Structural Capacity of Masonry Walls under Horizontal
Loads
Antonio Fortunato*, Fernando Fraternali*, Maurizio Angelillo*
SUMMARY – In the present work we describe a procedure for determining the ground acceleration that activates
the in-plane mechanism of a wall panel with openings. The method, based on the assumption that the material is
unilateral (namely a No-Tension material in the sense of Heyman), leans on the kinematic theorem of limit analysis.
By working with the kinematic theorem, we admit singular strains representing concentrated fractures; in other words
we allow for strong discontinuities in the displacements. In recent papers we adopted finite elements with strong
discontinuities and search for local minima of the energy (in proximity of equilibrium trajectories) by minimizing
the energy through descent, both with respect to the displacements and with respect to the position of the jump
set. In this paper we propose a similar, though simplified, strategy to explore compatible mechanisms having free
discontinuities. The numerical implementation of the proposed approach is discussed through illustrative examples,
on examining collapse mechanisms of masonry structures subject to vertical and seismic loads.
Keywords: Structural capacity, Masonry walls, Limit analisis, Free discontinuities.
1. Introduction
In the present work we describe a procedure for determining the ground acceleration that activates the inplane mechanism of a wall panel with openings.
The method, based on the assumption that the material is unilateral (namely a No-Tension material in the
sense of Heyman [Heyman, 1995]), leans on the kinematic theorem of limit analysis. The method generalizes and extends a numerical model proposed by Como
and Grimaldi [Como and Grimaldi, 1985].
Indeed, the most basic assumption that can be made,
in view of the small and often erratic value of the tensile strength of masonry materials, is that the material
behaves unilaterally, that is only compressive stresses
can be transmitted (No-Tension assumption).
It is generally recognized that such an assumption is
the first clue for the interpretation of masonry behaviour; on adopting and applying it, fracture patterns can
be predicted; fracture being the masonry most peculiar
manifestation, representing the way in which masonry
buildings relieve and can often survive to radical, and
sometimes dramatic, changes of the environment.
The unilateral model for masonry, that, though in
a mathematically unconscious way, has been known
since the nineteenth century [Moseley, 1833], was first
rationally introduced by Hewyman [Heyman, 1966] and
divulgated and extended in Italy, thanks to the effort of
Salvatore Di Pasquale [Di Pasquale, 1984] and other
* Department of Civil Engineering, University of Salerno, Fisciano
(SA), Italy.
Anno XXXI – N. 1 – gennaio-marzo 2014
members of the Italian school of Structural Mechanics,
such as the Romano brothers [Romano and Romano,
1979], Baratta [Baratta and Toscano, 1982], Del Piero
[Del Piero, 1989], Como [Como, 1992] and Angelillo
[Angelillo, 1993].
A research group at the University of Salerno (of
which the present authors have been a leading part)
has contributed to the debate on the unilateral models
for masonry, with a number of papers, since the early
works of Angelillo & Giliberti[Angelillo and Giliberti,
1988], Angelillo & Rosso [Angelillo and Rosso, 1995]
and Angelillo & Olivito [Angelillo and Olivito, 1995],
till the more recent articles of Angelillo & Fortunato
[Angelillo and Fortunato, 2002], Fortunato [Fortunato,
2010] and [Angelillo et al., 2010], [Angelillo et al.,
2014a], [Angelillo et al., 2013b] on plane structures
and [Angelillo et al., 2013a] on vaults. A general treatment of internal constraints on stress can be found in
[Angelillo and Fortunato, 2001] and a review of the
unilateral models for masonry in the recent book edited
by Angelillo [Angelillo Ed.].
In Section 2 we formulate the Boundary Value Problem (BVP) for unilateral masonry materials as Rigid
No-Tension (RNT) materials for which the latent strains
(fractures) satisfy a normality condition with respect to
the admissible stresses.
The first tool that can be introduced for applying the
unilateral No-Tension model to masonry structures is
the systematic use of singular stress and strain fields,
within the framework defined by the two theorems of
Limit Analysis [Angelillo et al., 2014b], [Angelillo
Ed.].
41
In Section 3 the concept of compatible loads and
distortions is introduced, and the validity of the two
theorems of Limit Analysis, admitting singular stress
and discontinuous displacements, is discussed in Section 4.
The use of singular equilibrated stresses for approximating plane equilibrium problems can be traced to the
work of Fraternali [Fraternali et al., 2002; Fraternali,
2010, 2011], and Block [Block and Ochsendorf, 2006].
In particular, in the application we present, we focus on the kinematic approach. By working with the
kinematic theorem, we admit singular strains representing concentrated fractures; in other words we allow for
strong discontinuities in the displacements.
The research group of Salerno has developed in the
last decade a computer model for fracture nucleation
and propagation in 2D brittle solids, based on variational fracture [Angelillo et al., 2006], [Angelillo et al.,
2012], [Angelillo et al., 2005]. Essentially the analysis
is based on the variational formulation of Griffith-type
fracture, formulated by Francfort and Marigo [G.A.
Francfort, 1998], the main difference being the fact that
we rely on local rather than on global minimization.
Nucleation and propagation of fracture is obtained by
minimizing in a step by step process a form of energy
that is the sum of bulk and interface terms.
Recent attempts of producing numerical codes for
variational fracture [Bourdin, 2007], [Del Piero et al.,
2007], are based on the approximation of the energy,
in the sense of C-convergence, by means of elliptic
functionals [Ambrosio and Tortorelli, 1990].
Finite elements with strong discontinuities and
search for local minima of the energy (in proximity
of equilibrium trajectories) by minimizing the energy
through descent, both with respect to the displacements
and with respect to the position of the jump set, were
instead adopted by the Authors in [Angelillo et al.,
2006], [Angelillo et al., 2012], [Angelillo et al., 2005],
In the present paper a similar, though simplified,
strategy to explore compatible mechanisms having free
discontinuities was adopted The practical implementation of the proposed approach is illustrated through a
collection of numerical examples dealing with masonry
structures subject to vertical and horizontal loads.
decomposed additively as follows: E (u) = E* + E, $ E ,
being the effective strain of the material.
We point out that the masonry structure is identified
with the set: X , 2XD , i.e. it is considered closed on
and open on ∂XD the rest of the boundary.
We consider that the body is composed of Rigid NoTension material (RNT), that is the stress T is negative
semi-definite
T ! Sym-, (1)
the effective strain E* = E (u) - E is positive semidefinite
E* ! Sym+, (2)
and the stress T does no work for the corresponding
effective strain E*
T $ E* = 0 (3)
In order to avoid trivial incompatible loads (s, b) , we
assume that the tractions s satisfy the condition
s $ n 1 0, or s = 0, 6x ! 2XN (4)
Notice that in the plane case (n = 2) conditions (1),
(2), can be rewritten as
trT # 0, det T $ 0 (5)
trE* $ 0, det E* $ 0 (6)
2.2. Statically admissible stress fields
An equilibrated stress field T (a stress field T balanced with the prescribed body forces b and the tractions s given on ∂XN) satisfying the unilateral condition
(1), is said statically admissible for a RNT body. The
set of statically admissible stress fields is denoted H
and is defined as follows
! X
H ! *T S ( ) s.t. div T + b = 0,4 (7)
Tn = s on 2XN , T ! Sym-
2. The boundary value problem for Rigid No-Tension
materials
2.1. Constitutive restrictions and equilibrium problem
S(X) being a function space of convenient regularity.
Since for RNT materials, discontinuous and even singular stress fields will be considered, one can assume
S(X) = M(X), that is the set of bounded measures.
For Elastic No-Tension materials (ENT) a sensible
choice is S(X) = L 2(X), the function space of square
summable functions. Space M(X) is much larger than
L 2(X), making the set of functions, which compete
for equilibrium, richer for RNT than for ENT materials. For this reason for RNT materials the search
of statically admissible (s.a.) stress fields appears
easier.
It is assumed that the body X ! R n (here n = 2)
loaded by the given tractions s on the part ∂XN of the
boundary, and subjected to given displacements u on
the complementary, constrained part of the boundary
∂XD, is in equilibrium under the action of such given
surface displacements and tractions, besides body loads
b and distortions E (the set of data being denoted:
(u, E; s, b) and undergoes small displacements u and
strains E(u). When eigenstrains are considered, under
the small strain assumption, the total strain E(u) is
42
Anno XXXI – N. 1 – gennaio-marzo 2014
2.3. Kinematically admissible displacement fields
A compatible displacement field u, defined as a displacement u matching the given displacements u on
∂XD for which (E (u) - E) ! Sym+ , i.e. such that the effective strain satisfies the unilateral conditions (6), is
said to be kinematically admissible (k.a.) for a RNT
body.
The set of kinematically admissible displacement
fields is denoted K and is defined as follows:
K !)
u ! T (M
X) s.t. u = ur on 2X D,
3 (8)
(E (u) – E) ! Sym +
where M
X = X , 2X D and T (M
X) is a function space of
convenient regularity. Since for RNT materials discontinuous displacements can be considered, it can be
assumed that T( M
X ) = BV( M
X ). The considered subset is
then BV( M
X ), the set of functions of bounded variation,
consisting of displacement fields u having finite jumps
on a finite number of regular arcs. It will actually be
demonstrated that only the discontinuous functions u
whose jump set is the union of a finite number of segments need to be considered.
3. Compatibility conditions
3.1. Compatibility of loads and distortions
The data of a general Boundary Value Problem
(b.v.p.) for a RNT body can be divided into two parts
, * (s, b) . loads,
(9)
, * * (u, E) . distortions.
The equilibrium problem for RNT materials,
namely the search of admissible stress or displacement fields for given data, are essentially independent,
in the sense that they are uncoupled but for condition
(3).
It has to be pointed out that, for RNT bodies, there
are non-trivial compatibility conditions, both on the
loads and on the distortions. The existence of statically
admissible stress fields for given loads, and the existence of kinematically admissible displacement fields
for given distortions, is subjected to special conditions
on the data (for a thorough study of compatibility conditions on the loads see [Del Piero et al., 2007] and
[Angelillo and Rosso, 1995]).
The definition of compatible loads and distortions is
rather straightforward:
Fig. 1. Compatible and equilibrated singular solution for a stone arch.
The strain is singular at the supports and at the key-stone; the stress is
singular being a concentrated axial force along the arches, dashed in (a)
and (b). In (a) the abutments spread by a given amount, in (b) they get
closer by the same amount, and the arches form in two different ways.
Soluzione singolare compatibile ed equilibrata per un arco in pietra. La
deformazione è singolare ai supporti e alla chiave di volta. Lo sforzo
è singolare essendo una forza assiale concentrata lungo gli archi (linea
tratteggiata in (a) e (b)).
In (a) gli appoggi si allontanano di una certa quantità. In (b) si avvicinano della stessa quantità.
stress field or a k. a. displacement field (as it is done
in the next examples).
To prove the existence of a solution to the b.v.p. for a
No-Tension body, the compatibility of ℓ and ℓ* is necessary but not sufficient, since the condition T · E*(u) = 0,
must be satisfied (this is the material restriction (3)).
A possible solution to the b.v.p. is then given if a s.a.
stress field and a k.a. displacement field which are reconcilable in the sense of condition (3), exist.
In the examples of Fig. 1a and Fig. 1b, the possible solution of two mixed b.v.p. for RNT material is
pictorially presented. It must be noticed that both the
stress solution and the displacement solution present
singularities. These examples [Heyman, 1966] testify
the need, in order to solve a b.v.p for Rigid No-Tension
materials, to consider at the same time singular stress
and strain fields, and call for an extended formulation
of the theorems of Limit Analysis.
3.2. Incompatibility of loads and distortions
The way to verify the incompatibility of the data
is less straightforward, requiring the definition of two
new sets
T 0 ! S (X) s.t. div T 0 = 0,
3, (12)
H0 ! ) 0
T n = 0 on 2X N , T 0 ! Sym –
and
K0 ! )
u 0 ! T (M
X) s.t. u 0 = 0, on 2X D,
3 (13)
E (u 0) ! Sym +
" , is compatible, + " H is void,, (10)
" ,* is compatible, + " K is void, . (11)
Both H0 and K0 can reduce to H00 and K00 corresponding to null stress and strain fields, depending on
the geometry of the boundary, of the loads and of the
constraints. The fact that H0 - H00 can be void and that
Therefore the more direct way to prove compatibility, both for loads and distortions, is to construct a s.a.
Anno XXXI – N. 1 – gennaio-marzo 2014
43
K0 - K00 can be non-void is kind of peculiar of RNT
materials; indeed we are used to think of 2D continua
as overdetermined and deprived of rigid, zero-energy,
internal modes.
One way to see overdeterminacy is to add to any
s.a. stress field a non zero, self balanced stress field
T0. The fact that H0 - H00 can be void means that
overdeterminacy depends on the loads. The structure
can also become statically admissible under certain
loading conditions. The fact that the overdeterminacy/
underdeterminacy of the structure depends on the load
is typical also of discrete structures with unilateral
constraints.
The absence of degrees of freedom is proved, for
discrete structures, by denying the possibility of zero
energy mechanisms. u0 ! K0 - K00 is indeed a non
trivial mechanism requiring, for the RNT body, zero
energy expended. The underdeterminacy of the structure, descending from the fact that K0 - K00 can be
non-void, demands for non trivial compatibility conditions on the loads.
The incompatibility of the data can be assessed as
follows
4.1. Theorems of Limit Analysis
Given the definition of RNT materials, we observe
that the restrictions (2) and (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 [Del Piero, 1998]. In order to avoid the
possibility of trivial incompatible loads (and simplify
the formulation of the two theorems), we assume that
the tractions s applied at the boundary are either compressive or zero (Eq. 4).
The rigorous proof of the two theorems of Limit
Analysis requires to set the problem in proper functions spaces. For RNT materials is appropriate and convenient to define the sets of statically admissible stress
fields H and kinematically admissible displacement
fields K , as follows
T ! S (X) s.t. div T + b = 0,
3 (16)
H !)
Tn = s on 2X N , T ! Sym –
K ! "u ! T (X) s.t. u = 0, on 2XD, E (u) ! Sym+ , (17)
" ,* is incompatible , % "7u0 ! K s.t. ,, u0 2 0,, (14)
where a convenient choice for the function spaces S(X)
and T(X) is
" ,* is incompatible , % "7T0 ! H s.t. ,*, T0 2 0,, (15)
S (X) = SMF (X) (18)
T (X) = "u s.t. grad u ! SMF* (X), (19)
where ,, u0 , ,*, T0 represent the work of the loads
and distortions for u0, T0, respectively.
It must be noted that the incompatibility of a given
set of loads means that equilibrium is not possible and
that acceleration of the structure must take place. A
trivial compatibility condition for all kind of bodies,
under pure traction conditions, is load balance. Load
balance is only a necessary compatibility condition for
unilateral bodies.
The incompatibility of a given set of distortions
means that the given kinematical data cannot be accommodated with a zero energy mechanism and demand for
more complex material models (i.e. elastic NT, elastic
NT-plastic, etc.).
4. Limit Analysis
For RNT bodies, both force and displacement data
are subject to compatibility conditions. In other words,
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. Here we concentrate
on necessary or sufficient conditions for the compatibility of a given set of loads ^s, bh , restricting to the
case of zero kinematical data (u, E) . 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.
SMF(X) being the set of Special Measures (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 [Ambrosio
et al., [2000]. With SMF*(X) we denote the subset of
SMF(X) for which the support of the singular part is
restricted to a finite number of (n − 1)d segments.
In order to formulate the theorems of Limit Analysis
the following definitions are needed:
On denoting ,, u , the work of the load , = (s, b)
for the displacement u, the load can be classified as
follows:
(, is a collapse load) + (7u* ! K s.t. ,, u* 2 0) (20)
(, is a limit load) + ( ,, u* # 0,
(21)
6u ! K and 7u* ! K s.t. ,, u* = 0)
(, is a safe load) + ( ,, u* 1 0, 6u ! K) (22)
A stress field H such that trT < 0 and detT > 0,
∀x∈X, is said to be strictly admissible. If T is strictly
admissible, then for each point of X, (that is the open
set X to which the fixed part of the boundary ∂XD is
added) it is:
v1 < 0, v2 < 0
(23)
v1, v2, being the eigenvalues of T at point x.
44
Anno XXXI – N. 1 – gennaio-marzo 2014
The Kinematic Theorem states that if ℓ is a collapse
load (Eq. 20) then H is void.
By the Static Theorem, if a strictly admissible stress
field T exists, then the load ℓ is safe (Eq. 22).
Consistently with the Limit Theorem, if H is not
void and there exists u* ! K s.t. ,, u* = 0 , then the load
ℓ is limit (in the sense of Eq. 21).
For the proof of these theorems we refer to the paper [Del Piero, 1998] even though the proofs provided
by Del Piero refer to a similar function space for the
displacement but to a different functional setting for
the stress (namely L2(X)).
In the present paper we assume that these theorems
are still valid in the present larger setting for the stress.
For general stress and strain fields the internal work
# T $ E is not defined. Given the restrictions for the
sets H and K , only when both stress and strain are
singular on the same line C, the line is curved and
there is a stress discontinuity in the direction of the
normal M to C, the internal work is of difficult computation. A way to avoid this is to allow stress singularities on curved lines but to assume that the support of
the jumps of u is a line formed by the union of a finite
number of straight arcs (segmentation).
4.2. Formalization of the kinematical problem
With the RNT model the two problems of displacement and equilibrium appears as distinct, in the sense
that each problem has its own data (forces/distortions)
and there is seemingly no trace of strain in the equilibrium problem or of stress in the kinematical problem.
The only equation that couples equilibrium and geometry is the orthogonality condition (3).
In this section the formalization of the kinematical
problem, under the effect of kinematical data (such
as settlements and distortions), is presented. It is thus
proposed how the possible displacement of the structure can be expressed, particularly when such data are
compatible ( K is not void).
By restricting to displacement fields characterized by
strain fields that are purely line Dirac deltas with support on a finite number of segments, the body X can be
divided into a finite number, n, of domains Xi (forming
a partition of X) each exhibiting a rigid body motion.
Under the assumption of small strains, for each element Xi, such rigid body motion ui of any point of
Xi, can be described in terms of three displacement
parameters ui, oi, zi, as:
u i1 (x 1, x 2) = u i – z i x 2,
(24)
u i2 (x 1, x 2) = o i + z i x 1
Under these restrictive assumptions the generalized
displacement of the structure, denoted ut , is a vector of
3n components, namely the three displacement parameters per each element Xi:
ut = " u 1, v 1, z 1, ..., u i, v i, z i, ..., u n, v n, z n , (25)
Anno XXXI – N. 1 – gennaio-marzo 2014
Fig. 2. A simple plane wall with regular openings is depicted in (a).
Thick solid lines represent the boundary; thin solid lines represent fixed
interfaces; diagonal lines represent moving interfaces. The movement
of the interfaces is controlled by the movement of their end nodes constrained along fixed interfaces and boundary segments. In (b) the constraints that are assumed along the interfaces are reported.
Una parete semplice con aperture regolari è mostrata in (a). Le linee
solide grosse rappresentano i contorni fissi; le linee diagonali rappresentano le interfacce mobili. Il movimento delle interfacce è controllato
dal movimento dei nodi di estremità confinati lungo interfacce fisse e
segmenti di contorno. In (b) sono indicati i vincoli assunti lungo le
interfacce.
5. A method for kinematic collapse analysis
In this section a method for the analysis of the effect of loads and distortions on masonry-like structures,
considered as structures composed of no-tension material in the sense of Heyman, is presented. As described
in section 2, the three basic assumptions of Heyman’s
model for masonry are:
– Tensile stresses (forces) are forbidden and, therefore, the material can separate freely (with zero energy
expended).
– The material can withstand stresses of infinite
intensity (infinite strength to compression, i.e. no possibility of crushing)
– Friction coefficient is infinite: no sliding on separation lines.
The restriction to small strains and displacements
is introduced. Based on the previous material assumptions, the infinitesimal strain can be a bounded measure. In the present paper we restrict to deformations
which are special bounded deformations, namely to
strains that are measures represented by line Dirac
deltas with support on a finite number of segments.
As a consequence of this restriction, the structure is
composed of non-intersecting, rigid polygonal pieces
connected by unilateral constraints along the common
interfaces.
The shape of this pieces, that is of the skeleton of
the interfaces (a segmentation of the domain), is actually unknown, and part of the sought solution.
45
In Figure 2a, a simple case concerning a plane wall
with openings is reported, as an illustration of the way
in which the kinematical problem can be approximated.
The idea is to consider a partition of the domain consisting of Polygonal pieces (quadrilateral in the example of Figure 2a), cut by fixed interfaces (thin solid
lines in Figure 2a) and moving interfaces (grey lines
in Figure 2a).
The unilateral constraint considered along the interfaces, incorporates the incompenetrability condition. The normality condition (2) and (3) (non-sliding
assumption along the interface) are enforced by the
bilateral pendulum (see Figure 2b).
For such a structure two kinds of data can be considered:
Kinematical data: (u, E) , that is given displacements
at the constrained boundary, given eigenstrains.
Statical data: (s, b) , that is given tractions at the
loaded boundary, given body forces.
The first set (u, E) , represents the datum for the kinematical problem of the given structure, the kinematical
problem being the search of a generalized rigid body
displacement ut of the polygonal pieces composing the
structure S, compatible with the unilateral and bilateral constraints described above (see Figure 2b). Such
constraints represent the inter-connections between the
pieces and the connections of the pieces with the constrained boundary.
The set of all the possible rigid body displacements
satisfying the constraints is denoted K and is called
the set of kinematically admissible displacements. As
already remarked and discussed in Section 3, in particular cases, K can be void, with the meaning that the
structure (for this shape and fixing conditions) cannot
accommodate the given kinematical datum with a rigid
body displacement. It must deform.
The second set (s, b) represents the datum for the
statical (equilibrium) problem of the given structure,
the statical problem for the structure S being the search
of the constraint reactions Rt , arising at the unilateral
and bilateral constraints inter-connecting the pieces
and connecting the pieces to the constrained boundary
(restricted by the “compression” assumption at the unilateral constraints), in equilibrium with the given loads.
The set of all the possible reactions satisfying the
”no-tension” constraint and in equilibrium with the
loads is denoted H and is called the set of statically
admissible reactions.
This set also, in particular cases, can be void (see
section 3). In this specific case the reactions (for this
structure and fixing conditions) cannot balance the
given loads under the no-tension assumption, and the
pieces composing must change either their linear or
their angular momentum.
The kinematical and statical problems for such a
structure are coupled, in the sense that, given the assumption of zero dissipation on any interface, the work
of the reactions for the displacements both at the internal and at the boundary interfaces must be zero.
In general there will be infinite elements ut ! K and
infinite elements Rt ! H , and the no-work assumption
gives a criterion to select (may be not uniquely), among
46
them, a couple (ut0, Rt 0) that is called solution of the
kinematical and statical problem. It must be noted that,
restricting to a finite number of rigid blocks, with fixed
or moving interfaces, the sets H and K become finite
dimensional. There is a way to select variationally such
a couple. The idea is to introduce the potential energy
of the structure i.e. the scalar product of the loads and
couples applied at the centroids of the pieces, collected
in a generalized force vector ft , for the generalized
displacement ut collecting the parameters of translation
and rotation of each piece of S:
E (ut) = - ft, ut is introduced as a linear function of the generalized
displacement ut of the structure and then minimized
over the set K :
E (ut0) = min ut ! K E (ut) (27)
This is a linearly constrained minimization problem
for a linear function if the interfaces are not moving.
In such a simplified case the problem can be solved by
using the Simplex Method. If both the load data and
the distorsion data are fixed, the minimum criterion selects, among all the kinematically admissible displacements ut the displacement ut0 that is more convenient
on an energetical ground. If the load is assigned with
a load parameter m (e.g. the vertical component of the
load is fixed and the horizontal component is gradually increased with m), at each stage of the loading
program (that is at any given value of m) the minimal
displacement can be calculated through the minimum
condition. The limit value m 0 of the load parameter is
obtained when a mechanism (an indefinite increase of
the displacement) for which the loads perform zero
work is detected.
5.1. A simple case study
As a first case study, we analyze the masonry wall
in Fig. 3 under the action of the self-weight and a settlement of the central base panel. On assuming that the
adopted mesh is fixed, we regard the analyzed structure
a set S of n = 8 rigid bodies connected by unilateral
and bilateral constraints (Fig. 3-left). An arbitrary generalized displacement of the wall is given by
ut = " u (1), o (1), z (1), ..., u (n), o (n), z (n) , (28)
with the displacement parameters being referred to the
centroid of mesh elements. The corresponding generalized dual force is:
Ft = " H (1), V (1), M (1), ..., H (n), V (n), M (n), (29)
Assuming pure dead loads due to the self-weight of
the wall: H(i) = 0, V(i) = P(i), M(i) = 0, for any i = 1, 2,
..., n, where P(i) is the self-weight of the panel i. The
bilateral and unilateral constraints depicted in Figure 2b
are considered to be active on all the mesh interfaces,
Anno XXXI – N. 1 – gennaio-marzo 2014
Fig. 3. A simple case study: a) Geometry and notation, b) collapse mechanism.
Un esempio semplice: (a) geometria e notazione, (b) meccanismo di collasso.
Fig. 4. Masonry structure covered with a barrel vault (left) and corresponding collapse mechanism (right).
Struttura in muratura con volta a botte (sinistra) e corrispondente meccanismo di collasso (destra).
which leads us to the following system of equality and
inequality constraints:
5.2. Collapse mechanism of a barrel vault subject to
seismic loading
C l ut = 0 (30)
Cllut # d (31)
A second example deals with the collapse analysis of
the monumental masonry structure shown in Fig. 4-left,
consisting of a barrel vault resting on thick buttresses
[Ascione et al., 2005]. The loading condition is represented by the self-weight p of the structure (masonry
unit weight equal to 17 KN/m3), and horizontal forces
mp (static seismic loading, see Fig. 4). We estimate the
collapse multiplier m0 of the horizontal forces by analyzing a 1.0 m long slice of the structure. Fig. 4-right
shows the collapse mechanism obtained for the present example through the self-adaptive mesh depicted
in the same figure (m0 = mc = 0.1053 [Ascione et al.,
2005]). Such a mechanism shows four opening hinges
(cracks) in the masonry, one of which is located in a
buttress and the other three in the vault (“semi-global”
mechanism).
d being the vector of the applied settlements. In the
present case, the latter accounts for a translational vertical settlement the center base panel, which we assume
equal to –0.66L. The set of kinematically admissible
generalized displacement is given by
K = " ut : Clut = 0, Cllut # d , (32)
and the collapse mechanism ut0 is found via a simplex
algorithm, by minimizing the following objective function
E (ut) = -Ft $ ut (33)
over K (Fig. 3-right).
Anno XXXI – N. 1 – gennaio-marzo 2014
47
Fig. 5. Masonry wall with openings subjected to fixed vertical loads and variable horizontal forces (left) and corresponding collapse mechanism (right).
Parete in muratura con aperture soggetta a carichi verticali fissi e carichi orizzontali variabili (sinistra) e corrispondente meccanismo di collasso (destra).
5.3. Collapse mechanism of a multi storey masonry wall
subject to seismic loading
Our final example is concerned with a three storey masonry wall subjected to fixed vertical loads and
variable horizontal forces (Fig. 5-left). The wall is
made up of tufe stones with 1.0 m thickness (constant
over the height) and 18 kN/m 3 unit weight. A permanent loading of 7.5 kN/m is acting in correspondence
with each storey level. The base values of horizontal
forces are: F1 = 33.82 kN; F2 = 45.58 kN; F3 = 70.31
kN [Ascione et al., 2005]. Fig. 5-right shows the employed mesh and the corresponding collapse mechanism (m0 = 3.9827).
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49
Curva di capacità di pareti forate in muratura soggette a
carichi orizzontali
A. Fortunato, F. Fraternali, M. Angelillo
È un’osservazione forse banale che le strutture murarie del patrimonio storico-monumentale siano dotate di
un’eccezionale stabilità in rapporto ad azioni repentine
quali quelle associate a cause atmosferiche o ai terremoti e soprattutto alle sollecitazioni più subdole provocate da assestamenti o cedimenti lenti delle fondazioni.
Le motivazioni di questo comportamento peculiare, riconducibile essenzialmente alla così detta resistenza
per forma delle strutture murarie, e determinato dalla
risposta essenzialmente unilaterale del materiale murario, si trovano spiegate in una serie di pubblicazioni
ed articoli che originano essenzialmente dal lavoro di
Heyman [1]1 e sono esposte in modo magistrale, con
stile più divulgativo, nel più recente libro dello stesso
autore [2].
In anni recenti la progettazione strutturale è divenuta
sempre più una questione di verifica della sicurezza
in rapporto a “stati limite” definiti in base ad un certo
numero di criteri.
Alcuni dei criteri che non possono essere trascurati
per le strutture metalliche o in calcestruzzo armato,
quale il limite consentito alla corrosione o all’ampiezza
delle fessure, sebbene talvolta possano giocare qualche
ruolo anche per alcuni tipi di muratura, sono certamente secondarie per le costruzioni murarie tradizionali.
Ciò non è sorprendente e, comunque, non è una
condizione peculiare delle murature: che vi sia una
scala di priorità nella verifica dei criteri è un concetto
del tutto accettato. Il problema nasce quando si cerca
di applicare alle struttura murarie i criteri base dell’analisi strutturale, ossia i criteri relativi alla resistenza, alla
deformabilità e alla stabilità dell’equilibrio. In molti casi
infatti si può constatare che la resistenza, la deformabilità e la stabilità, criteri cardine dell’analisi strutturale,
hanno poca o nessuna rilevanza ai fini della valutazione
della sicurezza della compagine muraria, la quale dipende essenzialmente dalla “forma” della struttura, dalle
sue proporzioni, non dall’intensità dei carichi.
Il modello che descrive tale tipo di comportamento
è stato razionalizzato da Heyman nel lavoro del 1967
sopra citato, ed è definito dalle seguenti condizioni.
1. Tensioni (e forze) di trazione non sono consentite
e, pertanto,il materiale si può separare (fratturare) a
costo energetico nullo.
2. Il materiale può sostenere tensioni di compressione infinite (ovvero non si schiaccia).
3. Il coefficiente di attrito è infinito, ossia non è possibile che avvenga uno slittamento su una superficie
compressa.
Per semplicità si introduce l’ipotesi di piccole deformazioni, una approssimazione accettabile nella maggior parte dei casi concreti. Sulla base delle precedenti
1
J. Heyman, The Stone Skeleton (paper), International Journal of
Solids and Structures.
50
restrizioni materiali la deformazione infinitesima è una
misura limitata (ovvero può presentare delle singolarità). Nel lavoro presente ci si limita a considerare
deformazioni che sono speciali misure limitate, ovvero
deformazioni rappresentate solamente da delta di Dirac
concentrate su un numero finito di segmenti. Di conseguenza la struttura resta suddivisa in un numero finito
di pezzi rigidi di forma poligonale. La forma di questi
pezzi è realmente incognita e fa parte della soluzione
che si cerca in termini di spostamenti.
Per il materiale definito dalle restrizioni 1-3 si può
dimostrare che valgono i due teoremi dell’analisi limite,
in base ai quali si può valutare la sicurezza di un dato
schema di carico.
Il primo teorema (statico) asserisce che la struttura
è sicura, cioè è in grado di sostenere i carichi esterni
senza collassare, se esiste almeno un campo di tensione staticamente ammissibile, di pura compressione
biassiale, in tutti i punti del corpo.
In base al secondo teorema (cinematico) la struttura
collassa se esiste un cinematismo virtuale di pura frattura, senza slittamenti, compatibile con i vincoli, per
il quale i carichi esterni compiono lavoro virtuale non
negativo.
Nel presente articolo si descrive una procedura per
definire l’entità della accelerazione al suolo necessaria
ad attivare il meccanismo di collasso nel piano di una
parete in muratura con aperture. Il metodo sfrutta il
teorema cinematico dell’analisi limite.
Lavorando con il teorema cinematico, si ammettono
deformazioni locali singolari che rappresentano fratture
concentrate su linee; in altre parole si ammettono discontinuità forti nel campo di spostamento. In lavori
recenti il nostro gruppo di ricerca ha costruito ed impiegato codici ad elementi finiti con discontinuità concentrate su linee, ricercando minimi locali di energia
con tecniche di discesa, minimizzando contemporaneamente rispetto al campo di spostamenti ed alla posizione dell’insieme di salto.
Nel presente lavoro si adotta una simile, sebbene
notevolmente semplificata, strategia, al fine di esplorare
cinematismi indotti da distorsioni assegnate e meccanismi di collasso prodotti dall’eccesso di carico, entrambi
descritti da campi di spostamento con discontinuità libere.
L’energia da minimizzare è la sola energia potenziale,
un funzionale lineare degli spostamenti, che, nell’approssimazione considerata, diviene una funzione lineare
dello “spostamento rigido” dei pezzi che compongono
la struttura. L’energia, essendo l’insieme di salto, ossia
lo scheletro dei pezzi, variabile, dipende però in modo
non lineare dalla posizione delle interfacce, e viene
minimizzata per discesa.
Il modo nel quale il metodo può essere implementato
è descritto attraverso alcuni esempi nella parte conclusiva del lavoro.
Anno XXXI – N. 1 – gennaio-marzo 2014
Se i dati di cedimento e carico sono assegnati, il
criterio di minimo seleziona lo spostamento rigido dei
pezzi della struttura che risulta più conveniente dal
punto di vista energetico. Viceversa, se i cedimenti
sono assegnati ma il carico è affetto da un parametro
di carico l (per esempio i carichi verticali sono fissi e
quelli orizzontali sono gradualmente incrementati con
Anno XXXI – N. 1 – gennaio-marzo 2014
il parametro l), ad ogni passo della storia di carico
(ovvero per ogni valore di l) si può ricavare lo spostamento rigido minimizzante. Il valore limite l 0 del parametro (moltiplicatore di collasso) si ottiene quando
lo spostamento minimizzante diviene un meccanismo
compatibile per il quale le forze applicate compiono
lavoro nullo.
51