Singular stress fields in masonry structures: Derand was right

Abstract

The idea of a no-tension (NT) material underlies the design of masonry structures since antiquity. Based on the NT model, the safety of the structure is a problem of geometry rather than of strength materials, in the same spirit of the “rules of proportion” of the medieval building tradition. The use of singular stress fields for equilibrium problems of NT materials in 2d, has been recently proposed by Lucchesi et al. to produce statically admissible stress fields; here we introduce a simple way to construct singular stresses, based on the Airy’s stress formulation. We interpret the singular part of such stress fields as axial contact forces acting on ideal 1d structures arising inside the body, in the same spirit of Strut and Tie methods. A number of simple problems of equilibrium concerning typical walls, arches and portals, is solved in terms of stress fields having regular and singular parts, by adopting the direct and the stress function formulation. The validity of the rules of proportion described by Derand and Gil is also verified.

Appendices

Appendix 1: Limit analysis for NT materials

For RNT materials, both force and displacement data are subject to compatibility conditions, that is the existence of a statically admissible stress field and the existence of a kinematically admissible displacement field, are subordinated to some necessary or sufficient conditions on the given data. The problem of the definition of sufficient conditions for load compatibility is studied in Angelillo and Rosso [8] and Del Piero [17]. Here we concentrate on necessary or sufficient conditions for the compatibility of a given set of loads ($\mathbf{s},\mathbf{b}$), restricting to the case of zero kinematical data $(\underline {\mathbf{u}},\underline {\mathbf{E}}),$ that is to zero displacements $\underline {\mathbf{u}}$ on the constrained boundary $\partial_D\varOmega$ and zero distortions $\underline {\mathbf{E}}$ in $\varOmega.$ The definition of safe, limit and collapse loads are given first, and the propositions defining the compatibility of the loads, that are essentially a special form of the theorems of limit analysis, are then discussed. For RNT materials, we observe that the restrictions (2), (3) are equivalent to a rule of normality of the total strain to the cone of admissible stress states. Normality is the essential ingredient allowing for the application of the two theorems of limit analysis (see Del Piero [17]). In order to avoid the possibility of trivial incompatible loads (and simplify the formulation of the two theorems), we make assumption (4), that is the tractions $\mathbf{s}$ applied at the boundary are either compressive or zero.

1.1 Admissible fields

The rigorous proof of the two theorems of limit analysis requires to set the problem in proper function spaces. For RNT materials is appropriate and convenient to define the set of statically admissible (s. a.) stress fields $\mathcal{H}$ as follows

$$ {\mathcal{H}}=\left\{ {\mathbf{T}}\in S(\varOmega) \;\hbox{s.t.}\; \textnormal{div}{\mathbf{T}}+{\mathbf{b}}={\mathbf{0}}\;,\;{\mathbf{T}}{\mathbf{n}}={\mathbf{s}} \;\hbox {on}\; \partial\varOmega_N\;,\;{\mathbf{T}}\in Sym^- \right\}\;, $$

(17)

where a convenient choice for the function space $S(\varOmega)$ is

$$ S(\varOmega)=SMF(\varOmega) \;, $$

SMF being the set of special measures (that is measures with null Cantor part) whose jump set is finite, in the sense that the support of their singular part consists of a finite number of regular (n − 1)d arcs, (see [1, 3, 4]). Notice that, depending on the geometry of the structure $\varOmega$ and on the given loads, the set $\mathcal{H}$ can be void. If $\mathcal{H}$ is void the loads ($\mathbf{s},\mathbf{b}$) are incompatible, in the sense previously specified (no possibility of equilibrium with purely compressive stresses). Instead, the set of kinematically admissible displacement fields is denoted $\mathcal{K}$ and is defined as follows:

$$ {\mathcal{K}}=\left\{{\mathbf{u}}\in T(\overline{\varOmega}) \;\hbox {s.t.}\; {\mathbf{u}}={\mathbf{0}} \;\hbox {on}\; \partial_D\varOmega\;,\;{\mathbf{E}}({\mathbf{u}})\in Sym^{+} \right\}\;, $$

(18)

where $\overline{\varOmega}=\varOmega \cup \partial_D\varOmega$ and $T(\overline{\varOmega})$ is a function space of convenient regularity. Since for RNT materials discontinuous displacements can be considered, one can assume $T(\overline{\varOmega}) = BV(\overline{\varOmega}),$ that is the set of functions of bounded variation (the functions whose gradient belongs to $M(\overline{\varOmega}),$ i.e. functions $\mathbf{u}$ admitting finite discontinuities). We restrict to the subset of $BV(\overline{\varOmega}),$ consisting of displacement fields $\mathbf{u}$ having finite jumps on a finite number of regular arcs. Actually, one needs only to consider discontinuous functions $\mathbf{u}$ whose jump set is the union of a finite number of segments.

1.2 Strictly admissible stress fields and load classification

In order to formulate the Theorems of Limit Analysis, we need to introduce the following definitions. On denoting $\left\langle \ell,\mathbf{u}\right\rangle$ the work of the load $\ell=({\mathbf{s}},\mathbf{b})$ for the displacement $\mathbf{u},$ the load can be classified as follows:

• (i) (ℓ is a collapse load) ⇔ ($\exists \mathbf{u}^{*}\in \mathcal{K}\; \hbox {s.t.}\; \left\langle \ell,\mathbf{u}^*\right\rangle > 0$),

• (ii) (ℓ is a limit load) ⇔ ($\left\langle \ell,\mathbf{u}\right\rangle \leq 0, \;\forall \mathbf{u}\in \mathcal{K} \; \hbox {and} \;\exists \mathbf{u}^{*}\in K \;\hbox {s.t.}\; \left\langle \ell,\mathbf{u}^{*}\right\rangle =0$),

• (iii) (ℓ is a safe load) ⇔ ($\left\langle \ell,\mathbf{u}\right\rangle <0, \; \forall \mathbf{u}\in \mathcal{K}$).

We now introduce an useful definition. A stress field $\mathbf{T}\in \mathcal{H}$ such that $tr \mathbf{T} < 0 \hbox { and } det \mathbf{T} > 0 \;\hbox { in } \overline{\varOmega}$, is said to be strictly admissible. Notice that, if $\mathbf{T}$ is strictly admissible, then at each point of $\overline{\varOmega}$ (that is the open set $\varOmega$ to which the fixed part of the boundary $\partial_D\varOmega$ is added) it results:

$$ \sigma_{1}<0\;,\;\sigma_{2}<0\;, $$

$\sigma_1,\;\sigma_2$ being the eigenvalues of $\mathbf{T}$ at any point of $\varOmega.$

1.3 Theorems of limit analysis

• Kinematic Theorem If ℓ is a collapse load (in the sense of item i. above) then H is void.

• Static Theorem If a strictly admissible stress field $\mathbf{T}$ exists, then the load ℓ is safe (in the sense of item (iii) above).

• Limit Theorem If H is not void and there exists $\mathbf{u}^{*}\in K \;\hbox {s.t.}\; \left\langle \ell,\mathbf{u}^{*}\right\rangle =0,$ then the load ℓ is limit (in the sense of item (ii) above).

For the proof of these Theorems we refer to the paper by Del Piero [17]. We have to warn the reader that the proofs given in Del Piero [17] refer to a similar function space for the displacement but to a different functional setting for the stress (namely $L^2(\varOmega)$). In the present paper we assume that these theorems are still valid in the present larger setting for the stress, and smaller setting for the strain.

Appendix 2: Construction of stress function for the example of $\S$3.2

To obtain a statically admissible stress field inside the body we proceed as follows: The boundary is parametrized with the arc length s. The data m(s) and n(s) are defined in terms of the load on the segments 0–1 and 1–2 (see Fig. 3b–c). From them the curve carrying the datum for F and for its gradient ∇F at the boundary are computed.

Segment 0–1.

Parametrization: ${\bf x}(s)=\{-s,\frac{3}{2}\};$

Moment: $m(s)=-\frac{qs^{2}}{2};$

Axial force: n(s) = 0;

Curve of boundary data: ${\bf X}(s)=\{-s,\frac{3}{2},-\frac{qs^{2}}{2}\};$

Gradient of F at the boundary: ∇F ⁰(s) = { − m′(s), n(s)} = {qs, 0}.

Segment 1–2.

Parametrization: $\mathbf{x}(s)=\{-\frac{1}{2},\frac{3}{2}-s\};$

Moment: $m(s)=-\frac{q}{8}-\frac{qs}{4};$

Axial force: $n(s)= -\frac{q}{2};$

Curve of boundary data: ${\bf X}(s)=\{-\frac{1}{2},\frac{3}{2}-s,-\frac{q}{8}-\frac{qs}{4}\};$

Gradient of F at the boundary: $\nabla F^0(s)=\{-n(s),-m'(s)\}=\{\frac{q}{2},\frac{q}{4}\}$.

The data are extended inside the body with a uniaxial prolongation. Such a prolongation is obtained by constructing on each part γ _i of the boundary, a ruled surface having as generating curve, the curve $\mathbf{X}(s)$ carrying the Dirichlet data, and formed by the straight lines r, directed as the given loads, and whose slope is specified by ∇F(s). If the load is zero, the direction of the lines r is taken as the inward normal $-\mathbf{n}$.

Segment 0–1

Load direction: $\mathbf{k}=\{0,-1\};$

Propagation vector: $\mathbf{v}(s)=\{k_{1}, k_{2}, \nabla F\cdot \mathbf{k}\}=\{0,-1,0\};$

Parametric form of $F^{1}: \mathbf{y}(s, \nu)=\mathbf{X}(s)+ \nu \mathbf{v}(s)= \{-s,\frac{3}{2}-\nu, - \frac{qs^{2}}{2}\};$

F ¹ in terms of $x_{1}, x_{2}: F^{1}(x_{1},x_{2})= -\frac{1}{2} q x^{2}_{1}$.

Segment 1–2

Load direction: k = {1,0};

Propagation vector: ${\bf v}(s)=\{k_{1}, k_{2}, \nabla F\cdot {\bf k}\}=\{1,0,\frac{q}{2}\};$

Parametric form of F ²: y(s, ν) = X(s) + ν v(s) = 

 $\{-\frac{1}{2}+\nu,\frac{3}{2}-s, - \frac{p}{8}(1- 2qs+ 4 \nu q)\};$

F ² in terms of $x_{1}, x_{2}: F^{2}(x_1,x_{2})= -\frac{q}{4} (1- 2x_{1}-x_{2})$.

Once the surfaces F ¹, F ² are obtained from the boundary data, their intersection can be easily obtained as described in $\S$3.2. Notice that, with this procedure, the stress is uniquely determined by the boundary data. This is actually the consequence of assuming that the stress field is uniaxial inside the body.