The objective of the present work is to develop an automated numerical method for
the analysis of thin masonry shells. The material model for masonry that we adopt is
the so-called “normal rigid no-tension” (NRNT) material; and for such a
material, the kinematical and the safe theorems of limit analysis are valid. The
present study focuses on the application of the second theorem to masonry
vaults and domes, being devoted to the determination of a class of purely
compressive stress regimes, which are balanced with the load. The mere existence
of such a class is a proof that the structure is safe, and members of this
class may be used to assess the geometric degree of safety of the structure
and to estimate bounds on the thrust forces exerted by the structure on
its boundary. The problem is reduced to the equilibrium of a membrane
and
can be formulated in terms of projected stresses defined on the planform
of
. The
search of the stress reduces to the solution of a second-order pde, in terms of the stress
potential . In order that
the membrane stress on
be
compressive, the potential
must be concave. As for the thrust line in an arch, the surface
is
not fixed and may be changed, given that it remains inside the masonry.
Under these simplifying assumptions, the whole class of equilibrated
stress regimes for a masonry shell is obtained by moving and deforming
inside
the masonry, and also, for any fixed shape, by changing the boundary data for
, that
is the distribution of thrust forces along the boundary. The search for a feasible
stress state on a convenient membrane surface, to be chosen with a trial
and error procedure, requires a substantial effort and may be unrewarded.
Then, the main object of the present work, is to produce a computer code
that can handle numerically the interplay of the shape controlled by a
function , and of
the stress potential
,
by developing a convergent optimization scheme able to give a safe state under
the given material and geometrical constraints, namely the concavity of
and the
inclusion of
within the masonry. Two simple cases, are exposed in detail to illustrate the
method.
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