Elastic Curve


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elastic curve

[i′las·tik ′kərv]
(mechanics)
The curved shape of the longitudinal centroidal surface of a beam when the transverse loads acting on it produced wholly elastic stresses.

Elastic Curve

 

in strength of materials, the curve along which the axis of a beam is bent under the action of a load (the axis of a beam is understood as the line connecting the centers of gravity of the beam’s cross sections). If the equation for the elastic curve is known, the differential equations of the theory of bending can be used to determine the amount of deflection for any section of a beam, as well as the angle of rotation, the bending moment, and the transverse force. The equation of the elastic curve is derived from the approximate differential equation for the axis of a bent beam, which may be solved by either the analytic or the graphic-analytic method. The latter is particularly convenient when it is sufficient to find the deflection or angle of rotation at isolated points of the beam, in which case there is no need to derive analytic expressions for the elastic curve.

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Then, we compute the energy of the elastic curves using a variational method in Bishop vectors according to different cases in E41.
We obtain significant differences both on the conditions that have to be satisfied by elastica and on the energy of elastic curves by using Lorentzian metric for different type of curves in [E.sup.4.sub.1].
The interpolating elastic curve provides a uniquely accessible model of complexity: it is physically one-dimensional and is easily conceptualized and exactly calculable.
This is the only example of numerical instability, as opposed to structural instability, which we have encountered in our investigation of interpolating elastic curves using the algorithm of Edwards [1992] for solving such curves, and of Figure 2 herein for their eigenanalysis.
The group of bank customers, either borrowers or depositors, with the least elastic curve will receive the greatest interest rate benefit from any subsidy leakage.
Other borrowers and most depositors are likely to have more elastic curves given the presence of wide nonbank deposit substitutes.
(The reproductions of the sketches of elastic curves by Euler can also be found in an interesting historical survey by Truedell [14]).
However, unlike the conservation laws for the rigid body, the conservation laws for the elastic problem imply several geometric types of solutions on the Lie algebra of G, which account for the bifurcations in the elastic curves and explain their diversity.
If the potential [phi] is constant, then extremals of the above energy [E.sub.[phi]] are known as elastic curves (see [29]).
In particular, if the potential [mu] is constant say [lambda], then the extremals of [A.sub.[lambda]] are the classical elastic curves (which we will call here [lambda]-elasticae).