3.1 Set up of the problem
When addressing problems relevant to the static analysis of masonry vaults one should handle the two problems of equilibrium with applied loads an admis-sibility with respect to the masonry material which, as well known, exhibits a very reduced capacity of resisting tensile stresses (Baratta 1984,1991, Baratta & Corbi 2005, 2006, 2007).
So the above reported treatment can be used, on one side, for describing the real vault geometry, i.e. the geometry of its extrados and intrados surfaces z1 and z2 bounding its thickness and the geometry of its mid-surface, and, on the other side, for describing the geometry of its membrane surface z, enveloping the local membrane stress resultants in equilibrium with the applied loads (Baratta & Corbi 2006a, 2007).
In this latter case both the equilibrated local stress resultants and the membrane surface z are unknown.
So the approach for the static study of the vault could consist of searching for the membrane surface both satisfying equilibrium and material admissibility.
As regards to equilibrium, considerations about stresses and external loads acting on the element of Figure 10 yield, as usual, three equations of equilibrium.
The external load which may be acting on the element has been resolved into the three perpendicular components p^,pe, pn., while stresses are those shown in the figure, yielding, after some algebraic developments, the three equilibrium conditions respectively in the tangential direction (direction of 9), in the tangential direction (direction of 9), and in the radial direction (i.e. normal direction), as follows
In the case when one considers an axial-symmetric loading the shear stress resultant N99 is zero everywhere and one has
When referring to the equilibrium conditions under not axial-symmetric Eqs (32) and axial-symmetric load Eqs (33), one should remember that, as mentioned in the above, the involved geometric quantities (angles and radii of curvature) depend on the variable z expressing the equation of the membrane stress surface. Therefore, besides the unknown membrane stress resultants, one also should search for the membrane surface expressions both satisfying equilibrium (in the forms of Eqs (32) or (33) and material admissibility (which implies that the surface must be contained in the vault profile z1 < z < z2).
After individuating the set of admissible membrane surfaces (in terms of admissibility and equilibrium), a possible approach in order to find the solution in terms of stresses is to set up a Complementary Energy problem to be formulated as a kind of extension to masonry vaults of the classical analogous energetic approach for linearly elastic structures.
This kind of approach has been already successfully applied to masonry arches modeled by the Not Resisting Tension (NRT) assumption (Baratta 1984,1991,
Baratta & Corbi 2005a, 2006, 2007, Heyman 1977), showing to produce in its numerical implementation results in excellent agreement with experimental data (Baratta & Corbi 2005b, 2006b).
In order to undertake this approach the expression of the Complementary Energy embedded in a masonry vault element should be evaluated. In case of NRT assumption, one should consider that the generic element appears to be partially resistant; this means that, denoting by u the distance of the membrane surface from the upper profile of the vault element, the position of the neutral surface (i.e. characterized by null stresses) is located at a distance of 3u from the extrados of the element.
For a bi-linear distribution of the normal stress on the volume element, when neglecting the shear stress component, the elastic energy can be calculated as follows
with Ar, A9 the areas of the compressed part of the volume element respectively in the two cross sections containing dsr and ds9, er and dor respectively the eccentricity of the solicitation centre with respect to the mass centre Gr , and the distance of the mass centre from the neutral plane in the two cross sections, and E and v respectively the Joung modulus and the Poisson coefficient.
The final Complementary Energy functional C expression is given by adding to the elastic energy term L, the energy related to the work developed by the constraint reactions R, as C = L + R.
The solution in terms of stresses can then be searched for by numerically implementing the minimization of the Complementary Energy functional under the condition that the solution itself is respectful of the above individuated equilibrium equations and of admissibility.
As an alternative an inverse approach can be successfully outlined also leading to the analytical expressions of the membrane stress functions for the different vault typologies. In this case the spatial surface equilibrium problem can be coupled to the one projected in a selected plane according to the classical Pucker's approach. One can then refer to the 2D-projected problem, and introduce a stress function which is built up in such a way to a priori satisfy some of the equilibrium conditions.
The key of the problem lays then in the coupling of the stress function with the membrane function.
So the stress function should be modeled in such a way to recognize, after imposing the remaining equilibrium conditions and admissibility, the membrane stress functions that can be assumed as the membrane surfaces relevant to different vault geometries.
Some practical applications relevant to the cases of the barrel vaults with indefinite length, the barrel vaults confined at their extremity cross-sections, and the spherical domes are developed in details in papers by the author (Baratta & Corbi 2007a, 2007b; Baratta et al. 2008).
To this regard, one should emphasize that this is not at all a trivial objective to be pursued. Even simple applications, which means the specialization of the described approach to vaults of simple shapes, require a pretty hard work for setting up and checking the expressions of the relevant stress functions.
As a counterpart, this approach allows to obtain solutions in analytical form, which represents its major result.
Thereafter the solution for each case can be completed by setting up the energetic approach as mentioned in the above.
Definitively this approach is pretty complex since it requires to hypothesize and test a number of analytical functions for any vault typology, which represents a consistent effort even for simple vaults geometries; anyway it has the big advantage and original research result, never available in literature before, to produce the explicit analytical expressions of the stress and membrane functions for the single cases, i.e. to give the solutions in analytical form.
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