On the statics of No-Tension masonry-like vaults and shells: solution domains, operative treatment and numerical validation

Abstract

In the paper, structural analysis of masonry vaults is based on the fundamental assumption that the material cannot resist tensile stresses. For the vault to work as a No-Tension (NT) structure, it is recognized that a membrane surface completely included in the thickness of the vault exists [Heyman (1977) Equilibrium of shell structures. Oxford University Press, Oxford], designed in way to resist applied loads by purely compressive membrane forces. After reducing the problem to a plane-stress problem, the stress function Ψ(x,y) is introduced, like in the classical Pucher’s approach. Statically admissible solutions are proven to be ruled by the non-homogeneous Monge-Ampere equation. Anyway purely admissible stress fields often yield non-compatible and non-credible results with reference to fractures and strains. A path aimed at generating sets of solutions related to given load shapes is outlined, as a preliminary basis for solutions fully accomplishing not only equilibrium and admissibility, but also compatibility with strains and fractures. Further analytical developments, set up of operative procedures, validation by means of numerical campaign, investigation of special characters of approximations when dealing with particular load shapes, i.e. all topics related to operative features of the proposed approach are presented in the second part of the paper. Starting from the path outlined for identifying solution domains, a constrained optimization is set up to minimize a suitably defined square error function calculated between the assigned objective load function and the membrane surface expression, under the condition that the solution membrane function is contained in the vault profile. Some analytical management is needed in order to reduce the dependence of the problem on the number of parameters, and to partially simplify and expedite the subsequent numerical simulation. Thereafter, ad hoc calculus codes are built up for implementing the problem and performing the numerical investigation. Final validation of the approach is given by demonstrating, with some numerical applications, the effectiveness of the method even for load shapes difficult to be handled.

Appendices

Appendix 1

In this section the symbol “>” means that the argument is strictly larger than zero at least in a subdomain of X. Dependence on the variables (x, y) is implicit in the expressions of the functions.

Hypotheses

$$ z_{p} = \sum\limits_{i = 1}^{M} {\rho_{i} z_{pi} } ;\quad \rho_{i} \ge 0\quad \forall i = 1, \ldots ,M $$

(A1.1)

with

$$ \left\{ {\begin{array}{*{20}c} {z_{pi,xx} \ge 0;\quad z_{pi,yy} \ge 0} \hfill \\ {H_{{z_{pi} }} = p_{i} \ge 0} \hfill \\ \end{array} } \right.\quad \forall i = 1, \ldots ,M $$

(A1.2)

it follows (Thesis)

$$ H_{{z_{p} }} = z_{p,xx} z_{p,yy} - \left( {z_{p,xy} } \right)^{2} \ge 0 $$

(A1.3)

Proof

Since the Hessian operator of z _i (x,y) equals p _i (x,y) ≥ 0, one gets

$$ z_{pi,xx} z_{pi,yy} - \left( {z_{pi,xy} } \right)^{2} \ge 0 \Rightarrow \left\{ \begin{gathered} z_{pi,xx} \,z_{pi,yy} \ge 0 \hfill \\ \left| {z_{pi,xy} } \right| \le \sqrt {z_{pi,xx} \,z_{pi,yy} } \hfill \\ \end{gathered} \right.\quad \forall i = 1, \ldots ,M $$

(A1.4)

If p _i  > 0 the relevant inequality holds strictly.

From Eq. A1.1

$$ z_{p,xx} \,z_{p,yy} - \left( {z_{p,xy} } \right)^{2} = \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\rho_{i} \rho_{j} z_{pi,xx} \,z_{pj,yy} } } - \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\rho_{i} \rho_{j} z_{pi,xy} \,z_{pj,xy} } } $$

(A1.5)

and

$$ \begin{aligned} z_{p,xx} \,z_{p,yy} - \left( {z_{p,xy} } \right)^{2} & = \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\rho_{i} \rho_{j} \,\left( {z_{pi,xx} \,z_{pj,yy} - z_{pi,xy} \,z_{pj,xy} } \right)} } \\ & = \sum\limits_{i = 1}^{M} {\rho_{i}^{2} \left( {z_{pi,xx} \,z_{pi,yy} - z_{pi,xy} \,z_{pi,xy} } \right)} \\ & \quad + \sum\limits_{i = 2}^{M} {\left[ {\sum\limits_{j < i} {\rho_{i} \rho_{j} \left( {z_{pi,xx} \,z_{pj,yy} \, - z_{pi,xy} \,z_{pj,xy} } \right)} + \sum\limits_{j > i} {\rho_{i} \rho_{j} \left( {z_{pi,xx} \,z_{pj,yy} - z_{pi,xy} z_{pj,xy} } \right)} } \right]} \\ \end{aligned} $$

(A1.6)

Considering that H _zpi  = p _i ∀i = 1, …, M and exchanging the indices in the second term in the parentheses, one gets

$$ z_{p,xx} \,z_{p,yy} - \left( {z_{p,xy} } \right)^{2} = \sum\limits_{i = 1}^{M} {\rho_{i}^{2} p_{i} } + \sum\limits_{i = 2}^{M} {\left[ {\sum\limits_{j < i} {\rho_{i} \rho_{j} \left( {z_{pi,xx} \,z_{pj,yy} + z_{pj,xx} \,z_{pi,yy} - 2z_{pj,xy} \,z_{pi,xy} } \right)} } \right]} $$

(A1.7)

From Eq. A1.4

$$ \left. \begin{gathered} \left| {z_{pi,xy} } \right| \le \sqrt {z_{pi,xx} \,z_{pi,yy} } \hfill \\ \left| {z_{pj,xy} } \right| \le \sqrt {z_{pj,xx} \,z_{pj,yy} } \hfill \\ \end{gathered} \right\} \Rightarrow - \left| {z_{pi,xy} } \right|\left| {z_{pj,xy} } \right| \ge - \sqrt {z_{pj,xx} \,z_{pj,yy} } \sqrt {z_{pi,xx} \,z_{pi,yy} } $$

(A1.8)

whence

$$ \begin{gathered} z_{pi,xx} \,z_{pj,yy} + z_{pj,xx} z_{pi,yy} - 2z_{pj,xy} z_{pi,xy} \ge z_{pi,xx} \,z_{pj,yy} + z_{pj,xx} \,z_{pi,yy} - 2\left| {z_{pj,xy} \,z_{pi,xy} } \right| \hfill \\ \ge z_{pi,xx} \,z_{pj,yy} + z_{pj,xx} z_{pi,yy} - 2\sqrt {z_{pj,xx} \,z_{pi,xx} \,z_{pi,yy} \,z_{pj,yy} } = \left[ {\sqrt {z_{pi,xx} \,z_{pj,yy} } - \sqrt {z_{pj,xx} \,z_{pi,yy} } } \right]^{2} \ge 0 \hfill \\ \end{gathered} $$

(A1.9)

and finally, since the coefficients ρ _i are non-negative

$$ z_{p,xx} \,z_{p,yy} - \left( {z_{p,xy} } \right)^{2} \ge \sum\limits_{i = 1}^{M} {\rho_{i}^{2} p_{i} } + \sum\limits_{i = 2}^{M} {\left[ {\sum\limits_{j < i} {\rho_{i} \rho_{j} \left( {\sqrt {z_{pi,xx} \,z_{pj,yy} } - \sqrt {z_{pj,xx} z_{pi,yy} } } \right)^{2} } } \right] \ge 0} $$

(A1.10)

where the first inequality holds with the sign “>” if at least on of the p _i ’s is strictly larger than zero.

In the particular case that all p _i ’s are null

$$ z_{p,xx} \,z_{p,yy} - \left( {z_{p,xy} } \right)^{2} \ge \sum\limits_{i = 2}^{M} {\left[ {\sum\limits_{j < i} {\rho_{i} \rho_{j} \left( {\sqrt {z_{pi,xx} \,z_{pj,yy} } - \sqrt {z_{pj,xx} z_{pi,yy} } } \right)^{2} } } \right] \ge 0}. $$

(A1.11)

Appendix 2

The set ${\tt{S}}$ is bounded.

In fact consider that the initial point c _o has finite modulus, and it is in the interior of ${\text{S}}$, so that a number β > 0 exists such that all points in a sphere with centre c _o and radius Δc > β are feasible. In other words two positive ordinates k ₁ and k ₂ exist such that

$$ k_{1} + \bar{z}_{1} \left( {x,y} \right) \le z_{o} \left( {x,y} \right) \le \bar{z}_{2} \left( {x,y} \right) - k_{2} \quad \forall \left( {x,y} \right) \in {\text{X}};\quad k_{1} > 0;\quad k_{2} > 0 $$

(A2.1)

with z _o (x,y) = z(x,y|c _o ).

Consider any direction t starting from c _o and a feasible point c* ≠ c _o on t in the neighbourhood of c _o , with |c ^*−c _o | ≥ β. It follows that

$$ - s\left( {x,y} \right) < - s\left( {x,y} \right) + k_{2} \le z^{*} \left( {x,y} \right) - z_{o} \left( {x,y} \right) \le s\left( {x,y} \right) - k_{1} < s\left( {x,y} \right)\quad \forall \left( {x,y} \right) \in {\tt{X}};\quad k_{1} > 0;\quad k_{2} > 0 $$

(A2.2)

whence

$$ \left| {z^{*} \left( {x,y} \right) - z_{o} \left( {x,y} \right)} \right| = \left| {\sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right)} } \right| < s\left( {x,y} \right)\quad \forall \left( {x,y} \right) \in {\tt{X}} $$

(A2.3)

and

$$ {\mathbf{c}}^{*} \in {\tt{S}} \Rightarrow \mathop {\max }\limits_{{\left( {x,y} \right) \in {\tt{X}}}} \left[ {\left| {z^{*} \left( {x,y} \right) - z_{o} \left( {x,y} \right)} \right|} \right] = \Updelta z^{*} \left( {\mathbf{t}} \right) < \mathop {\max }\limits_{{\left( {x,y} \right) \in {\tt{X}}}} s\left( {x,y} \right) = s_{\max } $$

(A2.4)

Since ${\tt{X}}$, the basis of the vault, is a closed and bounded (i.e. compact) set of ${\tt{R}}^{2}$, and the function

$$ h_{t} \left( {x,y\left| {{\mathbf{c}}^{*} ,{\mathbf{c}}_{o} } \right.} \right) = \left| {\sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right)} } \right| $$

(A2.5)

is a continuous non-zero function on ${\tt{X}}$, by the Weierstrass theorem a point $\left( {x_{t}^{*} ,y_{t}^{*} } \right)$ of ${\tt{X}}$ exists yielding a positive maximum value

$$ \Updelta z^{*} \left( {\mathbf{t}} \right) = \mathop {\max }\limits_{{\left( {x,y} \right) \in X}} \left[ {\left| {\sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right)} } \right|} \right] = \left| {\sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x_{t}^{*} ,y_{t}^{*} } \right)} } \right| > 0\quad \forall {\mathbf{t}} $$

(A2.6)

Consider any other c ∈ t [c = c _o  + λ(c ^* − c _o ), λ ≥ 0]

$$ \begin{gathered} z\left[ {x,y\left| {\mathbf{c}} \right.} \right] - z_{o} \left( {x,y} \right) = \lambda \sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right);\quad \lambda \ge 0} \hfill \\ \mathop {\max }\limits_{{\left( {x,y} \right) \in X}} \left[ {\left| {\lambda \sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right)} } \right|} \right] = \lambda \mathop {\max }\limits_{{\left( {x,y} \right) \in X}} \left[ {\left| {\sum\limits_{i = 1}^{M} {\left( {c_{i}^{*} - c_{oi} } \right)z_{i} \left( {x,y} \right)} } \right|} \right] = \lambda \Updelta z^{*} \left( {\mathbf{t}} \right) \le \mathop {\max }\limits_{{\left( {x,y} \right) \in X}} s\left( {x,y} \right) = s_{\max } \hfill \\ \end{gathered} $$

(A2.7)

One concludes that on any direction t starting from c _o

$$ \lambda_{t} \le \frac{{s_{\max } }}{{\Updelta z^{*} \left( {\mathbf{t}} \right)}} $$

(A2.8)

After considering the minimum $\Updelta z_{\min }^{*} > 0$ along all directions t, the feasible domain remains included in the sphere with center c _o and radius Λ

$$ \Uplambda = \frac{{s_{\max } }}{{\Updelta z^{*}_{\min } }} $$

(A2.9)

which is a finite number.