On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

Formal analysis of elastically supported beam-columns

Summary A method is presented for the systematic analysis of elastically supported beam-columns or tie-bars in which formal notational devices simplify the handling of complicated discontinuous lateral loads. Various relationships between axial load and support modulus are treated and three particular cases (fixed-fixed, pinned-pinned, and free-free ends) are exhibited.

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Übersicht Es wird ein Verfahren zur systematischen Untersuchung von elastisch gelagerten Balken-Säulen oder Verbindungs-Stäben angegeben, wobei die Berücksichtigung komplizierter Querlasten durch einen besonderen Rechenformalismus erleichtert wird. Verschiedenartige Beziehungen zwischen der Längslast und der Bettungsziffer werden untersucht; drei Spezialfälle sind diskutiert, wobei die Enden fest-fest, gelenkiggelenkig bzw. frei-frei gelagert sind.

     

On the buckling of a girder, elastically supported and elastically clamped in a number of equidistant points

Summary Lately the authors have been engaged in an inquiry with respect to the safety of scaffolding constructions, as used in the erection and the repair of large buildings. A question of detail was presented by the determination of the buckling load of a prismatic girder, elastically supported and elastically clampled in a number of equidistant points. The present paper gives an answer to this problem with the understanding that the endpoints of the girder — though elastically clamped — are fixedly supported.     

Non-conservative stability of intermediate spring supported columns with an elastically restrained end support

The non-conservative stability of an intermediate spring supported uniform column clastically restrained at one end and subjected to a follower force at the other unsupported end is studied. It is found that when the intermediate spring support is far from the unsupported end, the instability mechanism is flutter. As the intermediate spring support approaches the unsupported end, the instability mechanism is changed from flutter to divergence with the increase of intermediate spring stiffness. For the hinged-intermediate and guided-intermediatc spring supported columns, the critical buckling load of flutter instability will first decrease, then increase as the intermediate spring stiffness is increased. Nevertheless, when the instability mechanism is divergence, the critical buckling load depends on the location of the intermediate spring support only, whereas for the clamped-intermediate spring supported column the critical buckling load of divergence instability decreases monotonically to a fixed value as the intermediate spring stiffness is increased. Finally, the influence of elastic end restraints on the stability of the column is also investigated.     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.     

Nonlinear analysis and buckling of elastically supported circular shallow arches

Arches are often supported elastically by other structural members. This paper investigates the in-plane nonlinear elastic behaviour and stability of elastically supported shallow circular arches that are subjected to a radial load uniformly distributed around the arch axis. Analytical solutions for the nonlinear behaviour and for the nonlinear buckling load are obtained for shallow arches with equal or unequal elastic supports. It is found that the flexibility of the elastic supports and the shallowness of the arch play important roles in the nonlinear structural response of the arch. The limiting shallownesses that distinguish between the buckling modes are obtained and the relationship of the limiting shallowness with the flexibility of the elastic supports is established, and the critical flexibility of the elastic radial supports is derived. An arch with equal elastic radial supports whose flexibility is larger than the critical value becomes an elastically supported beam curved in elevation, while an arch with one rigid and one elastic radial support whose flexibility is larger than the critical value still behaves as an arch when its shallowness is higher than a limiting shallowness. Comparisons with finite element results demonstrate that the analytical solutions and the values of the critical flexibility of the elastic supports and the limiting shallowness of the arch are valid.     

Transition radiation in an elastically supported string. Abrupt and smooth variations of the support stiffness

Generation of waves due to a constant load moving uniformly along an infinite string resting on an inhomogeneous elastic foundation is studied. Two types of inhomogeneity are considered: (1) an abrupt change of the foundation stiffness; (2) a smooth change of the foundation stiffness. It is shown that transition radiation arises as the load passes the region of inhomogeneity. Expressions for the spectral density of the radiation energy forwards (in the direction of the load motion) and backwards are found and analysed as a function of the load velocity, the ratio of the foundation stiffness and a characteristic length of the inhomogeneity. To visualise the process of radiation the transient vibrations of the string are determined for an abrupt change of the stiffness.     

Influence of shear deformation on the buckling of elastically supported beams

Summary The influence of shear deformation on the buckling behavior of a beam supported laterally by a Winkler elastic foundation is studied. A full investigation of the bifurcation points at which, under axial load, the beam becomes critical with respect to one or two simultaneous buckling modes is made. The configurations and stabilities of the equilibrium paths that bifurcate from the critical points are derived. From the results of theoretical analysis, it becomes evident that shear deformation has a considerable effect upon the equilibriums and stabilities of the post-buckling of the beam. The results for the Bernoulli-Euler beam can be obtained as a limiting case for those of the present beam by letting the shear stiffness tend to infinity.Supported by the National Natural Science Foundation of China     

Transverse vibration of rectangular plates elastically supported at points on edges

This paper studies transverse vibration of rectangular plates with two opposite edges simply supported, other two edges arbitrarily supported and free edges elastically supported at points. A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular plates elastically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourier series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given. Inherent frequencies and mode shape with any accuracy can be gained.     

Radiation emitted by a constant load moving uniformly in a circle on an elastically supported membrane

Wave radiation is studied which is due a constant load moving with a constant speed along a circular path over an unbounded membrane on a elastic foundation. The steady-state solution of the problem is obtained, showing that the radiation occurs for all load velocities. It is shown that the elastic field radiated by the supercritically moving load is confined in a spiral-like apex. The membrane displacements at the boundaries of this apex are discontinuous. The radiated energy per period of load rotation is calculated showing a discrete energy spectrum. For increasing load velocities, the total amount of radiated energy becomes larger. It also turns out that the major part of the radiated energy follows the direction of the load motion.     

Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force

The symmetric and asymmetric buckling of an initially curved micro beam subjected to an axial pre-stressing load and transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam used as the base functions. The criteria of symmetric limit point buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beam and the axial load. Two symmetry breaking conditions, defining the relations between the axial load and the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one, are obtained. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric response curve. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict symmetric and non-symmetric buckling in electrostatically actuated curved pre-stressed micro beams. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. The nature of the latter and the relations between it and the symmetry breaking criteria are found to go through a prominent qualitative change as the initial distance between the beam and the electrode, characterizing the electrostatic force, changes.     

The buckling load of an elastically supported cantilevered column with continuously varying cross section and distributed axial load

Summary The solution for the buckling load of an elastic supported cantilevered column with continuously varying cross section and distributed axial load is developed. Boundary conditions are formulated for fixed and elastic support. The critical loads are tabled for an usual variation in cross section and axial load distribution as function of the ratio between respectively rigidities and axial forces at fixed and free end, the variation of the rigidity and axial forces given by their degrees of the polynomial variation and rotation stiffness of the elastic support.

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Übersicht Es wird die Knicklast für eine elastisch gelagerte, freitragende Säule mit stetig veränderlichem Querschnitt und stetig verteilter Axiallast berechnet. Die Randbedingungen sind für feste sowie für elastische Lagerung formuliert worden. Für einen übliczen Bereich von Querschnitten und Axiallasten werden die kritischen Lasten in Tabellen angegeben. Dabei wurde das Verhältnis zwischen der Steifigkeit und den Axiallasten am festen und freien Ende variiert.

     

Dynamic coefficient evaluation for an algebraic subgrid stress model

Static model coefficients for an algebraic subgrid stress (SGS) model are determined using a dynamic approach, based on results from simulations of isotropic decaying turbulence. The study was motivated by the discrepancies in energy transfer predictions using the previously documented coefficients (Bhushan and Warsi, Int. J. Numer. Meth. Fluids 2005; 49 : 489–519). The discrepancies are identified to be due to inconsistent filter functions used in the analytic estimates and the simulations. The study emphasizes that SGS model development should use filter functions compatible with those inherent in CFD application solvers. The dynamic approach predicts consistent model and transfer coefficients for different grid resolutions and is judged to be a reliable basis for model coefficient adjustments. The predicted Leonard's stress coefficient and associated energy transfer coefficients agree very well with the analytic estimates using a Gaussian/cutoff combination filter. This suggests that the modeling of Leonard's stress term using a truncated Taylor series expansion is robust and may not benefit significantly from dynamic modeling. Validation simulations were performed for turbulent channel flow at Re_τ = 180 and 590. The dynamic approach was found to be reliable only for the lower log‐layer of the Re_τ = 590 case, where the scale invariance condition was satisfied. Nonetheless, in this narrow range, the model and transfer coefficients compare well with the isotropic case. The static coefficient algebraic model with new adjusted coefficients shows improved predictions compared with the previous coefficients, for both isotropic decaying turbulence and channel flow cases. Copyright © 2013 John Wiley & Sons, Ltd.     

Rocking instability under ground motion of large statues freely standing atop elastically supported cantilevers

The rocking (overturning) instability of freely standing large statues atop the capital of elastically supported cantilevers subjected to ground motion is thoroughly discussed. The present analytical solution deals with a monolithic slender cantilever carrying a freely standing rigid block (simulating the statue) instead of the actual multi-drum cantilever of ancient columns that exhibit the loss of energy due to impact and sliding between drums. Hence, owing to the omission of the dissipation of energy, the results of this analysis based on a monolithic cantilever are quite conservative. Attention is focused on establishing the minimum-amplitude ground acceleration causing overturning of the rigid block with or without impact. Rocking instability criteria for determining such a minimum amplitude, which leads through the vanishing of the angular velocity to an escaped motion in the phase-plane portrait, are properly established by including the effect of time-dependent initial conditions. Closed-form solutions confirmed via numerical simulation bring to light important findings regarding the individual and combined effect of various parameters on the rocking instability of the rigid block.     

Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment

Free vibration analysis of functionally graded (FG) thin-to-moderately thick annular plates subjected to thermal environment and supported on two-parameter elastic foundation is investigated. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle based on the first order shear deformation theory (FSDT). The initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with the available solutions in the literature for isotropic and FG circular and annular plates. The effects of the temperature rise, elastic foundation coefficients, the material graded index and different geometrical parameters on the frequency parameters of the FG annular plates are investigated. The new results can be used as benchmark solutions for future researches.