Equilibria of a clamped Euler beam (Elastica) with distributed load: Large deformations

We present some novel equilibrium shapes of a clamped Euler beam (Elastica from now on) under uniformly distributed dead load orthogonal to the straight reference configuration. We characterize the properties of the minimizers of total energy, determine the corresponding Euler–Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers. Moreover, we prove some sufficient conditions for stability and instability of solutions of the Euler–Lagrange, that can be applied to numerically found curled shapes.

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Exact Penalization in Stochastic Programming—Calmness and Constraint Qualification

Advances in Decision Sciences ◽

10.1155/2014/569458 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Martin Branda

Keyword(s):

Stochastic Programming ◽

Error Bound ◽

Constraint Qualification ◽

Sufficient Conditions ◽

Discrete Distributions ◽

Exact Penalty ◽

Asymptotic Equivalence ◽

Exact Penalization ◽

Local Minimizers ◽

Constrained Problems

We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

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Analysis of the Strong and Weak Monotonic External Stability of the Resonance in a Perturbed Dynamical System

WSEAS TRANSACTIONS ON FLUID MECHANICS ◽

10.37394/232013.2021.16.17 ◽

2021 ◽

Vol 16 ◽

pp. 180-191

Author(s):

Vladislav V. Lyubimov

Keyword(s):

Dynamical System ◽

Sufficient Conditions ◽

Strong Stability ◽

New Classification ◽

External Stability ◽

Fast Phase ◽

Stability And Instability ◽

Long Time ◽

Independent Variable

A perturbed dynamical system involving two ordinary differential equations is under review. Whereupon, the differential equation for determining the fast phase contains the ratio of the two frequencies. When these frequencies coincide for a long time, a resonance is implemented in this system. The aim of this paper is to obtain the conditions of monotonic external stability and instability of this resonance. The sufficient conditions for the external stability and instability of the resonance defined in this paper assume that the signs of the analyzed derivatives remain unchanged in the non-resonant section of the change in the independent variable. This paper gives a new classification of the phenomenon of external stability of resonance, which includes weak, linear, and strong stability. It should be noted that the conditions of monotonic external stability and instability of the resonance presented in this paper can be used in various scientific and technological problems, in which resonances are observed. Particularly, the concluding part of the paper considers the application of the results obtained within the framework of the problem of the perturbed motion of a rigid body relative to a fixed point.

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Study of Fire Behavior of a Profiled Sheet-Ceramsite Concrete Composite Floor

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.243-249.5231 ◽

2011 ◽

Vol 243-249 ◽

pp. 5231-5235

Author(s):

Xin Tang Wang ◽

Ming Zhou ◽

Hai Jiang Wang ◽

Zhi Guo Xie

Keyword(s):

Bearing Capacity ◽

Fire Behavior ◽

Experimental Results ◽

Dead Load ◽

Fire Response ◽

Fire Load ◽

Distributed Load ◽

Obvious Change ◽

Bending Capacity

In order to study the fire behavior of the profiled sheet-ceramsite concrete composite floor subjected to fire load, research on fire response and post-fire bearing capacity of a profiled sheet-ceramsite concrete composite floor subjected to dead load, which has no shearing nails, is carried out here through experiment. Based on the experimental results, the fire behavior and post-fire bearing capacity of the floor after exposure to fire are analyzed. It is shown that the failure form of the profiled sheet-ceramsite concrete composite floor after exposure to fire has obvious change compared with the floor not subjected to fire load, but the composite floor subjected to fire still exhibits higher bending capacity, and the ultimate value of the equivalent distributed load is up to 35kN/m2, which may be used as basis of strengthening and repairing of the profiled sheet-ceramsite concrete composite floor after exposure to fire.

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GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

International Journal of Biomathematics ◽

10.1142/s1793524508000436 ◽

2008 ◽

Vol 01 (04) ◽

pp. 503-520 ◽

Cited By ~ 3

Author(s):

ZHIQI LU ◽

JINGJING WU

Keyword(s):

Global Stability ◽

Local Stability ◽

Sufficient Conditions ◽

Positive Equilibrium ◽

Delayed Response ◽

Qualitative Properties ◽

Chemostat Model ◽

Stability And Instability ◽

Discrete Delays ◽

Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

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Stability Conditions for a Class of Distributed-Parameter Systems

Journal of Basic Engineering ◽

10.1115/1.3645882 ◽

1966 ◽

Vol 88 (2) ◽

pp. 475-479 ◽

Cited By ~ 4

Author(s):

R. E. Blodgett

Keyword(s):

Equilibrium State ◽

Local Stability ◽

Direct Method ◽

Sufficient Conditions ◽

Chemical Reactor ◽

Distributed Parameter Systems ◽

Distributed Parameter ◽

Stability Conditions ◽

Stability And Instability ◽

Liapunov's Direct Method

The purpose of this paper is to obtain stability conditions for a class of nonlinear distributed-parameter systems by using a generalization of Liapunov’s direct method. Sufficient conditions for local stability and instability of the equilibrium state are derived. An application is given in which conditions are obtained for stability of a chemical-reactor process.

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Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

Journal of Vibration and Acoustics ◽

10.1115/1.2838656 ◽

1995 ◽

Vol 117 (B) ◽

pp. 145-153 ◽

Cited By ~ 29

Author(s):

D. S. Bernstein ◽

S. P. Bhat

Keyword(s):

Asymptotic Stability ◽

Lyapunov Stability ◽

Sufficient Conditions ◽

Second Order ◽

Necessary And Sufficient Conditions ◽

Viscous Damping ◽

Stability And Instability ◽

The Stability ◽

Second Order Systems ◽

Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

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Eigenvalue problems for the wave equation with strong damping

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023805 ◽

1997 ◽

Vol 127 (4) ◽

pp. 755-771 ◽

Cited By ~ 4

Author(s):

Pedro Freitas

Keyword(s):

Stationary Solution ◽

Wave Equations ◽

Eigenvalue Problems ◽

Sufficient Conditions ◽

Stationary Solutions ◽

Linear Operators ◽

Stability And Instability ◽

The Stability ◽

Necessary And Sufficient ◽

Strongly Damped

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

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MULTIPLE SOLUTIONS FOR INEXTENSIBLE ARCHES

Transactions of the Canadian Society for Mechanical Engineering ◽

10.1139/tcsme-1993-0001 ◽

1993 ◽

Vol 17 (1) ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

A.W. Lipsett ◽

M.G. Faulkner ◽

V. Tam

Keyword(s):

Present Method ◽

Nonlinear Boundary ◽

Large Deformations ◽

Multiple Equilibrium ◽

Equilibrium Solutions ◽

Initial Value ◽

Shooting Technique ◽

Systematic Fashion ◽

Equilibrium Shapes ◽

Large Deformation Problems

The multiple equilibrium solutions of both deep and shallow inextensible arches is investigated through the use of a segmental shooting technique. The original nonlinear boundary value problem governing the large deformations of these arches is solved using a sequence of linear initial value problems which converge iteratively to the required boundary conditions. This method has proved successful in previous studies to analyse these large deformation problems when there is initially only one unknown at the beginning of the arch that must be assumed. Extension of this method to the problem with two initial unknowns is presented here. In either case all equilibrium solutions can be found in a systematic fashion. When there are three initial unkowns the present method is not able to determine the totality of equilibrium shapes for a given problem and loading. However, in this case, it is possible to determine a number of equilibrium shapes using engineering judgement for the values of the initial unknowns.

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