Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
Some results on the strength evaluation of
vaulted masonry structures
D. Abruzzese, M. Como, G. Lanni
Department of Civil Engineering, University of Rome
'Tor Vergata \ Italy
Abstract
Within the framework of the studies on masonry structures that use the notension model for the masonry material, the paper aims to analyze the
strength of some vaulted masonry structures, very frequent in the context of
historical buildings.
The analysis has been first focused to the case of the vertical loads,
examining both the possibility of symmetric and non symmetric collapse
mechanism. The effect of increasing horizontal forces has been then
analyzed.
Some useful results have been obtained particularly in the case of the
circular and the pointed arch with abutment piers.
1 Introduction
There is an increasing need to understand the structural behaviour of historical
masonry structures, mainly for their repair and maintenance. The assumption
of a consistent model of the "masonry material" behaviour is a fundamental
starting point because very seldom the traditional linear elastic analysis can be
useful. The response of the masonry to the applied stresses has an unilateral
nature: masonry, in fact, can sustain also high compressive stresses but can
resist only feeble tensions. In this context the pioneering studies of Heyman
[1,2,3] on the masonry arches and vaults still today stimulate further research
and analysis.
According to the Heyman approach four constitutive assumptions are
Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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Architectural Studies, Materials & Analysis
made:
- sliding failure cannot occur;
- masonry has not tensile strength;
- masonry has infinite compressive strength;
- masonry is rigid under compressive stresses.
These assumptions reveal particularly useful to analyze the strength of
masonry structures when the failure of the structure is more due to cracks
openings than to material crushing. With the above assumptions many
problems of the behaviour of masonry arches and cylindrical vaults have been
studied by Como et al. [4,5,6] in the framework of the Limit Analysis. In all
the cases the failure mechanisms come out from the occurrence in the arches
and piers of no dissipating hinges.
In this context a classical problem is the evaluation of the thrust of the
vaults loaded by their weight, in order to design abutments able to stand firm
against overturning, or to analyze the actual behaviour of old masonry
monuments. The determination of the failure mechanism of these vaulted
structures can be considered as another aspect of the same problem.
For a
symmetric vaulted structure, with abutments piers and
symmetrically loaded, the occurrence of a symmetric failure mechanism is
usual taken for granted. This symmetric failure mechanism is characterized by
the presence of two hinges symmetrically placed at the intrados of the vaulted
arch, of a key hinge at the extrados of the vault and by the outside overturning
of the piers (fig.l).
Also a nonsymmetric mechanism could, on the other hand, occur at the
failure. This nonsymmetric mechanism involves, on the contrary, overturning
of a single pier and the presence of three hinges in the arch, as drawn in fig.2.
fig.l
fig.2
A second problem of relevant interest in seismic areas is the evaluation
of the strength of the vaulted structures with abutments piers, loaded by their
weights and by horizontal actions proportional to the same weights. In this
case a non symmetric global mechanism as drawn infig.2generally occurs at
Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
Architectural Studies, Materials & Analysis 433
the failure. Only if the piers are unusually squat a local failure of the only arch
can occur.
In the next the cases of the circular and the pointed arch will be
particularly examined. Some simplified formulas will be presented in order to
evaluate the collapse multiplier of the vertical loads acting on the arch and that
one of the horizontal actions applied on the whole structure. Moreover the
effect of a reinforcing chain will be also evaluated.
2 The circular masonry arch with abutment piers loaded by
vertical forces.
The scheme of the considered structure is drawn infig.3.Let R be the inner
radius of the arch and S its thickness; let also B and H be respectively the
lower width and the height of the abutment piers. The loads are only vertical
and are represented by the weights of the arch Q», distributed along the span,
and the weight Qp of the piers, concentrated in the centers of gravity Gp. We
will assume, on the other hand, that the forces acting on the vault, but not
those acting on the piers, are increasing with the load multiplier X.
iTTTTTTTTTTTV
B\
—
A f: *AB/\I/
°a
A
0 (i)
fig.3
"A
pB
•
"V.
8H
X
fig.4
2.1 Symmetric collapse
A single parameter % will define the symmetric mechanism, i.e. the position
of the hinge localized at the intrados of the arch (fig.4). Let also AB and BC
be the two portions of the half arch divided by this hinge, having weights
respectively QAB and QBC.
The two centers of gravity GAB and GBC of the two portions AB and
BC are indicated in fig.4. They have distances from the centers of the
respective chords given by:
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Architectural Studies, Materials & Analysis
^Hv"°2-°°'2.
where cp is the characteristic angle of the portion of arch (angles
offig.4),and have coordinates x given by:
TC
>.
i
i
s
cosIAB+-
clk- YC
sin^f-
(1)
and
(3)
where XA....YC are the coordinates of A,...,C in the local references system
O.xy.
The mechanism centers of rotation (!),(!,2) and (2) are shown infig.4:
in particular the coordinate x of (2) is:
(2),=
d,2),
H + R. + -
(4)
If the part I rotates around the center (1) of an unitary angle, the part II
rotates around (2) of an angle a given by:
a=—
(1,21
(5)
With all these elements we can write now the virtual work equation and
obtain the expression of a kinematics multiplier for the symmetric mechanism:
x:=-
(6)
The research of the minimum of the kinematics multiplier V varying the
parameter cpe yields the symmetric collapse multiplier A,OS of the weights
acting on the arch. A numerical analysis has shown that the hinge B forms at a
distance from left springing given by 0.1L, where L is the span of the arch.
With these assumptions we can produce easily the following simplified
expression of the symmetric collapse multiplier A^,:
Q. K.
(7)
Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
Architectural Studies, Materials & Analysis 435
where:
(8)
R
(9)
2.2 Non-symmetric collapse
In this case two parameters, the position angles cpe and cpc of the two hinges
occurring inside the arch, define the mechanism (fig.6). Let so AB, BC and
CD be the three portions in which the arch is divided by the internal hinges,
having weights respectively QAB, QBC, QCD
The three centers of gravity GAB, GBC and GCD of the three portions
AB,BC and CD are indicated infig.6.They have distances from the centers of
the respective chords given by (1) and coordinates x given by:
G_ =
MB
(10)
cos-"
.cos
.....
....
(11)
+(%B-%c) _
COS
(Pc
(12)
/Yc
where XA,. YD are the coordinates of A,...,D in the local references system
0,xy
The mechanism centers of rotation (1),(1,2),(2), (2,3) and (3) are shown
in fig.6 : in particular the coordinates of (2) are:
(3)y(2,3)*-(3)%(2,3).
(1.2)y-(l)y(2.3)y-(3)y
(2)y =
(13)
(14)
(1,2)*-0)*
0,2),-OX
If the part I rotates around the center (1) of an unitary angle, the part II
rotates around (2) of a and the part III rotates around the center (3) of (3
These angles are respectively:
Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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Architectural Studies, Materials & Analysis
'
"
(3)x-(2,3)x
With all these elements we can write now the virtual work equation and obtain
the expression of the kinematics multiplier for the nonsymmetric mechanism:
_
_
" "~ QAB[GABX - (Ox] + QBC<*[GBC* " (2) J + QcnP[GcD* ~ (3)*]
The research of the minimum of the kinematics multiplier A** varying
both the parameters cpe and cpc yields the non symmetric collapse multiplier
of the weights acting on the arch. A numerical analysis has shown that the
hinges B and C form at a distance from the left springing equal respectively to
0.5 L and 0.9 L. With these assumptions we can produce easily the following
simplified expression of the nonsymmetric collapse multiplier Ao^
where:
K. =
b + 02
(0 46C - 0.6) - 0.2(b 4- 0.02)
(18)
and
__
h + 0.6
-— — •
b + 0.2
where h, b, s are defined by the (8).
2.3 Conclusion
A numerical analysis has been performed proving that, even if the geometric
parameters h, b, s change, it always occurs:
K,>Kn
(20)
Thus it can be stated that the collapse mechanism for the circular arch
with piers is symmetric as well the weight on the arch arises, and the
collapse multiplier A,o is given by the (7).
Empirical rules did exist for the design of the abutments: the seventeenth
century architects knew the "Blondels" rule, for instance. (Heyman,[3J). This
rule, on the other hand, did not involve neither the thickness of the arch or the
Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
Architectural Studies, Materials & Analysis 437
height of the piers, both independently defined by some fixed ratios with the
span of the arch. On the contrary, the simple formula obtained (7) puts clearly
in evidence the influence of the various geometrical factors on the strength of
the arch.
3 The circular masonry arch with abutment pier loaded by
horizontal actions
The scheme of the considered structure is drawn infig.5. The vertical loads
are represented by the weight of the arch Qa, distributed along the span, and
the weight Qp of the piers, concentrated in the gravity centers Gp. The
horizontal actions are proportional to the vertical ones and increase with the
multiplier X.
fig.5
fig.6
In this case two parameters, the position angles (pA and cpe of the two
hinges occurring inside the arch, define the mechanism. Let so AB, BC and
CD the three portions in which the arch is divided by the internal hinges,
having weights respectively QAB, QBC, QCDThe three centers of gravity GAB, GBC and GCD of the three arcs AB,BC
and CD are indicated infig.6.They have distances from the centers of the
respective chords given by (1), coordinates x given by (10),(11),(12) and
coordinates y given by:
71
-Z-VB
(21)
sm-
TIBC
sin
(22)
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Architectural Studies, Materials & Analysis
sin %
(23)
where XA,...YD are the coordinates of A,...,D in the local references system
0,xy
The mechanism centers of rotation (1),(1,2),(2), (2,3) and (3) are shown
in fig.6, and in particular the coordinates of (2) are given by (13),(14).
If the part I rotates around the center (1) of an unitary angle, the part II
rotates around (2) of a and the part III rotates around the center (3) of p.
These angles are respectively given by (15).
With all these elements we can write now the virtual work equation and
obtain the expression of the kinematics multiplier K^ for the global
mechanism:
% = - "<ABl^AB*
x
BC BCx
X
CD CD*
X
pK_l^
~
^-(l)J+QBc[G^-(2)Ja + Q^[G^-(3)Jp + Q/""
(24)
The research of the minimum of the kinematics multiplier X*/ varying
both the parameters cps and cpc yields the global collapse multiplier ?IOH of the
horizontal actions. A numerical analysis has shown that the hinges B and C
form at a distance from the left springing equal respectively to 0.5 L and 0.9
L. With these assumptions we can produce easily the following simplified
expression of the global collapse multiplier XOH:
pb Q.K,
8h Q Sh
*"*"{$
where:
K, =——[0.61-0.47C]-0.20(b + 0.02)
C —1.8
(26)
K^ = —^-[0.58-(0.62 + 0.30 s)C] + 0.20(h + 0.30)
\^ — l.o
(27)
and C is given by (19).
It is useful to observe that the term pb/8h which appears in the (25)
represents the strength of the isolated pier respect to the overturning.
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Architectural Studies, Materials & Analysis 439
4 The pointed arch with abutment piers
The results obtained in 2. and 3. have been extended to the case of the
pointed arch, particularly frequent in the gothic architecture. A pointed arch is
formed by two portions of circular arch, each having the inner radius R equal
to the span L. The two portions meet exactly at the top of the pointed arch.
The collapse mechanism under increasing vertical loads on the only arch
is symmetric (fig.l) and the hinges localized at the intrados of the arch form at
a distance from the springings equal to 0.05 L
The collapse multiplier is given by a formula quite similar to the (7) if
the expression (9) is substituted by the following one:
L
.
n
b + 0.24-(h + l.l s+0.87)^-j-^YJ-0.15(b + 0.02)
(28)
The collapse mechanism under horizontal actions increasing proportionally to the weights is non symmetric(fig.2) and the hinges in the arch form at
a distance from the left springing equal respectively to 0.4 L and 0.9 L
The collapse multiplier is given by a formula quite similar to the (25) if
the expressions (26), (27) and (19) are substituted by the following ones
b + Olf
004C
1
h + 044
1 + s Ih +_
0.44
_
1-1.5 s b + 0.10
5 The effect of the reinforcing chain
Very often the vaulted structures with abutments piers are reinforced with an
iron chain which connects the springings. The effect of this additional
elements on the collapse under horizontal or vertical actions is evaluable very
easily.
If we consider both a symmetric and a non symmetric collapse
mechanism (fig.l and fig.2) and denote with 8 the rotation of the pier, the
resistant work due to the uplifting of a pier is equal to QppB8, where p is the
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Architectural Studies, Materials & Analysis
geometric eccentricity of the pier weight respect to the external foot. If we
add a reinforcing chain, this generally yields at the failure of the structure and
the additional resistant work is equal to TO H8 if To is the yielding tension.
Then the collapse multiplier formulas (7) and (25) for the circular arch and the
corresponding ones for the pointed arch are completely usable substituting the
real eccentricity p with a fictitious eccentricity p* equal to:
(32)
References
1. Heyman, J., The stone skeleton, Int. Journ. of Solids and Struct., 2, 1966
2. Heyman, J., Equilibrium of shell structures, Clarendon Press, Oxford 1977.
3. Heyman, J. The masonry arch, Cambridge Press, Cambridge 1982
4. Como, M, A.Grimaldi, An unilateral Model for the Limit Analysis of
Masonry Walls, Int. Congr. on Unilateral Problems in Structural Analysis,
Ravello 1983, CISM, Springer Verlag, 1985
5. Como, M., A.Grimaldi, G.Lanni, New results on the horizontal strength
evaluation of the masonry buildings and monuments, 9th World Conf.
Earthquake Eng., Tokyo, 1988
6. Abruzzese, D , M.Como, G.Lanni, On the horizontal strength of the
masonry cathedrals, ECEE9, Moscow, 1990