1. Symposium in honour of Gianpietro Del Piero
Sperlonga
September 30th and October 1st 2011.
The static theorem of limit analysis for masonry
panels
M. Lucchesi, M. Šilhavý, N. Zani

2. Constitutive equations
In order to characterize a masonry-like material we assume
that the stress must be negative semidefinite and that the
strain is the sum of two parts: the former depends linearly
on the stress, the letter is orthogonal to the stress and
positive semidefinite,
X − Sym
I œ I/  I0
I 0 − Sym
X œ ‚ÒI / Ó
I0 † X œ !
where
I (?) œ " Ðf?  f?T Ñ
#
is the infinitesimal strain and ‚ is symmetric and positive
definite, i.e.,
2

3. E" † ‚ÒE# Ó œ E# † ‚ÒE" Óß
E † ‚ÒEÓ  ! for each E − Symß E Á !
Proposition
For each I − Sym there exists a unique triplet
ÐX ß I / ß I 0 Ñ that satisfies the constitutive equation.
Moreover, a masonry-like material is hyperelastic.
We write
AÐI Ñ œ " X ÐI Ñ † I
#
for the stored energy.
3

4. Notations
H reference configuration of the body
` H œ W  fß such that W  f œ g
? displacement field, such that ul W œ !
= À f Ä ‘$ surface traction
, À H Ä ‘$ body force
X À H Ä Sym stress tensor
s
S
b
Ω
D
4

5. Limit analysis
The limit analysis deals with a family of loads _Ð-Ñ that
depend linearly on a scalar parameter - − ‘,
_ Ð - Ñ œ Ð , - ß =- Ñ œ Ð , !  - , " ß = !  - = " Ñ
, ! , =! permanent part of the load
, " ß =" variable part of the load
- loading multiplier
,! and ," are supposed to be square integrable functions on
H, =! and =" are supposed to be square integrable functions
on f .
5

6. Let
Z œ Ö@ − [ "ß# ÐHß ‘$ Ñ À @ œ ! a.e. on W×
be the Sobolev space of all ‘$ valued maps such that @ and
the distributional derivative f@ of @ are square integrable
on H.
For each Ð-ß @Ñ − ‘ ‚ Z we define the potential energy
of the body
MÐ-ß @Ñ œ ( AÐI Ð@Ñ.Z  ( @ † , - .Z  ( @ † =- .E
H H f
where
( AÐI Ð@Ñ.Z
H
is the strain energy and
 _Ð-Ñß @  œ ( @ † , - .Z  ( @ † =- .E
H f
is the work of the loads.
6

7. Moreover, we define the infimum energy
M! Ð-Ñ œ inf ÖMÐ-ß @Ñ À @ − Z ×Þ
Proposition
(i) The functional M! À ‘ Ä ‘  Ö  _× is concave,
M! Ð+-  Ð"  +Ñ.Ñ +M! Ð-Ñ  Ð"  +ÑM! Ð.Ñ
for every -, . − ‘ and + − Ò!ß "Ó.
Therefore, the set
A À œ Ö- − ‘ À M! Ð-Ñ   _×
is an interval.
7

8. (ii) The functional M! is upper semicontinuous,
M! Ð-Ñ lim supM! Ð-5 Ñ
5Ä_
for every - − ‘ and every sequence -5 Ä -.
We interpret the elements of A as loading multipliers for
which the loads _Ð-Ñ are safe, i.e. the body does not
collapse.
8

9. Each finite endpoint -- of the interval A is called a
collapse multiplier with the interpretation that for - œ --
or at least for - arbitrarily close to -- outside A the body
collapses.
We say that a stress field X is admissible if
X − P# ÐHß Sym Ñ
and that X equilibrates the loads _Ð-Ñ œ Ð, - ß =- Ñ if
( X † I Ð@Ñ.Z œ ( @ † , .Z  ( @ † = .E
- -
H H f
for every @ − Z .
We say that the loads _Ð-Ñ are compatible if there exists a
stress field X that is admissible and equilibrates _Ð-ÑÞ
9

10. Proposition (Static theorem of the limit analysis)
The loads _Ð-Ñ are compatible if and only if
M! Ð - Ñ   _ .
That is the loads _Ð-Ñ are safe (i.e. - − A) if and only if
there exist a stress field X which
(i) is square integrable,
(ii) takes its values in Sym ,
(iii) equilibrates the loads _Ð-Ñ.
10

11. Rectangular panels
In the study of the static of masonry panels we verify that
the problem of finding negative semidefinite stress fields
that equilibrate the loads is considerably simplified if
instead of stress fields represented by square integrable
functions we allow the presence of curves of concentrated
stress.
p
o
x
y
λ
Ω+
γλ h
λ
Ω−
λq
b
11

12. Example
In the following example, firstly, we will obtain a stress
field which is a measure, whose divergence is a measure,
that equilibrate the loads in a weak sense. Then we will see
a procedure that, starting from this singular stress field,
allow us to determine a stress field which is admissible and
equilibrates the loads in the classical sense.
12

13. p
x O
y
γλ
λ
H œ Ð!ß ,Ñ ‚ Ð!ß 2Ñ § ‘# ß
W œ Ð!ß FÑ ‚ Ö2×ß
f œ ` H ÏW ß
, - œ !ß
Ú :4, on Ð!ß ,Ñ ‚ Ö!×
=- Ð<Ñ œ Û -3ß on Ö!× ‚ Ð!ß 2Ñ
Ü ! elsewhere
where :  !ß -  ! and < œ ÐBß CÑ.
13

14. p
x O
y
Ωλ
+ γλ
a
Ωλ
−
λ
The singularity curve # - can be obtained by imposing the
equilibrium of the shaded rectangular region with the
respect to the rotation about its left lower corner. We
obtain
# œ ÖÐBß CÑ − H À C œ Ê B×
- :
-
14

15. which divides H into the two regions H- ,
„
H- œ ÖÐBß CÑ − H À C  Ê
:
+ B×,
-
H- œ ÖÐBß CÑ − H À C  Ê B×Þ
:

-
The singularity curve # - has to be wholly contained into
the region H and this implies
!  -  --
with
:, #
-- œ # Þ
2
15

16. For the regular part of the stress field X<- (defined outside
# - ) we take
œœ
-
 :4 Œ 4 if < − H+
X<- Ð<Ñ
 -3 Œ 3 if < − H- Þ

The stress field defined on # - can be obtained by imposing
the equilibrium of the shaded rectangular region with the
respect to the translation. We obtain
X=- Ð<Ñ œ  È:-
<Œ<
k<k
Þ
<
k<k
We note that is the unit tangent vector to # - .
16

17. The stress field
T- œ X<-  X=-
has to be interpreted as a tensor valued measure. That is,
a function defined on the system of all Borel subsets of H
which takes its values in Sym and is countably additive,
T Œ  E3  œ ! T- ÐE3 Ñ
_ _
-
3œ" 3œ"
for each sequence of pairwise disjoint (borelian) sets.
T- is the sum of an absolutely continuous part (with
respect to the area measure) with density X<- and a
singular, part concentrated on # - , with density X=- . Both
the densities X<- and X=- are regular functions.
17

18. We can prove that the stress measure T- weakly
equilibrates the loads Ð, - ß =- Ñ, that is
( I Ð@Ñ. T œ ( @ † , .Z  ( @ † = .Eß
- -
H H f
for any @ − Z .
18

19. Integration of measures
By the static theorem of limit analysis, in order to assert
that the loads _Ð-Ñ are compatible we need an equilibrated
stress field that is a square integrable function and then
equilibrated stress measures are not enough.
Now we describe a procedure that in certain cases allows
us to use the stress measure T- to determine a square
integrable stress field X - . Crucial to this procedure is the
fact that both the loads Ð=- ß , - Ñ and the admissible
equilibrating stress measure T- depend on a parameter -.
19

20. p
x o
y
γ μ+ε
γμ γ μ−ε
μ
The idea is to take the average of the stress measure over
any set Ð.  %ß .  %Ñ, where %  ! is sufficiently small.
Averaging gives the measure
Tœ (
" .% -
T . -Þ
#% .%
20

21. It may happen that this measure, in contrast to T. , is
absolutely continuous with respect to the area measure with
a square integrable density.
For the previous example we obtain the following result.
If !  -  -- , then the loads _Ð.Ñ are compatible. In fact
if Ð.  %ß .  %Ñ § Ð!ß -- Ñ then the measure
Tœ (
" .% -
T .-
#% .%
is absolutely continuous with respect to the area
(Lebesgue) measure with density X .
X is a bounded admissible stress field on H that
equilibrates the loads _Ð.Ñ. We have
X œ X<  X = Þ
21

22. The densities X< and X= can be explicitly calculated. In
fact, we have
X< Ð < Ñ œ (
" .% -
X Ð<Ñ. - œ
#% .% <
Ú  :4 Œ 4 if < − H- ÏE
Û  .3 Œ 3
+
Ü Ð#%Ñ" Ð!Ð<Ñ3 Œ 3  " Ð<Ñ4 Œ 4 Ñ
if < − H- ÏE

if < − E
where
E œ ÖÐBß CÑ − H À .  %  :B# ÎC #  .  %×
is the shaded area in the figure, and
" # % %
!Ð<Ñ œ Ð: B ÎC  Ð.  %Ñ# Ñß
#
" Ð<Ñ œ :Ð.  %  :# B% ÎC % ÑÞ
22

23. Moreover,
Ú  : # B#
X= Ð<Ñ œ Û
<Œ< if < − E
Ü!
%C%
otherwise
It is easily to verify that the stress field
X œ X<  X =
is admissible, i.e. X a square integrable function which
takes its values in Sym , and that it equilibrates the loads
_Ð.Ñ,
X 8 œ =. on f , div X œ ! in H.
23

24. Panels with opening
a
p
c
x o
F1
y
γ2
h2 Ω1 γ1
Ω3 e
Ω2
B
γ3
h1 d
C A q
b2 b1 b2
24

25. p
a
c
h2 e
h1
qc
b2 b1 b2
h2 e
B
h1
A
C
b2 b1 b2
25

26. p
qc
a
h2
h1
b2 b1 b2
a
h2
B
Ι
h1
C A
b2 b1 b2
26

27. Panel under gravity
x O λ
y
b
γ
The panel is subjected to a side loads and its own gravity
on ˜!× ‚ Ð!ß LÑ,
=- Ð < Ñ œ œ
-3
! elsewhere.
27

28. The singularity curve is
# - œ šÐBß CÑ − H À C œ -,B# Î-›, - œ "Î#  È$Î'Þ
and
-- œ -,F # ÎLÞ
Ú  ,C 4 Œ 4 in H- ß
X<- Ð<Ñ œ Û

Ü  -3 Œ 3  ,B3  4  - 4 Œ 4
, # B# -
in H ß
with 3  4 œ 3 Œ 4  4 Œ 3Þ
28

29. X=- œ 5 - >- Œ >- ß
with
È$
-
5 Ð< Ñ œ  ,BÈB#  %C #
'
and
ÈB#  %C #
ÐBß #CÑ
>- Ð< Ñ œ
the unit tangent vector to # - .
29

30. By integration we obtain
Ú  ,C 4 Œ 4 -
X Ð<Ñ œ Û  .3 Œ 3  ,B3  4  , # B# Î-4 Œ 4
in H ÏE ß
Ü W Ð< Ñ
in H- ÏEß

in Eß
where
E œ Ö< œ ÐBß CÑ À ,-B# ÎC − Ð.  &ß .  &Ñ×
and
W Ð<Ñ œ  Ð#&Ñ š’   Ð .  % Ñ # “3 Œ 3 
, # B%
" "
"#C# #
’  ,BÐ.  &Ñ“3  4 
, # B$
$C
È$
’Š ‹, B  ,CÐ.  %Ñ“4 Œ 4 ›Þ
& CÐ.  &Ñ # #
  68
# ' ,-B#
30

31. References
[1] G. Del Piero Limit analysis and no-tension materials
Int. J. Plasticity 14, 259-271 (1998)
[2] M. Lucchesi, N. Zani Some explicit solutions to
equilibrium problem for masonry-like bodies Struc.
Eng. Mech. 16, 295-316, (2003)
[3] M. Lucchesi, M. Silhavy, N. Zani A new class of
equilibrated stress fields for no-tension bodies J.
Mech. Mat. Struct. 1, 503-539, (2006)
[4] M. Lucchesi, M. Silhavy, N. Zani Integration of
measures and admissible stress fields for masonry
bodies J. Mech. Mat. Struct. 3, 675-696, (2008)
[5] M. Lucchesi, C. Padovani, M. Silhavy An energetic
view of limit analysis for normal bodies Quart. Appl.
Math 68, 713-746 (2010)
[6] M. Lucchesi, M. Silhavy, N. Zani Integration of
parametric measures and the statics of masonry
panels Ann. Solid Struct Mech. 2, 33-44, (2011)
31