Arch Bridges
ARCH´04
P. Roca and E. Oñate (Eds)
 CIMNE, Barcelona, 2004
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*
Dipartimento di Scienze per l’Architettura (DSA)
University of Genoa
Stradone S. Agostino 37, 16123 Genoa, Italy
e-mail: [email protected], [email protected], web page: http://www.arch.unige.it/
**
Dipartimento di Ingegneria Strutturale (DIS)
University of Pisa
Via Diotisalvi 2, 56126 Pisa, Italy
e-mail: [email protected], [email protected], web page: http://www.dis.ing.unipi.it/
.H\ZRUGV masonry arch, collapse, non-linear elastic analysis, Durand-Claye’s method
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1
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
,1752'8&7,21
The historical development of the theory of the masonry arch is marked by two alternative
theoretical approaches corresponding to what we now call ‘limit’ and ‘elastic’ analysis.
According to the first approach, global VWDELOLW\ is the main question and the safety of the
arch is guaranteed as long as VRPH equilibrium condition preventing rigid mechanisms does
exist. This is the HTXLOLEULXP approach followed in the first 18th century studies on masonry
arches and subsequently developed during the early 19th century on the basis of Coulomb’s
method of maxima and minimai. Within this framework, the arch is considered as an
assemblage of rigid voussoirs subject to unilateral constraints and friction at the joints. When
the equilibrium of such an assemblage is impossible, the voussoirs undergo some relative
displacements which transform the arch into a mechanism.
According to the second approach, local VWUHVV and VWUDLQ become the objects of
investigation and the safety of the arch is insured if the DFWXDO stresses at each cross-section
are less than the admissible strength. This approach involves the complete set of the
HTXLOLEULXP, FRPSDWLELOLW\ and VWUHVVVWUDLQ equations. Taking up the groundbreaking work of
Signoriniii, some of the authorsiii,iv,v,vi have proposed a non-linear elastic constitutive relation
for masonry arches. Within this framework, collapse is conventionally attained when the
residual stiffness of the arch is less than a preset fraction of its initial value.
The alternative between limit and elastic structural approaches - VWDELOLW\ versus QRQOLQHDU
HODVWLFVROXWLRQ - is not so radical as it may seem at first sight. As a matter of fact, it may be
partially removed by following an intermediate approach proposed by the French scholar
Durand-Claye in 1867vii,viii. This approach aims at verifying, as the author wrote, “s’il existe
des solutions d’équilibre compatibles avec un effort-limite donné”. This methodology
preserves the fundamental concepts of the limit analysis and, at the same time, embodies
some aspects of the non-linear elastic analysis by imposing a restriction on the stress levelix,x.
This paper presents a parallel treatment of masonry arches in terms of both non-linear
elastic analysis and Durand-Claye’s method. The two procedures have been applied to pointed
arches subject to their own weight in order to determine the evolution of the non-linear elastic
solution and the stability area, for different values of thickness and friction coefficient.
'85$1'&/$<(¶60(7+2'$1'7+(121/,1($5(/$67,&02'(/
Durand-Claye’s method is a graphical procedure to define the so-called DUHDRIVWDELOLW\ at
the crown section of a symmetric arch, that is to say, the area within which the extremes of
the vectors representing the crown thrust must be included in order to respect the global
equilibrium of the structure and the strength of masonry.
Let us consider a symmetric masonry arch, with finite compressive strength σ and zero
tensile strength σ , and examine the ideal voussoir between the crown joint F0G0 and a generic
joint F G (Fig. 1). Let :θ be the weight of the voussoir F0G0G F , 1θ the normal force at
joint F G , 3 the thrust at the crown section and H the eccentricity of its application point.
Assuming a piecewise-linear law for the normal stress distribution along the section, we
can draw the area F ω G containing the admissible normal force 1θ with respect to the
strength σ and the corresponding hyperbolas α and β at the crown joint, defined by the
2
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
equilibrium of the voussoir F0G0G F . The admissible thrusts 3 with respect to rotational
equilibrium and strength at joint F G are then represented by horizontal vectors whose
extremes are contained within the area U V S T , delimited by the hyperbolas α and β , and the
curves F0ω0 and G0ω0ix,x
0
0
0
σt
σ
εc
εt
ε
σc
Figure 1: The stability area
Figure 2: Non-linear stress-strain relation
In order to account for the translational equilibrium along F G , it is sufficient to draw at a
generic point D the friction cone defined by the friction angle ϕ (Fig. 1). Let us take the
weight :θ applied at point D and consider the two horizontal thrusts which give a resultant
coinciding with the boundaries of the cone. These thrusts define two vertical lines at the
crown, between which the extremes of the vectors representing the admissible thrusts 3 for
the transational equilibrium along F G are comprised.
To assure both rotational and translational equilibrium at joint F G , the admissible thrusts at
the crown will be contained within the intersection $ of the area U V S T with the two vertical
lines + − and + + , corresponding to the translational equilibrium along F G .
By repeating the foregoing constructions for every joint L, the DUHDRIVWDELOLW\ $common
to all the areas $ can be defined. For a well-designed arch with a high friction coefficient, the
area of stability is a curvilinear quadrilateral of the type U V S T , in the sense that it
corresponds to the joints F G and F G . When the area of stability shrinks to a point, the limit
condition is attained, and a unique admissible thrust line exists. Let U V S T be the area of
stability for the admissible compressive strength σ (Fig. 1). By decreasing this strength, the
area U V S T will correspondingly shrink to a point for a value σ ,lim. The ratio Y = σ /σ ,lim
provides the safety coefficient of the real arch.
Unlike Durand-Claye’ s method, in the one-dimensional non-linear elastic model the arch is
viewed as an elastic curved beam. As commonly accepted in the theory of the bending of
beams, in any given cross-section the longitudinal strain is assumed to be linear along the
thickness. Furthermore, a piecewise-linear constitutive relation between longitudinal strains
and stresses is chosen to roughly describe the complex mechanical behaviour of masonry. The
3
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
relation is plotted in Fig. 2, where σ and σ are the material’ s tensile and compressive
strength, respectively. Consequently, the dimensionless axial force and bending moment,
Q = − 1 (Kε and P = − 2 0 (K 2ε , are non-linear functions of the dimensionless axial
strain and cross-sectional curvature, ξ = − ε 0 ε and η = − χK 2ε , where 1 is the axial
force, 0 the bending moment, ( is Young’ s modulus, K is the section thickness, ε ! is the
axial strain and χ is the curvature.
It is quite easy to determine the elastic domain of the cross-section, i.e., the set of all the (Q,
P) values that are compatible with the assumed constitutive relation (Fig. 3). The domain is
subdivided into six regions, each one corresponding to a different linear or non-linear
distribution of longitudinal stresses. A different constitutive relation holds between the
dimensionless internal forces Q and P and the cross-sectional kinematical parameters in each
region. As an example, in region C (where the stress distribution is non-linear under both
tension and compression and P > 0),
1 − W (W + 1)(W − 1 − 2ξ )
1 + W (W + 1)(1 − W + W 2 + 3(1 − W )ξ + 3ξ 2 )
Q=−
+
, P=
+
,
(1)
4
2
4η
12η 2
where W = − ε # ε " . Analogous relations, omitted here for brevity, hold for the other regions.
Let be V the curvilinear abscissa along the line of the arch’ s axis, X and Y the displacements
of points along the axis line in the tangential and radial directions, respectively, and ϕ the
rotation of the cross-section. Simple calculations, omitted here for the sake of brevity, show
that under the foregoing hypotheses and in the case of circular arch with radius 5, tangential
displacement X is a solution to the differential equation
X ′′′ + X ′ = 5ε 0′′ − 5 2 χ ,
(2)
where the primes denote differentiation with respect to θ= V5. The integral of (2) is
θ
X (θ ) = $ + % sin(θ ) + & cos(θ ) + ∫ [1 − cos(W − θ )][− 5 2 χ (W ) + 5ε 0′′(W )]GW
θ0
(3)
where $, % and & are constants. In turn, as Y = X ′ − 5ε 0 , and ϕ = (X + Y ′) / 5 , analogous
expressions hold for Y and ϕ. Under general load and constraint conditions, equation (3) and
the analogous ones for Y and ϕ lead to a non-linear set of equations in six unknowns: the three
constants $, % and & and the three end-reaction components. Due to the strong non-linearity
of the problem, the solution to the set of equations can be obtained in closed form only for
cases of relatively simple loads and geometries. In the general case, the solution is sought via
an iterative method. For the present context, an DG KRF numerical procedure has been used,
following a modified standard Newton-Raphson scheme. The arch’ s behaviour is assumed to
be linear during any given iteration. In this way we obtain a linear set of equations which is
solved by numerically performing the required integration. The curvature and axial strain are
updated and the procedure is repeated until the difference between the solutions at two
consecutive iterations becomes lower than a small fixed threshold value (more details will be
given in a forthcoming paper).
4
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
7+(32,17('$5&+81'(5,762:1:(,*+7
Let us now consider a symmetric pointed masonry arch (Fig. 4) whose span and radius referred to the axis line - are Oand5, respectively. Let K, µ, γ bethe constant thickness of the
cross-section, the friction coefficient and the specific weight of the material, respectively.
m
(t+1)/4
ϕ
C
δ
γ
α
A
(t+1)/6
D
B
β
E
F
(t - 1)/2
-1
n
t
D
B
C
Figure 3: The elastic domain.
Figure 4: The pointed arch
Let the circular intrados curve of the pointed arch be divided into 2Qequal arcs, the middle
point of the crown section be identified by angle θ0 and the generic joint F $ G $ by angle θ $ with
reference to the vertical line. If the cross sections are radial (apart from the vertical crown
joint), the bending moment 0(θ $ )can be written as a function of the weight :(θ $ ), the thrust 3
at the crown section and its eccentricity H:
0 (θ ' ) = − 3 H + 3 5 (cos θ 0 − cos θ ' ) − : (θ ' )(5 (sin θ ' − sin θ 0 ) − [ ( (θ '
))
(4)
where [G(θ $ ) is the distance of the centre of gravity of voussoir F0G0G $ F $ from the vertical line
of the crown joint. The normal force 1(θ $ ) is given by:
1 (θ ) ) = − 3 cos θ ) − : (θ ) )sin θ )
(5)
If in (4) we now set 0(θ $ ) = 0lim(θ $ ), we find the thrust 3at the crown section as a function
of the eccentricity H, corresponding to the limit value of the bending moment at joint F $ G $ .
From (5) we define the admissible thrusts for the translational equilibrium.
Let us consider the limit equilibrium of a generic voussoir F0G0G $ F $ with respect to
downwards sliding, upwards sliding, rotation about the intrados edge, and rotation about the
extrados edge. The corresponding values of the thrusts are:
%
3min (θ $ ): for downwards sliding;
&
3max (θ $ ): for upwards sliding;
5
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
+ *
3min, (θ $ ): for rotation about the intrados F $ , when the thrust is applied at the crown extrados G0;
- ,
,
3max
(θ $ ): for rotation about the extrados G $ , when the thrust is applied at the crown extrados G0;
+ )
3min, (θ $ ): for rotation about the intrados F $ , when the thrust is applied at the crown intrados F0;
- ,.
3max
(θ $ ): for rotation about the extrados G $ , when the thrust is applied at the crown intrados F0;
/
30 ,* (θ $ ): for rotation about the intrados F $ , when the thrust is applied inside the crown section;
/
31 ,* (θ $ ): for rotation about the extrados G $ , when the thrust is applied inside the crown section.
By fixing O = 10 m , 5 = 10 m , γ = 2000 daN/m 3 , σ 2 = 200 daN/ cm 2 , σ 3 = 0 , and varying
the thickness K and the friction coefficient µ, different collapse modes have been identified
and compared with the eight collapse modes of a symmetric arch given by Michonxi in 1857.
6OLGLQJFROODSVH0RGHDQG0RGH
When the range of admissible thrusts for the upwards and downwards translational
equilibrium shrinks to a single value, collapse can occur by Mode 5 (Fig. 5) or Mode 6 (Fig.
6). In particular, Mode 5 can occur for:
5
5
µ < 0.189248 ( ϕ < 10.7164 °), K > 127.799 cm ( max 3min (θ 6 ) = min 3max (θ 4 ), with θ 7 < θ8 ),
while Mode 6 can occur for:
5
5
µ > 0.189248 ( ϕ > 10.7164 °), K < 127.799 cm ( min 3max (θ 6 ) = max 3min (θ 4 ), with θ 7 < θ8 )
Figure 5: Collapse mode 5
Figure 6: Collapse mode 6
Figure 7: Transition 5 - 6
The transition between Mode 5 and Mode 6 (Fig. 7) takes place for:
µ = 0.189248 ( ϕ = 10.7164 °), K = 127.799 cm. The limit condition is given by:
9
;
:
min 3max (35.17° ) = max 3min (52.48° ) = min 3max (90° ) = 50.6483 daN (Fig. 8).
? @ A B C (37.35°)
?D@ A B E (69.52°)
-e
-e
200
200
20
40
60
80
100
P
P
120
20
40
60
80
-200
-400
-600
< = min (35.17°)
min < = max (35.17°)
max <>= min (52.48°)
min < = max (90°)
-200
< = max (52.48°)
? F min (34.42°)
-400
Figure 8: Transition 5 - 6. The stability area
min ? F max (34.42°)
max ? F min (51.97)
? F max (51.97)
Figure 9: Transition 6 - 4*. The stability area
6
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
By decreasing the thickness to K = 86 cm , it can be seen that for µ = 0.212878 the
straight line corresponding to the limit thrust for the translational equilibrium (Mode 6, with
θ = 34.42° and θ = 51.97°) passes through the point corresponding to 0 lim > 0 at joint θ =
37.35° and 0 lim < 0 at joint θ = 69.52° (Fig. 9). In this case, the transition mixed collapse
mode in Fig.10 can occur and the limit condition is:
G
H
H
J
min 3max (34.42° ) = 3I ,* (37.35° ) = max 3min (51.97° ) = 3K ,* (69.52°) = 32.18 daN
0L[HGFROODSVHVOLGLQJDQGURWDWLRQ0RGH
For 39.931 cm < K < 86 cm and 0.212878 < µ < 0.333738 , the straight line corresponding
to the admissible thrust for the upwards translational equilibrium at joint F L G L goes through the
point corresponding to 0 lim > 0 , at joint FM GM , and 0 lim < 0 , at joint FN GN , with θ L < θM < θN (Fig. 12). In this case, the mixed collapse mode in Fig.11 can occur, and the limit condition is:
O
Q
T
min 3max (θ K ) = 3R ,* (θ P ) = 3U ,* (θ S ), with θ L < θM < θN .
This collapse mode can be called Mode 4*, because it is similiar to Michon’ s Mode 4,
W
Y
Y
according to which min 3max (θ V ) = 3Z ,* (θ X ) = 3X ,* (θ ), with θ L < θM < θN .
` abc
` a b c d (θe )
f (θg )
-e
100
10
-100
` h
θf )
min (
20
30
` h
min
max
40
P50
(θ f )
-200
Figure 10: Transition 6 - 4*
Figure 11: Collapse mode 4*
Figure 12: Mode 4*. The stability area
For K = 39.931 cm and µ = 0.333738 , the area of stability shrinks to a single point, and a
transition mixed collapse mode can arise, for which the limit condition is:
[
] ,\
_ ^
(39.57°) = max 3min, (69.24°) = 18.7 daN
min 3max (33.64° ) = min 3max
7
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
5RWDWLRQDOFROODSVH0RGH
For K = 39.931 cm and µ > 0.333738 , the rotational collapse mode of Fig. 13 can occur.
] ,\
_ ^
(39.57°) = max 3min, (69.24°) = 18.7 daN.
The limit condition is: min 3max
-e
min i j k l max (39.57°)
max imj k l min (69.24°)
50
θ =θ0=30°
5
10
15
20
25
30
35
P
-50
θ =39.57°
-100
θ = 69.24°
Figure 13: Collapse mode 2 (left) and the stability area (right)
The foregoing results are summarised in Fig. 14 in terms of thickness and friction
coefficient.
µ 0.5
Mode 2
0.4
Mode 4*
0.3
Mode 6
0.2
Mode 5
0.1
n
50
100
150
200
250
300
Figure 14: Collapse modes in the (K, µ) plane
Figure 15: The structural scheme
Durand-Claye’ s method focuses on the static aspects of the mechanical behaviour of the
masonry arch. It seems interesting to approach the same problem from a different point of
view and to check which results could be obtained by using the non-linear elastic model of the
arch introduced in section 2.
Although the comparison of the two solutions is necessarily incomplete, as the elastic
model in its current version neglects shear deformability, it is worthwhile underlining that the
results of the non-linear elastic analysis match those obtained with the stability areas method.
In particular, this is evident for two limit cases corresponding to pure shear collapse
mechanism (very thick arch) and pure flexural collapse mechanism (very thin arch).
The non-linear elastic model has been applied to the structural scheme shown in Fig. 15.
The equilibrium problem has been solved for different values of the thickness, ranging from
45 cm to 150 cm. The results obtained for the two limit cases (thickness 45 cm and 150 cm)
are summarized in Fig. 16 and Fig. 17 in terms of internal forces and stresses, respectively.
8
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
For the thin arch, the position of the line of thrust (Fig. 16a) reveals that the arch attains a
limit condition that therefore can be considered as near collapse. Such a situation is confirmed
by the diagram of the normal stresses at the arch’ s extrados and intrados (Fig. 16c),
characterized by high compressive values. It is worthwhile noting that the hinge positions for
collapse mode 2 are fully compatible with the line of thrust in Fig. 16.
0,2
0,1
0,0
-0,1
T/N
-0,2
-0,3
-0,4
-0,5
-0,6
-0,7
Stress [MPa]
30
(D)
0,0
-0,5
-1,0
-1,5
-2,0
-2,5
-3,0
-3,5
-4,0
-4,5
-5,0
-5,5
40
50
60
70
Angle [deg]
80
90
(E)
z { r t usvmxmy
oqpsr t uwvmxmy
| }s~ w€
30
41.0°
40
70.3°
50
60
70
Angle [deg]
80
90
(F)
Figure 16: Line of thrust (D), shear over axial force ratio (E) and normal stress distribution (F) for K = 45 cm.
0,2
0,1
0,0
-0,1
-0,2
2,0
-0,3
-0,4
-0,5
-0,6
-0,7
30
40
50
60
70
Angle [deg]
80
90
(E)
0,0
Stress [MPa]
-0,1
‡ ˆ ‰ Š ‹ Œ„„Ž
-0,2
  ‰ Š ‹ Œ„„Ž
1,5
1,0
0,5
0,0
-0,5
0
-0,3
(D)
keystone displacement [cm]
T/N
50
100
150
200
arch thickness [cm]
45.0°
-0,4
30
40
50
60
70
Angle [deg]
73.3°
80
 ‚„ƒ … †
90
(F)
Figure 17: Line of thrust (D), shear over axial force ratio (E)
and normal stress distribution (F) for K = 150 cm.
9
Figure 18: Keystone vertical displacement
as a function of arch thickness.
Danila Aita, Riccardo Barsotti, Stefano Bennati and Federico Foce
For the thick arch, both the line of thrust and the stress diagram clearly reveal that the arch
is far from any flexural collapse mechanism (Fig. 17). Moreover, by looking at the plot of the
shear over axial force ratio values along the arch, it can be seen that a friction coefficient of
0.2 may represent a limit value also for the non-linear elastic analysis, under which a shear
collapse mechanism could take place. Also in this case, the non-linear elastic results thus
seem to confirm those found via the stability areas method.
Finally, Fig. 18 shows the values of the vertical displacement at the keystone plotted
against the arch thickness. The rapid increase in the displacement as the thickness approaches
45 cm once again confirms that in this case the arch is close to a collapse mechanism. It
should also be noted that the sign of the displacement for the two cases corresponding to 45
cm and 150 cm are consistent with the mechanisms determined via the stability areas method.
i
5()(5(1&(6
[i] Coulomb, C.A., Essai sur une application des règles de maximis et minimis à quelques
problèmes de statique, relatifs à l’ architecture, 0pPRLUHVSUpVHQWpVjO¶$FDGpPLH5R\DOH
GHV6FLHQFHVSDUGLYHUVVDYDQV, vol. 7, année 1773, Paris 1776, pp. 343-382.
[ii] Signorini A., Sulla pressoflessione delle murature, 5HQGLFRQWLGHOO
$FFDGHPLD1D]LRQDOH
GHL/LQFHL, serie IV, , 1925, pp. 484-489.
[iii] Bennati, S., Barsotti, R., Comportamento non lineare di archi in muratura, in $WWLGHO;,9
&RQJUHVVR1D]LRQDOH$,0(7$&RPRRWWREUH, 1999.
[iv] Bennati S., Barsotti R., Optimum radii of circular masonry arches, in C. Abdunur (ed.),
$UFK µ7KLUG ,QW &RQIHUHQFH RQ $UFK %ULGJHV, Presses de l’ École Nat. des Ponts et
Chaussées, Paris 2001, pp. 489-498.
[v] Bennati S., Barsotti R., Non-linear analysis and collapse of masonry arches, in Becchi A.
et al. (eds.), 7RZDUGVDKLVWRU\RIFRQVWUXFWLRQ, Birkhäuser, Basel, 2002, pp. 53-71.
[vi] Aita D., Barsotti R., Bennati S., Some explicit solutions for flat and depressed masonry
arches, in S. Huerta ed.), 3URF)LUVW,QW&RQJUHVVRQ&RQVWUXFWLRQ+LVWRU\, Inst. Juan de
Herrera, Madrid, 2003, vol. I, pp. 171-183.
[vii] Durand-Claye A., Note sur la vérification de la stabilité des voûtes en maçonnerie et sur
l’ emploi des courbes de pression, $QQGHV3RQWVHW&KDXVVpHV, , 1867, pp. 63-93.
[viii]Durand-Claye A., Note sur la verification de la stabilité des arcs métalliques et sur
l’ emploi des courbes de pression, $QQGHV3RQWVHW&KDXVVpHV, , 1868, pp.109-144.
[ix] Foce F., Sinopoli A., Stability and strength of materials for static analysis of masonry
arches: Durand-Claye’ s method, in C. Abdunur (ed.), $UFKµ7KLUG,QW&RQIRQ$UFK
%ULGJHV, Presses de l’ École Nat. des Ponts et Chaussées, Paris 2001, pp. 437-443.
[x] Foce F., Aita D., Between limit and elastic analysis. A critical re-examination of DurandClaye’ s method, in S. Huerta (ed.), 3URF )LUVW ,QW &RQJUHVV RQ &RQVWUXFWLRQ +LVWRU\,
Inst. Juan de Herrera, Madrid 2003, vol. II, pp. 895-905.
[xi] Michon, P.F., ,QVWUXFWLRQ VXU OD VWDELOLWp GHV YR€WHV HW GHV PXUV GH UHYrWHPHQW, Metz:
Lithographie de l’ École d’ Application, 1857.
10