects of end conditions, material gradient, Winkler parameter, tapering ratio, and aspect ratio on the critical buckling load of AFG tapered Timoshenko beam are studied in detail. The numerical outcomes reveal that the elastic foundation enhances the stability characteristics of axially non-homogeneous and homogeneous beams with constant or variable cross-section. Moreover, the results show that the influence of non-uniformity in the cross-section and axially inhomogeneity in material characteristics play significant roles in the linear stability behavior of Timoshenko beams subjected to different boundary conditions.

This study deals with the geometrically non-linear vibration analysis of concrete shallow funicular shells of rectangular plan with four clamped edges under impulse loads. The shape of a concrete funicular shell is such that the shell is subjected to pure compression under its dead weight. Following the existing method presented for the linear vibration analysis, the geometrically non-linear vibration analysis is considered through the use of non-linear shallow shells theory. Each displacement component is expanded in a double Fourier series and the kinetic energy, the elastic strain energy and the virtual work done by external forces are calculated in terms of the displacement components. Then, the equations of motion are obtained using the Lagrangian approach and are solved with the Runge–Kutta fourth-order method. The solution is verified against the results obtained with the finite-element method. The difference between the results of the linear and non-linear vibration analyses has been considered. Furthermore, it is indicated that under the considered dynamic loads, internal moments and consequentially tensile stresses are formed in the funicular shell and the shell does not behave purely as a funicular element. Finally, the plan aspect ratio effect on the time response of funicular shells has been shown.

In this paper, the Discrete Least Squares Meshless (DLSM) method is developed to determine crack-tip fields. In DLSM, the problem domain and its boundary are discretized by unrelated field nodes used to introduce the shape functions by the moving least-squares (MLS) interpolant. This method aims to minimize the sum of squared residuals of the governing differential equations at any nodal point. Since high-continuity shape functions are used, some necessary treatments, including the visibility criterion, diffraction, and transparency approaches, are employed in the DLSM to introduce strong discontinuities such as cracks. The stress extrapolation and $J$-integral methods are used to calculate stress intensity factors. Three classic numerical examples using three approaches to defining discontinuities in the irregular distribution of nodal points are considered to investigate the effectiveness of the DLSM method. The numerical tests indicated that the proposed method effectively employed the approaches to defining discontinuities to deal with discontinuous boundaries. It was also demonstrated that the diffraction approach obtained higher accuracy than the other techniques.

The NURBS method is widely used for representing the geometry of engineering and physics systems in computer aided design (CAD) software. However, no finite element formulations currently used in computer aided analysis (CAA) can produce complex geometry identical to NURBS representations. Most contemporary finite element methods, in fact, are at best only crudely capable of representing the complex shapes that can be represented using NURBS. However, finite elements based on the nonlinear absolute nodal coordinate formulation (ANCF) are known to utilize similar shape functions to those employed in Bezier and B-spline representations of geometry. In order to facilitate the integration of computer aided design and analysis (ICADA), this investigation proposes the creation of a new finite element formulation based on the ANCF that utilizes rational polynomials for shape functions. This will lead to the new rational absolute nodal coordinate formulation (RANCF). To demonstrate the feasibility of developing RANCF finite elements, it is shown that rational cubic Bezier curve elements can be converted through a simple coordinate transformation into a rational ANCF form, in which the coordinates are the nodal positions at the end points of the element and the gradient vectors at those nodal points. Furthermore, it is shown that because a cubic NURBS curve can be converted to a series of rational cubic Bezier elements that such a NURBS curve can be converted into a series of these RANCF elements. This investigation only proposes the kinematic description of this new RANCF element. The full development of the nonlinear dynamic equations of motion as well as the development of fully parameterized RANCF finite elements will be explored in future investigations.

Energy methods for calculating structures, which have become popular for a century, are based on the Lagrange principle and have the meaning of equality of work of external forces and internal forces. Having proved their effectiveness in the overwhelming majority of problems of structural mechanics, they became the dominant approach in formulating the problems of studying solid deformable systems and gave rise to the main methodology for solving problems. As a result, a situation has arisen that the internal potential energy of a deformed body remains insufficiently studied.
 The paper develops an approach to the study of the symmetric structure at critical levels of strain energy. The criterion of critical levels of strain energy, based on the concepts of "self-stress" ("self-balance") of a deformable body. Limiting values of the structure strain energy may get by varying the reactions and deflections in the nodal points. The extreme values of forces and displacements of the rods are calculated in matrix form from the values of nodal reactions (displacements).
 Methodology for studying the energy properties of a system is shown on the examples of the study of symmetric rod systems without involving the concept of external forces. The technique is based on matrix methods of structural mechanics and the mathematical apparatus of eigenvalue problems. The comparison of structural design and structural analysis solution of structural mechanics tasks by traditional methods and with the proposed methodology is carried out.

This paper investigates the lateral buckling of simply supported nonprismatic I-beams with axially varying materials by a novel finite element formulation. The material properties of the beam are assumed to vary continuously through the axis according to the volume fraction of the constituent materials based on an exponential or a power law. The torsion governing equilibrium equation of the simply supported beam with free warping is numerically solved by employing the power series approximation. To this end, all the mechanical properties and displacement components are expanded in terms of the power series to a known degree. Then the shape functions are obtained by representing the deformation shape of the axially functionally graded (AFG) web and/or flanges tapered thin-walled beam in a power series form. At the end, new [Formula: see text] elastic and buckling stiffness matrices are exactly determined from the weak form expression of the governing equation. Three comprehensive examples each of axially nonhomogeneous and homogeneous tapered beams with doubly symmetric I-sections are presented to evaluate the effects of different parameters such as axial variation of material properties, tapering ratio and load height parameters on the lateral buckling strength of the beam. The numerical outcomes of this paper can serve as a benchmark for future studies on lateral-torsional critical loads of AFG beams with varying I-sections.