Uniform shallow arches of minimum weight and minimum maximum deflection

The present paper serves several purposes. Besides presenting closed-form solutions to the problems in the title, it serves to illustrate the deduction of existence and nonexistence of solutions in this context, to point out some of the more or less known inadequacies of these design criteria and, finally, to provide the basis for a comparison with the natural structural shapes of shallow arches presented in another reference. The minimum weight and minimum maximum deflection criteria both yield, as one possible optimal design, an arch on the verge of failure. This is consequence of the fully-stressed design aspects of these criteria which, in this case, correspond to the maximum possible axial load. However, meaningful results are obtained for a prescribed axial load in the minimum weight problem and for a given weight in the minimum of the maximum deflection problem.     

Particle modelling of an elastic arch

A new particle-type of modelling has been developed for the simulation of phenomena related to the bending of an elastic arch. Application is made to the simulation of snap-through inversion and to the simulation of unstable equilibrium modes. Computer examples are described and discussed.     

The optimal elbow angle for acrobatics ‐ stability analysis of the elbow with a load

We consider a human elbow with a load and study the existence and stability of equilibrium positions without the use of reflexes. We observe that for each elbow angle there is a critical mass above which the equilibrium becomes unstable. We determine the optimal elbow angle, i.e., the angle such that the corresponding critical mass is as high as possible.     

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

Nonlinear in-plane buckling of fixed shallow functionally graded graphene reinforced composite arches subjected to mechanical and thermal loading

The nonlinear in-plane buckling analysis for fixed shallow functionally graded (FG) graphene reinforced composite arches which are subjected to uniform radial load and temperature field is presented in this paper. The arch is composed of multiple graphene platelet reinforced composite (GPLRC) layers with gradient changes of concentration of graphene platelets (GPLs) in each layer. The principle of virtual work, combined with the effective materials properties estimated by the Halpin-Tsai micromechanics model for GPLRC layer, is used to derive the nonlinear buckling equilibrium equations of the FG-GPLRC arch, and then the analytical solutions for the limit point and bifurcation buckling loads are obtained. Comprehensive parametric studies are conducted to explore the effects of various distribution patterns and geometries of GPL, temperature field and arch geometry on the nonlinear equilibrium path and buckling behavior of the composite arch. The influence of temperature on the geometric parameters which are defined as switches between limit point buckling, bifurcation buckling and no buckling are also discussed. It is found that a higher temperature field can increase the buckling loads of the FG-GPLRC arch but reduce the value of the minimum geometric parameters that switching the buckling modes. The results also show that even a small amount of GPLs filler content can increase the buckling loads of the FG-GPLRC arch considerably, and distributing more GPLs near the surface layers is the best pattern to enhance the buckling performances of FG-GPLRC arches.     

Convergence of mixed finite element approximations for the shallow arch problem

Summary The stability and convergence of mixed finite element methods are investigated, for an equilibrium problem for thin shallow elastic arches. The problem in its standard form contains two terms, corresponding to the contributions from the shear and axial strains, with a small parameter. Lagrange multipliers are introduced, to formulate the problem in an alternative mixed form. Questions of existence and uniqueness of solutions to the standard and mixed problems are addressed. It is shown that finite element approximations of the mixed problem are stable and convergent. Reduced integration formulations are equivalent to a mixed formulation which in general is distinct from the formulation shown to be stable and convergent, except when the order of polynomial interpolationt of the arch shape satisfies 1tmin (2,r) wherer is the order of polynomial approximation of the unknown variables.     

The effect of the batdorf parameter on the post-critical behaviour of an axially compressed imperfect cylindrical shell

The loss of stability and post-critical behaviour of a geometrically imperfect elastic cylindrical shell subjected to axial compression at moveable hinged endfaces are asymptotically analysed in the limit as Z → ∞ (where Z is the Batdorf parameter). The asymptotic behaviour of the eigenvalues and associated vectorial eigen-functions, linearized about a torqueless solution of the boundary-value problem are constructed when Z → ∞. The Lyapunov-Schmidt method is applied in the neighbourhood of each eigenvalue for which the asymptotic behaviour has been determined. For Z → ∞ equilibrium eigenshapes that are odd with respect to the axial coordinate are shown to be unstable (the Koiter parameter b < 0), and the even ones (b 0) are shown to be stable. It is shown that by an appropriate choice of initial imperfection the upper critical load for shell loss of stability (the limiting point) can be made to correspond to any of the close to (Z → ∞) critical loads for loss of stability of an ideal shell.     

Buckling of clamped circular arches subjected to a point load

Summary This note presents the critical values of the downward point load applied at the crown of clamped circular arches with inextensible centroidal line. The calculations are based on the Euler theory of the initially curved elastica undergoing large deflections. In contrast to the two-highed circular arches the buckling is not always of the sidesway type; for most practical dimensions of the clamped arch the instability is characterized by snap-through.

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Zusammenfassung Mit Hilfe der nichtlinearen Eulerschen Theorie wird die Stabilität des beidseitig eingespannten und durch eine mittige Einzelkraft belasteten Kreisbogenträgers untersucht. Man nimmt dabei an, der Bogen sei genügend schlandk und die Dehnung der Stabachse vernachlässigbar klein. Im Gegensatz zum Zweigelenkbogen zeigt sich die Instabilität des steilen, eingespannten Bogenträgers eher in einem Durchschlagen als im seitlichen Ausweichen.

     

Optimization of reliability and longevity of cylindrical shells subject to corrosive wear

The problem of the optimal (with respect to the minimum average rate of weight loss in a structure) design of a smooth cylindrical shell loaded by an axial compressive force and under corrosive wear, the rate of which depends exponentially on the stresses, is solved. The problem is solved analytically using apparatus of the method of Lagrange multipliers.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 59–63, 1987.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

Optimization of shallow arches against instability using sensitivity derivatives

In this paper, the problem of optimization of shallow frame structures, which involves coupling of axial and bending responses, is discussed. A shallow arch of given shape and given weight is optimized such that its limit point load is maximized. The crosssectional area, A(x), and the moment of inertia, I(x), of the arch obey the relationship I(x) = [A(x)]ⁿ, where N = 1, 2, 3 and is a specified constant. Analysis of the arch for its limit point calculation involves a geometric nonlinear analysis which is performed using a co-rotational formulation.

The optimization is carried out using a second-order projected Langragian algorithm, and the sensitivity derivatives of the critical load parameter with respect to the areas of the finite elements of the arch are calculated using implicit differentiation. Results are presented for an arch of a specified rise to span under two different loadings, and the limitations of the approach for the intermediate rise arches are addressed.     

Stabilization of the equilibria of dynamical systems

Based on Rumyantsev's method, a procedure is developed for stabilizing stable and unstable equilibria of dynamical systems by continuous and modulus-constrained control actions. It is shown that, when modulus constraints are imposed on the controls, and when the quadratic-form coefficients are reduced in modulus, the optimal stabilization, in terms of this method, approximates to time-optimal stabilization. A solution is obtained of the problem of stabilizing unstable equilibrium positions at which the potential energy of the system has neither a maximum nor a minimum (or, in particular, at which the potential energy is identically equal to zero).     

Dynamical stability and optimization of parameters of cylindrical shells under short-time influence of loads that increase in a power-wise way

Some criteria of dynamical nonstability are derived for smooth cylindrical shells under actions of axial contracting loads that increase fast in a power-wise way. In terms of nonlinear programming the optimization problem is formulated that determines parameters of shells of minimal mass under some restrictions with respect to local and global loss of stability and also with respect to strength. With the help of the Kuhn-Tucker theorem some variants of the optimal solution are obtained. As an example, we consider two characteristic forms of load: linearly increasing load, and a load that increases in the quadratic way. We study the influence of basic parameters of these loads on the mass, and geometric dimensions of the optimal shells. Some numerical examples are given.Translated from Dinamicheskie Sistemy, No. 8, pp. 47–53, 1989.     

Influence of load position on the stability of shallow arches

The influence of load position of lhe snap-through instability of shallow, circular, elastic arches is investigated. Results are obtained for pinned, clamped, and clamped-pinned end conditions. The worst load position is seen to depend on the arch rise.

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Zusammenfassung Der Eifluss der Kraftlage auf die Durchschgsinstabilität von flachen, elastischen Kreisbögen wird untersucht. Es sind Ergebnisse erhalten worden für Bögen mit gelenkigen eingespannten, und gelenkig-eingespannten Lagern. Die ungünstigste Kraftlage hängt von der Bogenhöhe ab.

     

Minimum Principle of Complementary Energy for Nonlinear Elastic Cable Networks with Geometrical Nonlinearities

A minimum principle of complementary energy is established for cable networks involving only the stress components as variables with geometrical nonlinearities and nonlinear elastic materials. The minimization problem of total potential energy is reformulated as a variational problem with a convex objective functional and an infinite number of second-order cone constraints; its Fenchel dual problem is shown to coincide with the minimization problem of the complementary energy. It is of interest to note that the obtained complementary energy attains always its minimum value at the equilibrium state irrespective of the stability of the cable networks, contrary to the fact that only stationary principles have been presented for elastic trusses and continua, even in the case of a stable equilibrium state.     

闭合圆柱壳在轴向冲击荷载下的某些动力计算问题

本文讨论了闭合圆柱形壳体在轴向冲击荷载作用下的某些动力计算问题,其中包括动应力的计算和稳定性问题.文中分析了冲击过程中动量及能量的变化,并计入冲击物和被冲击的闭合圆柱壳系统质量的影响;用相当质量法将整个圆柱形壳体的分布质量转化为只集中在壳体一端的“相当质量”,从而导出闭合圆柱形壳体在轴向冲击荷载作用下的动力因数,因而解决了在上述受力情况下计算动应力的问题和求出临界荷重的问题.     

Chaos and optimal control of cancer self-remission and tumor system steady states

This paper is devoted to study the problem of optimal control of cancer self-remission and tumor unstable steady-states. The stability analysis of the biologically feasible equilibrium states is presented using a local stability approach. The system appears exhibit a chaotic behavior for some ranges of the system parameters. The necessary optimal control inputs for the asymptotic stability of the positive equilibrium states and minimizes the require performance measure are obtained as nonlinear function of the system densities. Analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out for various parameters values and different initial densities.     

Snap-through buckling of a shallow arch resting on a two-parameter elastic foundation

In this paper, we study the static and dynamic snap-through of a shallow arch resting on a two-parameter elastic foundation under a point load moving at a constant speed. The deformation of the arch is expressed in Fourier series. We extend the previous works when the arch resting on a certain model of two-parameter elastic foundation. Each model of foundation has a different definition for the second foundation parameter, where the model is discussed in this paper is Pasternak model. For quasi-static analysis, it is noted that the first four modes in the expansion are sufficient to predict the response of the arch. Similarly, when the point load moves with a significant speed, we integrate the equation of motion numerically using the first modes in the expansion.     

Stabilization of a quasi-conservative system subjected to high frequency excitation

The existence and stability conditions for the steady motions and equilibrium positions of non-linear quasi-conservative systems with fast external perturbations having quasi-periodic and random components are investigated. A change of variables is proposed which reduces Lagrange's equations of the system to standard form. It is shown the averaged system of the first approximation has a canonical form and the effect of fast perturbations (not necessarily potential) is equivalent to a change in the system's potential. This leads to stabilization of unstable equilibrium positions and to the appearance of additional stationary points different from the equilibrium positions of the unperturbed system. The approach used is illustrated by examples; the stability of a pendulum on an elastic suspension when there is suspension point, and the steady motion of a sphere subjected to a high-frequency load. The critical loading of a double pendulum loaded by a pulsating tracking force is estimated. A form of wide-band random perturbations capable of stabilizing the system is considered.     

The stability of Riemann ellipsoids in the Lyapunov formulation

The problem of the stability of the Riemann ellipsoids of a rotating uniform self-gravitating ideal liquid is considered within the framework of the Lyapunov definition of the stability of the form of equilibrium [1]. The regions such that almost all the ellipsoids belonging to it are unstable forms of equilibrium, specified in explicit analytical form, are determined in parameter spaces of the first and second families of Riemann ellipsoids. The proof is based on the general fact (which is formulated and justified separately) that, when an unstable equilibrium position of an autonomous system is stable with respect to a certain function, the trajectory of this system, which belongs to a certain manifold, is obtained, and also on a consequence of this fact, which has a constructive form. The stability of the form of ellipsoidal figures of equilibrium, with the exception of special cases of Maclaurin and Jacobi ellipsoids, the stability of the form of which was investigated by Lyapunov himself, has not been investigated previously in the literature.