Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.

Some aspects of consideration of initial imperfections in the calculations of stability of thin-walled elements of open profile

Performed analysis of the initial geometric imperfections influence on the stability of the open C-shaped bars. Test tasks were solved in MSC Nastran, which is based on the finite element method. Imperfections are given in different formulations: the general stability loss of an ideal bar, of wavy bulging of walls and shelves, of deplanation of a bar. To model imperfections, has been developed a program which for the formation of new coordinates of the nodes of the "deformed" model, the components of a vector similar to the form of stability loss are added to the corresponding coordinates of the middle surface of the bar. In this way, you can set initial imperfections in the forms of stability loss of the bar with different amplitude. Researches made with different values of the imperfection amplitude and eccentricity of applied efforts. All tasks are performed in linear and nonlinear staging. The conclusion is made regarding the influence of initial imperfections form and imperfection amplitude on the critical force in nonlinear calculations. It was found that the most affected are imperfections, which are given in the form of total loss of stability. It was revealed the influence of the imperfection amplitude on the magnitude of the critical force for such imperfections. The influence of imperfections amplitude given in the form of wavy bulging walls and in the form of deplanations is not affected on the value of the critical force.

Parametric Optimization of Isotropic and Composite Axially Symmetric Shells Subjected to External Pressure and Twisting

The present paper is devoted to the problem of the optimal design of thin-walled composite axially symmetric shells with respect to buckling resistance. The optimization problem is formulated with the following constraints: namely, all analyzed shells have identical capacity and volume of material. The optimization procedure consists of four steps. In the first step, the initial calculations are made for cylindrical shells with non-optimal orientation of layers and these results are used as the reference for optimization. Next, the optimal orientations of layers for cylindrical shapes are determined. In the third step, the optimal geometrical shape of a middle surface with a constant thickness is determined for isotropic material. Finally, for the assumed shape of the middle surface, the optimal fiber orientation angle θ of the composite shell is appointed. Such studies were carried for three cases: pure external pressure, pure twisting, and combined external pressure with twisting. In the case of shells made of isotropic material the obtained results are compared with the optimal structure of uniform stability, where the analytical Shirshov’s local stability condition is utilized. In the case of structures made of composite materials, the computations are carried out for two different materials, where the ratio of E1/E2 is equal to 17.573 and 3.415. The obtained benefit from optimization, measured as the ratio of critical load multiplier computed for reference shell and optimal structure, is significant. Finally, the optimal geometrical shapes and orientations of the layers for the assumed loadings is proposed.

LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE

The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

An Increase in the Levels of Middle Surface Antigen Characterizes Patients Developing HBV-Driven Liver Cancer Despite Prolonged Virological Suppression

Hepatitis B virus (HBV) contains three surface glycoproteins—Large-HBs (L-HBs), Middle-HBs (M-HBs), and Small-HBs (S-HBs), known to contribute to HBV-driven pro-oncogenic properties. Here, we examined the kinetics of HBs-isoforms in virologically-suppressed patients who developed or did not develop hepatocellular carcinoma (HCC). This study enrolled 30 chronically HBV-infected cirrhotic patients under fully-suppressive anti-HBV treatment. Among them, 13 patients developed HCC. Serum samples were collected at enrolment (T0) and at HCC diagnosis or at the last control for non-HCC patients (median (range) follow-up: 38 (12–48) months). Ad-hoc ELISAs were designed to quantify L-HBs, M-HBs and S-HBs (Beacle). At T0, median (IQR) levels of S-HBs, M-HBs and L-HBs were 3140 (457–6995), 220 (31–433) and 0.2 (0–1.7) ng/mL. No significant differences in the fraction of the three HBs-isoforms were noticed between patients who developed or did not develop HCC at T0. On treatment, S-HBs showed a >25% decline or remained stable in a similar proportion of HCC and non-HCC patients (58.3% of HCC- vs. 47.1% of non-HCC patients, p = 0.6; 25% of HCC vs. 29.4% of non-HCC, p = 0.8, respectively). Conversely, M-HBs showed a >25% increase in a higher proportion of HCC compared to non-HCC patients (50% vs. 11.8%, p = 0.02), in line with M-HBs pro-oncogenic role reported in in vitro studies. No difference in L-HBs kinetics was observed in HCC and non-HCC patients. In conclusion, an increase in M-HBs levels characterizes a significant fraction of HCC-patients while under prolonged HBV suppression and stable/reduced total-HBs. The role of M-HBs kinetics in identifying patients at higher HCC risk deserves further investigation.

Stability of Layered Structures with Hybridized Configuration by Means of a Reddy-Type Higher-Order Finite Element Formulation

To make use of the merit of designability, each lamina in layered structures may possess diverse materials and geometry to realize specific application. For the hybridized structures, geometry and material properties relative to the middle surface are generally unsymmetrical, which have a significant impact on stability. Some models might lose capability to deal with such issues, so that these issues are less reported. Within the developed models, Reddy’s model possesses merit of simplicity and efficiency, so a Reddy-type higher-order zig-zag model is constructed by utilizing the proposed zig-zag function (ZZF). Instead of the standard finite element formulation using the principle of minimum potential energy, the three-field Hu–Washizu (HW) mixed variational principle is employed to acquire the finite element formulation which can meet the harmonious conditions of transverse shear stress at the interface of adjacent layers. By investigating buckling behaviors of hybridized structures, performance of the proposed finite element formulation is appraised by comparing with the results obtained from the three-dimensional (3D) model as well as other models. Effect of boundary conditions (BCs), material properties, and span-to-thickness ratio on the buckling loads is also studied in detail. Numerical results show that buckling loads of hybridized structures are significantly impacted by the chosen parameters. The results acquired from proposed model are in very good agreement with those obtained from the layerwise (LW) model and the 3D finite element results.

A new approach to curvature measures in linear shell theories

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

Stresses in the experiments with loading tubular specimens by internal pressure

When using different formulas for determination of axial and circumferential stresses in the experiments on loading thin-walled tubular specimens with internal pressure the radial stresses are neglected due to their smallness. We propose a novel procedure for determining stresses in the internal pressure loaded thin-walled tubular specimens. The distribution of stresses in the radial direction of a tubular specimen is studied both for the elastic state and for perfectly plastic state according to the Huber – von Mises criterion of an incompressible material. It is shown that the degree of heterogeneity of the stress state depends on the ratio of the wall thickness to the specimen diameter and on the elastic or plastic state of the material. The circumferential stresses are maximal on the inner surface of the specimen and the axial stresses are constant along the radius of the specimen in the elastic state, whereas in the plastic state circumferential and axial stresses are maximal on the outer- and inner surface of the specimen, respectively. The distributions of radial stresses in the elastic and plastic state of the material are almost identical, i.e., both are maximal on the inner surface and equal to zero on the outer surface of the specimen. The values of circumferential and axial stresses on the middle surface of a thin-walled tubular specimen normalized to the internal pressure almost do not depend on the elastic or plastic state of the specimen material thus providing a basis for determination of the mechanical properties of the material from the stress-strain state of the middle surface of the specimen using the Lame formulas for stress calculations. When determining the stress intensity, it is desirable to take into account the radial stresses, since it increases the accuracy of determining the mechanical properties of the material and reduces the sampling range of the yield point for different types of the stress state.

MOMENTAL METHOD FOR CALCULATING FORCES IN CYLINDRICAL SHELLS OF ORBITAL FACILITIES

The article is devoted to the torque method, according to the principles of which it is possible to determine the loads arising in the thin-walled shell of the hulls of orbital vehicles from the action of the internal pressure uniformly distributed over the area, normally oriented to the middle surface of the shell. In this case, the load on the shell consists of normal forces (longitudinal and circumferential), a transverse force that causes radial displacements in the shell, and a bending moment in the longitudinal plane of the object. The specified bending moment can occur during the manifestation of inertial forces (for example, the Coriolis force) during the transition of the orbital vehicle in height from one orbit to another in vertical directions normal to the orbit (rotational movements); the same rotary movements can cause inertial forces to appear when maneuvering spacecraft and rocket units at the same height (in conventionally horizontal directions). To determine the above loads and radial displacements of the shell, a mathematical algorithm is proposed based on the principles of the moment calculation method according to the scheme of an infinitely long shell, which is typical for orbital vehicle housings.

Dent Imperfections in Shell Buckling: The Role of Geometry, Residual Stress, and Plasticity

Departures of the geometry of the middle surface of a thin shell from the perfect shape have long been regarded as the most deleterious imperfections responsible for reducing a shell’s buckling capacity. Here, systematic simulations are conducted for both spherical and cylindrical metal shells whereby, in the first step, dimple-shaped dents are created by indenting a perfect shell into the plastic range. Then, in the second step, buckling of the dented shell is analyzed, under external pressure for the spherical shells and in axial compression for the cylindrical shells. Three distinct buckling analyses are carried out: (1) elastic buckling accounting only for the geometry of the dent, (2) elastic buckling accounting for both dent geometry and residual stresses, and (3) a full elastic–plastic buckling analysis accounting for both the dent geometry and residual stresses. The analyses reveal the relative importance of the geometry and the residual stress associated with the dent, and they also provide a clear indicator of whether plasticity is important in establishing the buckling load of the dented shells.