dy the length of the corresponding sides on the element A'B'C'D' projected in the xy-plane, and p and 0 denote the angles formed by the meridian sides AB and DC of the element with the x-axis and by the parallel sides AD and BC with the y-axis, respectively. As concerns equilibrium, hypothesizing that the vault is in a membrane state of stress, a correspondence can be set between forces acting on the element ABCD (stresses Nx, Ny, Nxy = Nyx and applied load for unit area, px, py, pz) and projected forces acting on the associated element A'B'C'D' (Nx, Ny in the xy-plane (Baratta & O. Corbi 2007).
In absence of horizontal loads and if the vertical load is not dependent on "y", as it happens when the vault is subject to only vertical loads due to the self-weight (i.e. pz = pz (x) > 0), and assuming that the vault has an indefinite length in the direction y, equilibrium may be expressed in the form
dy2 ik2
with
where z0 and z1 are arbitrary ordinates, conditioned by the fact that z(t) should be contained in the interior of the profile of the vault.
After this result, it is_poss_ible to calculate the internal forces Nx < 0, * * ~ " ~
Ny = Nxy = 0 and Nx < 0, which reduces the problem to the determination of stress function ^(y).
Assuming that the directrix curve of the vault is a circular arch (Fig. 10) of radius R, with constant thickness "s" and unit weight y, and imposing suitable constraint conditions, one yields the final solution (Baratta & O. Corbi 2007)
It is also possible to realize that the equilibrium solution allows the structure to behave as a sequence of identical independent arches. From this result, one may refer to the results reported in the previous section for the portal arch model, reinforced or not with some FRP strips, whose analytical problem implementation has been shown to give theoretical results in perfect agreement with the produced experimental data, also exhibiting very effective results in the reinforced case.
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