Nonconservative Loads Research Articles

Global stabilty analysis of a classical non-conservative system under a follower load

The bifurcations, critical conditions and global stability of the classical two-degree-of freedom Ziegler's dissipative model under a partial follower load are thoroughly discussed with the aid of a complete non-linear dynamic analysis. Attention is mainly focused on the range of values of the non-conservativeness loading parameter for which adjacent equilibria do not exist. Various characteristic findings are established by studying the nature of the Jacobian matrix eigenvalues associated with the above non-linear autonomous dissipative system. It is found that the system exhibits stable Hopf bifurcations (global stability) for the entire region of non-adjacent equilibria, while the classical linear stability analysis leads to flutter instability (local instability). Moreover, in an explicit general form, the significant effect of damping on the critical (dynamic bifurcation) load is established. On the basis of the latter, the critical loads for the case of vanishing damping are found to be completely different from those of the classical (local) stability analysis. All findings of this analysis are checked against numerical solution of the equations of motion.

The consistent liquid load operator of the uplifted strip

Recently a detailed analysis of an uplifted strip has been published in this fournal by Fischer, Rammerstorfer, and Schreiner [1]. Their nonlinear liquid load operator (given in the Appendix II of [1]) is not a potential one and not consistent with their liquid load potential (given in Chapter 2 of [1]). These discrepancies disappear, however, for small deformations—only this case is finally considered in [1]—where the load operator turns out to be configuration independent. Based on the author's articles [2] and [3] it is demonstrated in this paper that energetically consistent approximations of an originally nonlinear load operator have to be constructed via the corresponding potential; otherwise the potential quality may be lost simulating incorrectly a nonconservative pressure load.

Non-linear theory for elastic spatial rods

Starting with a three-dimensional non-linear formulation of elasticity and employing the variational procedure together with a suitable and proper strain and displacement field, general forms of kinematical relations, equilibrium equations, and a set of one-dimensional constitutive equations have been derived for spatial rods undergoing small finite deformations. These equations can be employed to study different types of spatial rod elements for different geometries, loading, and cross-sectional shapes. The formulations take account of the influence of the loading behavior, either conservative or non-conservative loading. The corresponding equations for the classical rod theory are extracted from the derived, general non-linear equations.

The influence of shear deformations on the stability of thin-walled beams under non-conservative loading

This paper analyses the influence of shear deformations on the stability of thin-walled beams of open section subjected to follower-type loads. The warping of the cross-sections is assumed to depend on the shear forces as well as on both primary and secondary torsion. The internal stresses are consistently deduced from the kinematic field. An algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique. An extensive comparison is performed with the results offered by the classical models.

Vibration and Stability of a Clamped-Elastically Restrained Timoshenko Column under Nonconservative Load.

The influence of shear deformation and rotatory inertia on the vibration and stability of a clamped-elastically restrained column subjected to a nonconservative force is studied in the present paper. The direction of an external force following the deflection of the column at its elastically restrained end is generally expressed in terms of the arbitrary angle dependent on the slope at the column end. In the analysis, the governing equations derived according to Hamilton's principle are used. The depen-dence of the eigenfrequency, and flutter and divergence loads on the slenderness ratio is numerically examined for various spring stiffnesses and follower force directions. It is shown that although the effect of shear deformation and rotatory inertia is almost to destabilize the system, there may be a case in which it is to stabilize the system in connection with the spring stiffness and follower force direction. Detailed discussion is also presented for the special case of a clamped-free column subjected to a fully tangential follower force in comparison with the other study.

The whiplash problem: an application of vector spherical harmonics

A brief discussion of an alternate method for finding the displacements of an elastic, homogeneous, isotropic sphere under a rotational loading is given. The method is shown to be applicable to nonconservative loadings and asymmetric geometry. Expressions for the strains and stresses are also given.

Some new instability aspects for nonconservative systems under follower loads

The mechanism of instability of nonlinear nonconservative discrete systems under follower loads with or without pre-critical deformation is thoroughly re-examined with the aid of a complete nonlinear dynamic analysis. Considering the stability of motion in the large, in the sense of Lagrange, the critical (divergence or dynamic) load is defined as the minimum load for which an unbounded (divergent) motion is initiated. Regions which have been considered (on the basis of linearized analyses) as of flutter instability are found (using a nonlinear dynamic analysis) dynamically stable. Some new instability phenomena contradict existing findings which have been widely accepted. Moreover, it is established that the divergence buckling loads, obtained by static methods of analysis, coincide with the nonlinear dynamic loads only in the case of no pre-critical deformation. Cases of random-like (or chaotic-like) motions for certain values of the nonconservativeness loading parameter are also revealed for autonomous non-dissipative structural systems.

The computer simulation of rubber components and composites

The numerical analysis of reinforced rubber tyres under different load conditions has been implemented in computer code. The so-called total Lagrangian formulation for a layered rubber shell is developed. The code permits the simulation of non-conservative loading and the effect of road contact which necessitates the provision of rigid wall contact conditions. The numerical solutions provide detailed information on profile deformation, distribution of stresses, strains and strain energies, which are in good agreement with available experimental data.

ELASTIC INSTABILITY AND NONLINEAR ANALYSIS OF THIN-WALLED MEMBERS UNDER NON-CONSERVATIVE FORCES

Elastic instability and the nonlinear finite displacement behavior for spatial thin-walled members under displacement-dependent loads are investigated. The displacement-dependency is reflected on the load stiffness matrix, of which the nonsymmetry indicates the nonconservative loads. A general formulation to derive the load stiffness matrix is presented. As important examples, the follower force and the wind forces are treated, and their load stiffness matrices are given explicitly. The mass matrix of thin-walled mambers is also derived to examine the dynamic instability. The nonlinear finite element analysis as well as the static and dynamic instability analyses are made as computational examples for the follower force and the wind forces, and some interesting features are discussed.

Effect of Storms and Gage Location on Tributary Load Estimate

Monthly and annual pollution loads delivered to Great Lakes receiving waters are made from data often collected far upstream of the mixing zone at the tributary receiving water confluence. The reliability of confluence loading estimates as determined from these upstream gages has not been established. Using data collected during June 1981 in Sandusky Bay/Lake Erie, a combined hydrodynamic and chloride transport model is used to synthesize the load at the confluence. This load is compared to that obtained with data collected during the same time period from the USGS Fremont, Ohio gage which is 29 km upstream of the confluence. The load at the confluence was 2.5 times greater than that estimated with the USGS data. This difference is, for the conservative substance sampled here, attributed to a timing or phasing error. However, the implication for nonconservative substance load estimates is that this difference would be more severe than that resulting from a timing problem.

Dynamic instability analysis of a non-autonomous system subjected to moving impulsive non-conservative load

When a force is applied tangentially on the main axis of beam-type structure, it is well known that the system becomes non-conservative. Dynamic instability of tube excited by moving follower force discretely and periodically is investigated. The effective stiffness of system becomes non-stationary due to time-dependent follower force. Floquet theory is employed to examine the heteroparametric instability regions in the system parameter space. It turns out that there exist two type of instability that are due to parametric and combination resonances, which are confirmed through numerical simulation. Unique features which occur in parametric vibration are also discussed.

Stability of Thin-Walled Open Beams Under Nonconservative Loads

ABSTRACT A finite element formulation for free-vibration analysis of straight prismatic beams of general thin-walled open cross-section, under conservative and nonconservative loads, is presented. The formulation is used to calculate the flutter load for a number of beam problems and is verified by comparison with pre-existing numerical solutions.

Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffnesses based on assumed stresses

Simple and economical procedures for large-deformation elasto-plastic analysis of frames, whose members can be characterized as beams, are presented. An assumed stress approach is employed to derive the tangent stiffness of the beam, subjected in general to non-conservative type distributed loading. The beam is assumed to undergo arbitrarily large rigid rotations but small axial stretch and relative (non-rigid) point-wise rotations. It is shown that if a plastic-hinge method (with allowance being made for the formation of the hinge at an arbitrary location or locations along the beam) is employed, the tangent stiffness matrix may be derived in an explicit fashion, without numerical integration. Several examples are given to illustrate the relative economy and efficiency of the method in solving large-deformation elasto-plastic problems. The method is of considerable utility in analysing off-shore structures and large structures that are likely to be deployed in outerspace.

A three-dimensional finite-strain rod model. part II: Computational aspects

The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.

Vibration and stability of elasticity supported circular plates under conservative and non-conservative loads

This paper deals with the vibration and stability of circular plates elastically restrained against translation and rotation along concentric intermediate circles as well as at the outer edge. The plate is subjected to a horizontal or tangential radial load at the periphery. Both axisymmetric and non-axisymmetric vibrations are considered. It is assumed that the plate is thin, elastic, and isotropic. The solutions are obtained in terms of Bessel functions for both the complete and annular plate sections. The effect of the support stiffnesses on the plate natural frequency and the divergence and flutter instability loads are studied in detail.

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.

Large deformations of in-plane beam

A theory of large deformations of straight slender in-plane beam is presented, based on the assumption of uniaxiality of the strain tensor. The integration of the compatibility conditions gives the parabolic variation of the axial component of the strain tensor over the cross-section of the beam. The integration of the kinematic equations gives linear variation of the displacement components over the cross-section. The equilibrium equations are written on a deformed configuration and numerically solved for linear elastic cantilever, subjected to concentrated or distributed conservative and nonconservative loads.

Extended variational formulations and F.E. models for nonlinear beams under nonconservative loading

Abstract For nonconservative mechanical systems, classical variational principles do not hold true; hence the principle of incremental virtual work is usually taken as a convenient foundation for finite element solution methods [2,8,9]. Recent results on the so-called ‘inverse problem’ of calculus of variations, however, lead to ‘extended’ variational formulations, which can be provided even for nonconservative systems. In the present paper, this approach is used for the particular case of a nonlinear beam subjected to a co-planar follower static loading. Suitable finite element models are deduced from the ‘extended’ variational principle, and their numerical performance is assessed by means of some meaningful examples.

Displacement dependent pressure loads in nonlinear finite element analyses

Often pressure loading is falsely identified as a nonconservative load leading automatically to nonsymmetric load stiffness matrices. The present paper discusses in detail the conditions when a pressure load is conservative and when it is not. The essential part is a clear classification of the load definition. Here either body attached or space attached loads are considered. The load stiffness matrices are derived for a pressure loaded curved surface in space. Several numerical examples are given; among these are linear and nonlinear buckling analyses of beams, rings and shells.

An updated Lagrangian formulation for Timoshenko beams including non-conservative loads for divergence-type systems

On etudie les forces non conservatives de type circulation pour les systemes de type divergence, et leur influence sur la formulation par elements finis des problemes de structures a poutres