On The Theory Of The Ellipse Of Elasticity

2.1 Fundamentals

The theory of the ellipse of elasticity can be considered as a main icon of the so-called Graphical Statics, the discipline which often characterised the resolving approach of practical design problems during the 2nd half of the 19th century. It represents a very elegant and practical method for the analysis of the flexural response of an elastic structure. It is based on an intrinsic discretisation of a continuous elastic problem. This theory is basically associated to the two outstanding figures of Culmann and Ritter, but also of people, like Giulio Rothlisberger, that were formed at the time at the Polytechnical Schools in Europe and that became later structural engineers and designers and largely contributed in the practical and effective application of the method.

The theory is based on the following main hypotheses (we refer here to the Italian text by Belluzzi 1942, which reports results from the technical literature of the time, basically ascribed to the two names of Culmann and Ritter): (a) linear elastic behavior of the material and the structure, which leads to the proportionality between acting forces and (reversible) displacements provoked by them (property that in turn implies the validity of the principle of superposition of effects); (b) existence of the ellipse of elasticity, referred to a section of a structure; (c) correspondence between the latter and the central ellipse of inertia of the distribution of the so-called elastic weight of the structure. This correspondence transforms the problem of the determination of the elastic response of a continuous structure to a task of pure geometry of masses. The latter can be feasibly handled by taking advantage ofthe assumed discrete character ofthe distribution of the elastic weight and is endowed with a visible interpretation of the elastic performance of the structure,

Figure 2. Representation of a force R applied along line of action r to a beam section A and the centre of rotation C of A. C is the symmetric of C with respect to the centre S of the ellipse of elasticity of beam AB referred to section A.

in view of its conception and design. Furthermore, the ellipse itself may actually play the role of an hidden, underlying, graphical construct. Indeed, the properties of projective geometry that are attached to that allow the elastic solution of the structure even without the explicit drawing of the ellipse itself. The methods are also said graphical-analytical because, in practice, main technical steps that are framed on the graphical constructions may be carried-out analytically, by working-out formulas that arise from the inspection of the drawings (Belluzzi 1942).

The concept of the ellipse of elasticity referred to a section of an elastic structure is achieved by inspecting the correspondence existing between the line of action r of a force R applied to a section A of a general, curvilinear, elastic beam (with little curvature and continuously-varying cross section) and the centre of rotation C of the same section (Fig. 2).

In particular, refer to a planar beam acted upon by forces laying in the same plane and cross sections of the beam that, during the beam's deformation, are assumed to remain plane and perpendicular to the geometric axis, also deforming in its original plane. The theory states that there exists an involutory relationship between the line of action r and the centre of rotation C of the section. Moreover, the ellipse of elasticity is the fundamental real conic of the polarity existing between the line of action r and the point C', which is the symmetric of C with respect to the centre S of the ellipse. In other words, the ellipse of elasticity can be defined as the fundamental conic with respect to which the lines of action r and the respective centres of rotation C correspond to each other through an antipolarity relationship.

The determination of the central ellipse of inertia of the distribution of the elastic weight of the structure, which coincides with the ellipse of elasticity above, is linked first to the definition of the general concept of elastic weight and then to the quantification of its distribution for the structure under consideration.The concept of elastic weight goes as follows. If on a section A of a beam, a moment M acts in the plane which contains the geometric axis, it causes a rotation 0 of A, around the centre S of the ellipse of elasticity. The rotation is proportional to the applied moment M as:

where G represents the so-called elastic weight of the beam. Thus, G can be defined as the angle of rotation 0 that is caused by the application of a unitary moment M = 1; it depends on the beam's geometrical and physical properties; it gives a global measure of the beam's aptitude to deform. In case of a straight cantilever beam of length l loaded by a moment M at its free end, composed by a linear elastic material with Young's modulus E and endowed with a constant cross section with moment of inertia J with respect to the axis perpendicular to the beam's plane, it turns out that, referring to the case of flexure of de Saint Venant, the rotation of the free end is:

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