Piecewise Constant Thickness Research Articles

Natural vibrations of circular nanoarches of piecewise constant thickness

The free vibrations of elastic circular arches made of a nano-material are considered. A method of determination of eigenfrequencies of nanoarches weakened with stable cracks is developed making use of the concept of the massless spring and Eringen's nonlocal theory of elasticity. The aim of the paper is to evaluate the sensitivity of eigenfrequencies on the geometrical and physical parameters of the nanoarch.

STRESS-DEFORMED STATE OF VERTICAL CILINDRICAL METAL SHELL UNDER TEMPERATURE CLIMATE IMPACT

Vertical cylindrical containers are widely used for storage of granular bulk materials. The enclosing side surface of such containers is made in the form of cylindrical metal circular shells with a wall thickness that is constant or piecewise constant along the height of the shell. Known designs for storage of bulk materials of the reservoir type with the installation of a cylindrical shell on the annular foundation with hinged fixed attachment of the shell to the foundation. Thin-walled shells are made suspended from the supporting structures of the storage facilities. To stabilize the cylindrical shape of the shell and its position in space, the suspended shells are pre-tensioned in the vertical direction. During operation, storage facilities are empty and filled with bulk materials, exposed to the environment in the form of wind and temperature climatic influences. The object of study of this work is the enclosing structures of storages of granular bulk materials in the form of vertical circular cylindrical thin-walled metal shells, in the general case of piecewise constant thickness, subject to temperature and climatic influences of the environment ‒ changes in the temperature of the outside air, direct and diffuse solar radiation. The subject of the study is the components of the stress-strain state of the shell due to changes in temperature and climatic influences. The performed studies of the temperature fields of storage shells on models and full-scale objects made it possible to substantiate the assignment of the temperature field of cylindrical storage shells by Fourier series. One-sided solar heating of cylindrical storage shells completely illuminated by the sun induces in the shell wall a flat temperature field symmetrical with respect to the normal of incidence of sunlight, which can be represented by a Fourier series with five terms of the cosine expansion series. In the presence of a structure located next to the shell, which covers half of the shell along the entire height in the circumferential direction, the temperature field is described by a Fourier series containing 10 harmonics of expansion in sines and cosines. The small thickness of the shell, the significant radius of curvature of the shells, the large thermal diffusivity of metals provides a small variability of the temperature of the shell over the thickness, the ability to describe the stress-strain state of the shell, the momentless theory and a simple edge effect. Formulas are obtained in Fourier series for the forces of a momentless state, the residuals of which at the joints of shell chords of different thicknesses and in the support zones are eliminated by a simple edge effect.

Natural vibrations of curved nano-beams and nano-arches

The natural vibrations of curved nano-beams and nano-arches are studied. The nano-arches under consideration have piecewise constant thickness; these are weakened with stable cracks located at re-entrant corners of the steps. A method of determination of natural frequencies is developed making use of the method of weightless rotating spring. The aim of the paper is to assess the sensitivity of the eigenfrequencies on the geometrical and physical parameters of the nano-arch. The results of the calculations favourably compare with similar works of other researchers.

Dynamic Stability of a Cylindrical Shell Reinforced by Longitudinal Ribs of Piecewise-Constant Thickness under Axial Loading

A model is proposed for studying the simultaneous parametric and longitudinal vibrations of an orthotropic cylindrical shell reinforced by longitudinal ribs of piecewise-constant thickness under the action of an axial time-harmonic load. The discrete location and changes in the thickness of the ribs are taken into account using generalized functions. The basic stages of solving the problem are considered with the combination of methods proposed. Using a numerical example, the dependences of the principal instability region on the geometric parameters of the ribs are plotted for the first time, and a comparison is made with the instability regions for the ribs with a mean-integral thickness. The model developed makes it possible to correct significantly the solutions of problems on calculation of the dynamic stability of orthotropic cylindrical shells with the ribs of piecewise-constant thickness.

Optimal thickness distribution of stepped nonlinear Gao beam

We present an optimal shape design problem of a transversely and axially loaded nonlinear stepped elastic beam with piecewise constant thickness distribution. The nonlinear beam model proposed by D.Y. Gao in 1996 is considered for modelling the state problem. The objective of the optimization is to find the thickness distribution to maximize the stiffness of the beam subjected to geometrical constraints, namely the preservation of volume is imposed. The stiffness is characterized by the compliance cost functional. The existence of solutions of the optimization problem and convergence properties of discretization using finite element approach is analysed. The theory is illustrated by numerical results.

Dynamic Stability of an Orthotropic Cylindrical Shell of Piecewise Constant Thickness under the Action of External Pulsating Pressure

For a shell with three sections, dependences of the critical frequency on the different thicknesses of these sections are obtained. The error of the assumptions accepted at constructing the mathematical model of a shell is evaluated.

Vibration analysis of non-uniform porous beams with functionally graded porosity distribution

In this paper, the effect of different profile variations on vibrational properties of non-uniform beams made of graded porous materials is studied. Timoshenko beam theory is used to present the mathematical formulation of the problem including shear deformation, rotary inertia, non-uniformity of the cross-section, and graded porosity of the beam material. Three different variations of porosities through the thickness direction are introduced. The beam is assumed with the clamped condition at both ends. To obtain a numerical solution, finite element formulations of the governing equations are presented. The non-uniform beam is approximated by another beam consisting of n elements with piecewise constant thickness to keep the volume and hence the total mass unchanged for each element. The beam response has been calculated for the first three modes of vibration. In each case, the results for different types of thickness variation and porosity distribution are compared with those obtained for a beam with uniform thickness. The effects of non-uniformity, taper parameters, and porosity distribution on the frequencies and mode shapes are investigated. It is observed that a considerable change in frequencies and mode shapes can be achieved by selection of different thickness variation and porosity distribution.

Optimal design of a cylindrical composite panel of piecewise constant thickness

PurposeThe purpose of this paper is to investigate the problem of optimizing geometrical and physical parameters of a stepped shell that maximize its rigidity and strength for given overall shell dimensions and fixed weight equal to the weight of a shell of constant thickness.Design/methodology/approachA mathematical model of the construction’s stress-strain state is described by solving a system of differential equations for each of the constituent parts of the shell, conjugation conditions on the division lines and boundary conditions. The stated optimization problem is reduced to a nonlinear programming problem, which is solved by the deformable polyhedron method in combination with the method of direct search and using the parallel computing package in the Wolfram Mathematica software application.FindingsAs follows from the results of the calculation, optimizing the shell parameters allows for a substantial increase in rigidity (decrease of the greatest deflection) and strength (increase of the load-carrying capacity) of the shell of constant stepwise thickness, as opposed to a shell of constant thickness, with constant weight and dimensions.Originality/valueA problem of optimal design of a cylindrical composite panel of piecewise constant thickness is solved in the presented work. Numerical examples demonstrate that a substantial increase in rigidity and strength of a stepped composite shell can be achieved by the optimal choice of its geometrical and physical parameters.

Calculation of a composite cylindrical shell of piecewise constant thickness with consideration of transverse shears

The optimal geometric and physical parameters of a stepped cylindrical shell are determined. These parameters are optimal in the sense that they ensure the maximum carrying capacity of the shell for its given overall size and weight. The shell is made of a composite material. The refined theory of anisotropic shells is used with consideration of transverse shear effects on the stress-strain state.

Buckling of Beams and Columns with Defects

Beams and columns subjected to the axial pressure are studied. Critical buckling loads are established for stepped beams clamped at one end and elastically fixed at the other end. The beams under consideration are of piecewise constant thickness and are weakened by cracks emanating from re-entrant corners of steps. The cracks are assumed to be stable part-through surface cracks. The influence of a crack on the stability of the beam is modeled by the method of distributed line spring known in the elastic fracture mechanics. Numerical results are presented for beams with a single step making use of different stress correction functions.

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger’s equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.

Axisymmetric Vibrations of Stepped Cylindrical Shells Made of Composite Materials. Part 21

Axisymmetric vibrations of stepped cylindrical shells are studied. The shells are made of elastic unidirectionally reinforced composite materials and have a piecewise constant (stepped) thickness. It is assumed that, at the reentrant corners of steps, stable circumferential cracks of constant depth are located. The method of a “massless rotational spring” developed in earlier papers for investigation of homogeneous shells made of isotropic materials is extended to the case of anisotropic composite materials. Analytical expressions are derived for shells with an arbitrary number of steps, and numerical results are presented for shells with one and two steps.

Vibrations of non-uniform functionally graded MWCNTs-polystyrene nanocomposite beams under action of moving load

Free and forced vibrations of non-uniform functionally graded multi-walled carbon nanotubes (MWCNTs)-polystyrene nanocomposite beams are investigated via Timoshenko beam theory. Different MWCNTs distributions in the thickness direction are introduced to improve fundamental natural frequency and dynamic behavior of non-uniform polymer composite beam under action of moving load. So, linear distribution patterns of carbon nanotubes (CNTs) in the thickness direction which can readily be achieved in practice are studied. The effects of shear deformation, rotary inertia, non-uniformity of the cross-section are also considered in the formulation. The finite element method is employed to obtain a numerical approximation of the motion equation. The non-uniform beam is approximated by another beam consisting of n elements with piecewise constant thickness so that the volume remains constant for each element. The effects of non-uniformity parameters, material distributions, velocity of the moving load and boundary conditions on the dynamic behavior are investigated. It is found that the symmetrical linear distribution of MWCNTs results in an increase in the fundamental natural frequency of nanocomposite beams which are higher than those of beams with uniform and unsymmetrical MWCNTs distributions.

Оптимизация прямоугольной пластинки кусочно- постоянной толщины, изготовленной из композиционного материала при изгибе.

Оптимизация прямоугольной пластинки кусочно- постоянной толщины, изготовленной из композиционного материала при изгибе.

Analysis of shells reinforced by massive stiffening ribs

The paper presents results of applying a heterogeneous mathematical model “elastic body–Timoshenko shell” to design shells with massive ribs. Numerical results are obtained for a cylindrical shell with ribs. They are compared with results obtained using the theory of elasticity and the theory of Timoshenko shells with piecewise-constant thickness

Optimization of plastic spherical shells of von Mises material

An optimization procedure is developed for spherical shells pierced with a central hole. The outer edge of the shell is simply supported whereas the inner edge is absolutely free. The material of the shell is assumed to be an ideal plastic material obeying the von Mises yield condition. Resorting to the lower bound theorem of limit analysis, shells with constant and piece-wise constant thickness are considered. The designs of spherical shells corresponding to maximal load carrying capacity are established for a given weight. Necessary optimality conditions are derived with the aid of variational methods from the theory of optimal control. The obtained set of equations is solved numerically.

Optimal design of plastic cylindrical shells of von Mises material

Optimal design of plastic circular cylindrical shells of von Mises material is studied. The optimization problem is stated as the maximization problem of the load carrying capacity for given weight of the shell. Shells with constant and piecewise-constant thickness are considered. The maximization problem is performed under the requirement that the material volume of the stepped shell is equal to the case of the reference shell of constant thickness. The material of the shell is assumed to be an ideal rigid plastic obeying von Mises yield criterion. The considered nonlinear problems are solved by using the CASes method.

Optimal design of stepped circular plates with allowance for the effect of transverse shear deformation

Presented herein is a canonical exact deflection expression for stepped (or piecewise-constant thickness) circular plates under rotationally symmetric transverse loads. The circular plates may be either simply supported or clamped at the edges. As the plates may be very thick or certain portions of the optimal design may become rather thick, the significant effect of transverse shear deformation on the deflections cannot be ignored. This effect was taken into consideration in accordance to the Mindlin plate theory. Based on the analytical deflection expression, necessary conditions are derived for the optimal values of segmental lengths and thicknesses that minimize the maximum deflection of stepped circular plates of a given volume. These optimality conditions are solved using the Newton method for the optimal segmental lengths and thicknesses. Local minima are observed for this nonlinear problem at hand and they may pose some difficulties in getting the solutions. The shear deformation effect increases the plate deflections, but interestingly it affects the thickness variation marginally.

Optimization of clamped plastic shallow shells subjected to initial impulsive loading

An optimization technique is suggested for shallow spherical shells made of a ductile material and subjected to initial impact loading. The shell under consideration is pierced with a central hole and clamped at the outer edge. The optimal design of the shell of piece-wise constant thickness is established under the condition that the maximal residual deflection attains its minimal value for given total weight. The material of the shell is assumed to obey the Tresca yield condition and associated flow law. By the use of the method of mode form motions the problem is transformed into a particular problem of non-linear programming and solved numerically.

Optimization of conical shells of Mises material

Conical shells made of a von Mises material are considered. The shells are subjected to unifromly distributed lateral loading and are simply supported at outer edges whereas inner edges are absolutely free. The shell wall is assumed to be of piece-wise constant thickness. Resorting to the lower bound theorem of limit analysis, optimal designs of shells are established under given weight (material volume of a shell) which corresponds to the maximum load carrying capacity.