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The stress induced in a loaded beam will not exceed some threshold, but also its maximum deflection, as for all the elastic systems, will be controlled. Nevertheless, the linear beam theory fails to describe the large deflections; highly flexible linear elements, namely, rods, typically found in aerospace or oil applications, may experience large displacements—but small strains, for not leaving the field of linear elasticity—so that geometric nonlinearities become significant. In this article, we provide analytical solutions to large deflections problem of a straight, cantilevered rod under different coplanar loadings. Our researches are led by means of the elliptic integrals, but the main achievement concerns the Lauricella

As early as 1691, Jakob Bernoulli proposed to find out the deformed centerline (planar “elastica”) of a thin, homogeneous, straight, and flexible rod under a force applied at its end, and ignoring its weight. Studying the geometry of its deflection, he established,

A rod is something of a hybrid, that is, a mathematical curve made up of material points. It has no cross section, yet it has stiffness. It can weigh nothing or it can be heavy. We can twist it, bend it, stretch it and shake it, but we cannot break it, it is completely elastic. Of course, it is a mathematical object, and does not exist in the physical world; yet it finds wide application in structural (and biological) mechanics: columns, struts, cables, thread, and DNA have all been modeled with rod theory.

The link between elliptic functions and elastica was always understood so close, to induce the elliptic functions to be plotted through suitable curved rubber rods (G. Greenhill:

Our object is the analytical solutions to large-deflection problems of a planar slender cantilever under coplanar loads. The ^{1}, generally credited of great potential for future space missions in order to bring down space debris (derelict spacecraft, etc.). Unlike the majority of tethers, which can be treated as cables, Space European Tether (SET, see [^{2}. All above provide enough motivations to implement exact solutions within the rod theory, see [

The exact governing equations for the finite displacement rod theory become highly nonlinear, and, hence, it is very difficult to solve these equations analytically. Thus, for practical purposes, problems of this kind are usually solved by the methods based on the finite elements approximations. However, the closed form solutions for these problems are still important to the practical point that the accuracy of the approximate methods can be precisely evaluated by these solutions, to say nothing of the mathematical importance.

It is well known why the elastic curve of a rod is essential, in fact not only the stresses, but also its maximum deflection will not exceed a fixed amount. Furthermore, (see Mattiasson [

When developing finite element methods, there is also a need for checking the results against an exact solution. In the case of finite element methods for geometrically nonlinear beam, frame and shell structures, elliptic integral solutions of large deflection beam and frame problems offer such exact solution.

Nevertheless, his approach is a bit obsolete nowadays because he computes the elliptic integrals numerically through the arithmetic-geometrical means.

We are going to analyze an elastic, thin, flexible rod, clamped at one end (cantilever) bent under different loadings whose plane coincides with that of the strained centerline. Equating the exact curvature in cartesian coordinates to the bending moment divided into the flexural rigidity, one is led to a second-order nonlinear and nonautonomous differential equation. Boundary conditions are given both on the clamped end (Cauchy problem), so that, for example, if the constraint is perfect, displacement and rotation have to be zero there. We discarded the illusion of a general system of loads and rods and restricted ourselves to only one configuration, charged by a single load and treating each single problem by means of the special functions of mathematical physics. The chosen loads are constant couple, evenly distributed load, shear at the tip, hydrostatic load, and sinusoidal load. All our force loads have to be understood as “dead load”, namely, always equal to itself and then not “follower” as are currently named the loads keeping a fixed inclination to the rod before and after the strain.

For one of the above loads, based on the rod's unextensibility, we computed in closed form the

We assume that the reader knows the elliptic functions (see [

The hypergeometric series first appeared in the Wallis's ^{3}; nevertheless, he does not seem to have known the integral representation ^{4}. For all this and for the Stirling contributions, see [

Several functions have been introduced in the 19th century for generalizing the hypergeometric functions to multiple variables, but we will mention only those introduced and investigated by Lauricella (1893) and Saran (1954). Among them our interest is focused on the hypergeometric function

Let us tackle a

the rod is thin, initially straight, homogeneous, with uniform cross-section and uniform flexural stiffness

the slender rod is charged by coplanar dead loads;

linear constitutive load: the induced curvature is proportional via

the shear transverse deformation is ignored, and the rod is assumed to keep constant its previous length (inextensibility);

a stationary strain field, by isostatic equilibrium of active loads and reactive forces, takes place, and, due to rest of static equilibrium, no rod element undergoes acceleration.

The key Bernoulli-Euler linear constitutive law connects flexural stiffness

In the sequel, we are going to use dimensionless variables

The clamped section is at

Horizontal cantilever rod inflected by a constant bending couple at its tip.

The classic elastica theory started by the “lamina”, namely, a

This problem… has shown considerable resistance to a closed form solution and all investigations dealing with it have employed approximate methods only.

Accordingly, he obtains elastica coordinates as parametric functions of the arclength

Let us consider at Figure

Horizontal cantilever rod bent by a uniform load.

By identification, we obtain the solution to our elastica through a Lauricella function of 3 variables

We got there three

if

if

if

Formula (

We refer to Figure

Horizontal cantilever rod bent under tip-concentrated upwards load.

We can see that, recognizing the Lauricella parameters ^{5} stops without reducing his elliptic integral to the Legendre standard form. It will be just the elliptic integral (

The sheared cantilever: a 3D dimensionless plot of deflection

From it, one could obtain the relevant contour plots cutting by planes of ^{6}. Mind that ^{7} comparison shows at the low loads an agreement which is becoming worst for increasing loads when the approximate theory is progressively failing.

The sheared cantilever: solution to (

It is worth noting that the power series expansion of (

Going again to formula (

On the purpose, it will be observed that in [

The modulus of elliptic functions

In fact, they do not provide the

We are looking for the shape of a

Submerged vertical rod under hydrostatic pressure.

By the usual identification, we obtain the solution to our elastica through the Lauricella function of three variables

Assumption A3 implies that our systems are ruled by differential equations affected by a geometrical nonlinearity due to the nonlinear link between the curvature

What above provides enough motivation for analyzing once for all, the effect of the sinusoidal load, whose a half wave only is taken, omitting all the possible variants or subcases. Seeing Figure ^{8}

Elastica of a cantilever rod bent by a sinusoidal continuous load.

Integral (

Considering the small deflections, the deformed centerline's curvature is approximately equated to the deflection's second derivative, assuming the rod deformation field to proceed along purely vertical displacements. If it is so, the free end

With large displacements all, this is not true any more, and a further unknown will appear, namely, the abscissa

Following a completely different method, in [

Anyway, the simplest criterion is the inextensibility assumption A4; the rod keeps its originary length^{9}. For the horizontal weightless cantilever inflected by a tip shear (see Figure

Consider the transcendental

Let

We can then use the Lemma to solve (^{10}

Solving to ^{11}, and then

Sheared rod: tip's abscissa as a function of the load.

Sheared rod: dimensionless tip deflection as a function of the load. Dotted line refers to the approximate theory

First, the (

Plugging (

Five elasticae have been analytically solved of a thin cantilever rod bent by different coplanar “dead” loadings^{12}, namely, whose direction remains unchanged during the deformation of the rod. They are concentrated couple, tip shear, evenly continuous load, hydrostatic load, and sinusoidal load. Each of them describes a single physical situation for which we obtain the large deflections in closed form through elliptic and/or hypergeometric functions. Some care has been put (tip sheared rod) in providing analytically, via the isometric assumption, the free end position after the strain, parametrizing its coordinates as a function of the load. Some of our solutions, specially that to the heavy rod by the Lauricella functions, have enriched the literature on the elastica.

The authors are indebted to their friend Aldo Scimone who drew some Figures of this paper; they hereby take opportunity for thanking him warmly. They wish also to thank the anonymous referee whose valuable criticism improved their article.

A space tether is a long cable used to connect spacecraft to other orbiting bodies such as space stations, boosters, payload, and so forth in order to transfer energy and momentum.

The problem of 3D-elastica is very old, but out of our purpose. We will say only that it first appeared in Lagrange's

We quote three works: (a)

A.M. Legendre,

In [

Under transversal load at the cantilever tip, this problem in strength of materials is solved in cartesian coordinates following the linear approximation; the solution is found (e.g., see [

Notice that the plots are coming after Mathematica's computation of our explicit solution (

Inserting in

Only in some special cases such a tip path description is quick. For instance for a rod loaded by a fixed couple (Section

Having for instance

Nothing to say, apart from the more strict link imposed by the transcendental inequality (linearity range), (

The contrary case is that of the