M Ej

Figure 3. Representation of the quantities reported in Eq. (4) for the calculation of the rotation and displacements of terminal section A caused by a force Q applied to the same section.

and horizontal and vertical displacements dx, dy of a terminal beam section A caused by a force Q applied to the same section along line of action q (Fig. 3), can be nominally written as follows:

This relation clearly illustrates the physical meaning of G as the global parameter that expresses the flexural elastic deformability of the structure.

Now, the idea arises of thinking at the structure as the assembly of a series of discrete elastic elements of length As, each with a proper elastic weight such that the total elastic weight of the structure is represented by the discrete distribution of these elastic weights. It is possible to demonstrate that such a sought distribution of G is univocally known only for a statically-determined structure, whereas for a statically-undetermined structure the distribution of elastic weights is not univocally defined. This is not surprising, due to the redundancy of equilibrium in an hyperstatic system. Despite this, an hyperstatic structure can still be solved, via the Forces Method (with hyperstatic quantities as unknown), through the superposition of effects on underlying isostatic structures and imposition of the corresponding compatibility conditions. As the underlying isostatic structure can also be analysed with a univocally-defined distribution of elastic weights, such distribution can also be used to solve the original hyperstatic structure. Thus, indirectly, its ellipse of elasticity can in essence be determined, so the corresponding ellipse of inertia of the distribution of elastic weights.

This allows one to write the so-called "theorems of the theory of the ellipse of elasticity" (Belluzzi 1942), as a function of the properties of the distribution of elastic weights. For example, the rotation 0

where G = AG represents the total elastic weight of the structure; Sq, Jxq, Jyq the static moment of G with respect to q and the centrifugal moments of inertia of G with respect to q and axis x, and q and axis y. These parameters depend only on the distribution of elastic weights and on the position of the applied load Q, and can be expressed as a function of the quantities xs, ys, us, ux, uy depicted in Fig. 3.

The point S in Fig. 3 represents the centre of gravity of the elastic weights of the structure; the points X,Y represent the antipoles of the reference system axes x, y with respect to the central ellipse of inertia of the elastic weights. It is then apparent that, once the position of points S, X, Y and total elastic weight G are found, the elastic response of the structure is determined. The coordinates xs, ys, xy, yx defining the position of these points can be evaluated by standard calculations of geometry of masses, once given the discrete distribution of elastic weights.

2.2 Application to a doubly built-in parabolic arch

In the SNOS Report (1889), the remarkable application of the theory of the ellipse of elasticity to the analysis of the arch of the bridge refers to the determination of the elastic response of a parabolic arch, built-in at the two extremities, which is subjected to a vertical load P (that can be put equal to 1) and acting in an arbitrary position along the arch, at a horizontal distance a from left extreme A (Fig. 4).

First, the position of the line of action of the reaction A needs to be determined. This can be solved by a

Figure 4. Calculation of line of action LO of left reaction A; the segments FL and VO that the left reaction A locates on the vertical lines from A and on load P, below and above the horizontal line from centre S are determined.

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