Assessment of Multispan Masonry Arch
Bridges. I: Simplified Approach
Antonio Brencich1 and Ugo De Francesco2
Abstract: A stepwise iterative procedure for the nonlinear analysis of multispan arch bridges, suitable for implementation by standard
programming of commercial finite element codes, is discussed. Relying on the plane section hypothesis, masonry is assumed elasticperfectly plastic in compression and no tensile resistant; a collapse condition is found when an ultimate strain is reached. The iterative
procedure is that of an elastic prevision and subsequent nonlinear correction of the nodal forces: tensile stresses are not allowed in the
mortar joint by adapting the effective height of the arch to its compressed part, while the plastic response is represented by additional
external fictitious forces accounting for the compressive plastic plateau. The procedure is first tested by comparison with experimental
data and then applied to sample bridges, pointing out how the collapse mechanism and the ultimate load depend on the geometric and
mechanical parameters.
DOI: 10.1061/(ASCE)1084-0702(2004)9:6(582)
CE Database subject headings: Arches; Bridges, arch; Brick masonry; Bearing capacity; Numerical analysis.
Introduction
A large number of arch bridges, mainly built in the mid-to-late
19th century, are still in service in Europe and on the local road
system of the northeast United States. Most of these bridges are
still in their original configuration and have to face loads they had
not been designed for. This is why reliable estimates of their
structural response and load bearing capacity are needed.
The first nonlinear incremental procedure (Castigliano 1879)
assumes a compressive elastic–no tensile resistant (NTR) model
for masonry. Relying on a simplified iterative scheme, the procedure takes into account the opening of the mortar joints and assumes that collapse takes place when the masonry compressive
strength is first attained.
More accurate iterative schemes and more detailed onedimensional (1D) (Crisfield 1984, 1985; Bridle and Hughes 1990;
Choo et al. 1991; Molins and Roca 1998a, b), two-dimensional
(2D) (Loo and Yang 1991; Falconer 1994; Boothby et al. 1998;
Owen et al. 1998; Lourenço and Rots 2000, among the latest
results), and three-dimensional (3D) models (Rosson et al. 1998;
Fanning and Boothby 2001) are the tools of the modern incremental analysis in which the local collapse condition for masonry
is usually derived from experimental tests or set according to
well-established theories. 1D models proved to be efficient in
assessment and design procedures for single and multispan
bridges, while 2D and 3D models may give detailed information
1
Assistant Professor, Dept. of Structural and Geotechnical Engineering, Univ. of Genoa, via Montallegro 1, 16145 Genova, Italy.
2
Research Assistant, Dept. of Structural and Geotechnical Engineering, Univ. of Genoa, via Montallegro 1, 16145 Genova, Italy.
Note. Discussion open until April 1, 2005. Separate discussions must
be submitted for individual papers. To extend the closing date by one
month, a written request must be filed with the ASCE Managing Editor.
The manuscript for this paper was submitted for review and possible
publication on February 15, 2002; approved on September 3, 2003. This
paper is part of the Journal of Bridge Engineering, Vol. 9, No. 6,
November 1, 2004. ©ASCE, ISSN 1084-0702/2004/6-582–590/$18.00.
on local phenomena at the expense of high computational costs.
A different approach relies on the static and kinematic theorems. In the first case, assuming an elastoplastic model for masonry and simplifying assumptions for the distribution of compressive stresses (Clemente et al. 1995) or on the basis of
experimental tests (Taylor and Mallinder 1993; Boothby 1997),
yield surfaces are obtained. The limit load is reached when the
axial thrust and the bending moment from the thrust line theory
lie on or outside the limit surface, see Boothby (2001) for an
application of this method.
The mechanism approach assumes a compressive rigid and no
tensile resistant model for masonry (Heyman 1982). Collapse is
attained when the number of pointlike hinges developed in the
arch are large enough to transform the arch itself into a mechanism. Assuming the position of the hinges as the unknowns of the
problem, the limit load turns out to be the lowest value of the live
load activating the mechanism (Harvey 1988; Blasi and Foraboschi 1994; Gilbert and Melbourne 1994; Hughes 1995; Como
1998). Assuming a NTR and compressive elastoplastic response
for masonry, the collapse mechanism is found when the thrust line
is about to be outside the arch (Crisfield and Packham 1988). This
approach gives reasonable estimates of the limit load when the
collapse mechanism is activated at relatively low stress levels,
i.e., for deep arches where the nonlinear compressive response of
masonry plays a minor role. When this is not so, i.e., for shallow
bridges, this approach seems to be more questionable. The procedure is hardly applicable to multispan bridges because of the large
number of hinges needed to transform the bridge into a mechanism.
The popular MEXE-MOT method (MEXE 1963; DOT 1993)
assumes that the limit load of single and multispan bridges can be
deduced from the load carrying capacity of the arch barrel considered as a pinned-elastic arch. On the basis of the classical
elastic theory, of the allowable stress approach and of some experimental results, the relevant parameters, such as masonry
strength, mechanical characteristics of the fill, span interaction,
pier stiffness, etc., are taken into account by means of corrective
factors of uncertain origin (Hughes and Blackler 1997).
582 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2004
Fig. 1. No tensile resistant model for the voussoirs’ interface
Fig. 2. No tensile resistant perfectly elastoplastic model in compression
In this paper, a NTR compressive elastoplastic model for masonry (Brencich et al. 2001) is extended setting a limit to inelastic
strains, i.e., to masonry ductility. The procedure, implemented in
a one-dimensional finite element (FE) procedure, is applicable to
both single- and multispan bridge models. The comparison with
experimental results and some parametric analyses prove the procedure to be quite accurate at low computational cost, pointing
out some relevant features of the nonlinear response of masonry
arches.
␧dem = ␧ / ␧el, and for the ultimate ductility the ratio between strain
at crushing ␧ul and ␧el, ␦ul = ␧ul / ␧el, masonry is found to be a
quasi-brittle material with ultimate ductility around 1.5 (Brencich
et al. 2002). For this reason an improved model should also take
into account a limit to the inelastic strains.
Under the hypothesis of a linear distribution of strains, i.e., of
plane section, Fig. 2, the constitutive equations for the elastoplastic, ductility controlled model are given as
b
NEP = f c 共x + y兲
2
Elastoplastic Model with Controlled Ductility
The brick/joint interface can be represented by means of an elastic unilateral contact surface. Experimental evidences (Brencich
et al. 2002) show that the common hypothesis of the plane section
can be applied also to solid brick masonry, assuming a linear
distribution of stresses on the section, Fig. 1. The constitutive
equations for the NTR model can be derived in terms of the
effective section height x
N E = ␴ ⬘c
M E = ␴ ⬘c
bx
2
冉 冊 冉 冊
bx h x
h x
−
= NE −
2 2 3
2 3
共1a兲
共1b兲
where the superscript E indicates that the forces are referred to a
NTR material elastic in compression, i.e., with no limit to compressive stresses at this stage. Other symbols are defined in Fig. 1.
In Castigliano’s original work (1879) this model is assumed to
represent dry assemblages of stone blocks or weak mortar joints
but it is somewhat questionable, in the first case, because it does
not take into account the spalling collapse of the compressed edge
(Taylor and Mallinder 1993) and, in the second case, because
mortar joints, subjected to a biaxial compressive stress state, can
exhibit significant plastic strains before collapse.
An improved mechanical model of the interface should assume a limit stress f c in compression beyond which plastic strains
are allowed, Fig. 2. In a perfectly elastoplastic model no limit is
set to the plastic strains; this would be in contrast with experimental evidence (Taylor and Mallinder 1993; Brencich et al.
2002) showing that the inelastic strains after the peak load are
quite limited. Retaining the common definition of ductility, and
assuming for the ductility demand ␦dem the ratio between the actual strain ␧ and the strain at the end of the linear response ␧el,
M EP = f c
共2a兲
冋
1
b h
共x + y兲 − 共x2 + xy + y 2兲
3
2 2
␦dem =
册
共2b兲
␧ ␴c⬘
x
艋 ␦ul
=
=
␧el f c x − y
共2c兲
where ␦ul = ultimate ductility of masonry; y = extension of the plastic plateau; and ␴⬘c stands for the maximum compressive stress of
a NTR model.
The plastic cutoff, Fig. 2, is the difference between the elastic
response of the effective section x, Eqs. (1), and the elastoplastic
response, Eqs. (2)
⌬NEP = NE − NEP = f c
⌬M EP = M E − M EP = f c
冋
b y2
2x−y
册
共3a兲
冉 冊
1 y3
b h y2
h y
−
= ⌬NEP −
2 2x−y 3x−y
2 3
共3b兲
Vice versa, the elastoplastic NTR model, Fig. 2, can be represented as a NTR model, Fig. 1, to which the plastic forces ⌬NEP
and ⌬M EP, resulting from the compressive stress cutoff, are to be
applied as external forces. Representing the plastic response of
the material, they are unbalanced forces and will be called external fictitious forces in what follows.
Stepwise Iterative Algorithm
The procedure can be applied to generic quasi-brittle arch-type
structures following a predictor/corrector iterative scheme in the
frame of a standard FE approach. A series of beam-type elements
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Fig. 3. Finite elements with constant height meshing the effective
arch
with constant height can be used to model the compressed part of
an arch, Fig. 3. This choice, herein adopted for simplicity of the
algorithms instead of tapered beams, is an approximation in the
nodal sections but is exact in the central one, where the internal
forces M i and Ni are computed by averaging the nodal ones Nl,
M l, and Nm, M m and where the extension of the compressed and
inelastic part of the section, xi and y i, respectively, may be calculated by means of Eqs. (2).
In order to set up an iterative procedure, let us consider a
generic arch-type structure, Fig. 4, modeled by means of a FE
mesh.
1. Let us assume that the 共k − 1兲th approximation of the solution
共i兲
is known, i.e., for each ith element the effective height hk−1
(not the initial geometric height), the external fictitious
共p兲
forces, transferred in the adjacent nodes l and m, ⌬Fk−1
EP−共p兲
EP−共p兲
共i兲
= 共⌬Nk−1 , ⌬M k−1 , p = l , m兲 the compressed parts xk−1, the
c共i兲
maximum compressive stress ␴k−1
, and the plastic plateau
共i兲
y k−1, if any, are known, Fig. 4(a). The load vector consists of
the external applied forces F共p兲
0 to which the fictitious forces
共h兲
共p兲
共p兲
⌬Fk−1
need to be added: Fk−1
= F共p兲
0 + ⌬Fk−1.
共p兲
2. Prediction. An elastic analysis of the structure under the Fk−1
loads is performed; the stress state is calculated in every
element, so that the compressed part x共i兲
k , the maximum compressive stress ␴c共i兲
,
and
the
plastic
plateau
y 共i兲
k
k , if any, of the
element central section are known, Fig. 4(b).
3. Correction. The geometry is updated so as to reduce the el共i兲
ement height to the estimated compressed part: h共i兲
k = xk . The
共h兲
EP−共p兲
EP−共p兲
external fictitious forces ⌬Fk = 共⌬Nk
, ⌬M k
,p
= l , m兲, Eqs. (3), due to the new stress state are computed
共p兲
共p兲
updating the external loads: F共p兲
k = F0 + ⌬Fk , Fig. 4(c).
4. Convergence check Convergence is checked.
5. Crushing check. Ductility is computed in each section and
compared to the ultimate value.
6. The starting step is given by the elastic response to the external loads of the structure in its initial full geometry.
In the generic case of a redundant structure, the internal forces
to be equilibrated in each section are not known; for this reason
the convergence criterion cannot be referred to the unbalanced
part of the external loads but has to be referred to the variation of
the plastic fictitious forces between two subsequent steps:
max
再冉
EP
⌬Nk+1
− ⌬NEP
k
NEP
k
冊冉
,
i
EP
⌬M k+1
− ⌬M EP
k
M EP
k
冊
冎
, ∀ i 艋 ␻ 共4兲
i
where the subscript i stands for the generic element of the model
and ␻ = accepted tolerance.
The outlined procedure (see Fig. 5) can be implemented by
external programming of commercial FE codes since it is based
on a series of elastic analysis of structures differing only by an
Fig. 4. Scheme of the stepwise algorithm: (a) 共k − 1兲th iteration;
(b) prediction; and (c) correction
updating geometry and varying external forces. Like in any FE
approach, the model elements, representing the average masonry
response, are not linked to the block/joint dimensions.
Comparison with Theoretical Results
Theoretical results for arch-type structures assuming a nonlinear
compressive response for masonry are limited to the roman arch,
i.e., to the straight lintel assuming a perfectly elastoplastic model
with no limits to the ductility. Even though rather simple, this
structure is rather challenging for any numerical procedure being
characterized by low values of the axial thrust.
In the case of a straight lintel loaded by external moments, Fig.
6, all the sections present the same internal forces. The cross
section is assumed rectangular 1 cm wide and 10 cm high with a
100 cm long span; masonry is described by 5,000 MPa, f c
= 3 MPa.
Being the exact solution given with no limit to material ductility, this comparison is carried out for a perfectly elastoplastic
model. Both the NTR and the elastoplastic models are compared
in Fig. 6 showing a good convergence speed.
When a concentrated force of 250 N is set in the central section, Fig. 7, a prestrain equal to 10−4 is needed in the lintel. The
exact solution (Zani 2001) is compared to the results by the
584 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2004
present procedure: in Fig. 8 vertical and horizontal displacements
are plotted as a function of the tolerance ␻, Eq. (4), showing the
CPU time needed for the analysis of a 100 elements model; Fig. 9
shows the same quantities as a function of the model elements for
a fixed value of the tolerance, ␻ = 10−3.
The error of the procedure never exceeds 2.2% on the vertical
displacements for a rough accuracy ␻ = 10−2, while higher accuracy is obtained for the horizontal ones. Table 1 shows that convergence is much faster if referred to the internal forces, which
are more significant for design purposes.
Comparison with Experimental Results
The two-ring 1:5 scale model of Fig. 10 was loaded through its
whole width at 84 cm from the springing (0.28 times the span) by
means of a 20 cm wide device (Melbourne and Gilbert 1995).
Since the test showed that no separation took place up to collapse,
and being the spandrel walls detached from the arch barrel, this
test fits well the hypotheses of the discussed model. Table 2 summarizes the main mechanical properties of the materials.
The comparison of the experimental and model response, Fig.
11, shows rather good agreement. It has to be noted that in this
example, due to the high value of the compressive strength, the
material nonlinear response is expected to have reduced importance; in fact the maximum allowed ductility of 1.5 has not been
reached.
The Bridgemill arch at Girvan, Scotland (Hendry et al. 1985)
is a shallow single arch bridge made of 62 sandstone blocks with
a significant span of 18.29 m and an 8.30 m width; the fill in
crown is only 20 cm thick, Fig. 12. The load was applied on a
concrete strip 75 cm wide crossing the whole bridge width and
centered at 1/4 of the bridge span, in the weakest position for the
arch. Deflections were measured directly below the load; both the
arch barrel and the spandrel walls were in good condition. The
main parameters of the bridge and of the adopted model are summarized in Table 3.
Two models of the bridge were tested and compared to the
experimental data: Model A, with E = 4,000 MPa and f c
= 15 MPa, and Model B, with E = 4,000 MPa and f c = 7 MPa. Different values for the compressive strength of brickwork were considered in order to allow a direct comparison with the numerical
results obtained by other writers (Crisfield 1985). In Fig. 13, representing the load displacement curves, the vertical displacement
just below the load is plotted on the left-hand side against the
sustained load, while diagrams on the right-hand side represent
the deflection of the key-stone, which is lifted upwards in the
collapse mechanism. The circles mark the points where the compressive strength has first been attained, while the squares show
when the maximum allowed ductility 共␦ul = 1.5兲 has been reached,
Fig. 5. Flow chart of the iterative procedure
Fig. 6. Convergence of the procedure for a straight lintel
Fig. 7. Sample lintel loaded with a concentrated load
JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2004 / 585
Fig. 8. Displacements of the straight lintel as a function of the
tolerance: (a) vertical and (b) horizontal displacements of the beam
(half lintel only is represented)
i.e., the points where collapse is expected. It can be observed that,
for this geometry, a limited ductility allows a significant increase
of the load carrying capacity; besides, both Crisfield’s and the
present approaches fail in representing the experimental response
with good approximation. This is probably due to the collaboration of the arch barrel with the spandrel walls, which is not con-
Table 1. Reaction Forces versus Tolerance—100 Elements Model
Tolerance ␻
Rx [N]
Ry [N]
M [N cm]
0.01
0.001
0.0001
839.6
127.6
3,185.9
856.5
127.6
3,184.4
876.4
127.6
3,181.5
Table 2. Main Parameters of the Arch of Fig. 8
Property
Value
Property
Fig. 9. Displacements of the straight lintel as a function of the
number of elements: (a) vertical and (b) horizontal displacements of
the beam (half of the lintel is represented)
sidered in the simplified monodimensional model of the arch
herein proposed, and to the small displacement assumption of the
proposed model. Besides, the comparison with experimental data
shows that the proposed procedure is conservative.
Elastoplastic Response of a Single Arch
In order to get a deeper insight into the mechanical parameters
affecting the limit load of an arch, mainly compressive strength,
load position, and arch geometry, a sample arch has been considered, Fig. 14, representing a typical shallow arch derived from the
scale models tested by Melbourne et al. (1995) and typical of a
large number of European single- and multispan bridges.
Table 3. Main Parameters of the Bridgemill Bridge
Value
300 cm
Width
288 cm
Span s
21.5 cm
1 / 14⬵ 0.07
Ring thickness d
d/s
30 cm
60/ 43⬵ 1.39
Fill (crown) f
f /d
75 cm
1 / 4 = 0.25
Rise r
r/s
Arch density
22 kN/ m3
Fill density
22.2 kN/ m3
Compressive
27 MPa
Young’s
15,900 MPa
strength
modulus
Engineering bricks masonry—1:2:9 mortar
Property
Span s
Ring thickness d
Fill (crown) f
Rise r
Arch density
Compressive
strength
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Value
1,829 cm
71.1 cm
20.3 cm
284 cm
21 kN/ m3
5 – 8 MPa
Property
Width
d/s
f /d
r/s
Fill density
Young’s
modulus
Masonry type: sandstone blocks
Value
830 cm
1 / 11.7⬵ 0.08
1 / 3.5⬵ 0.28
5 / 32= 0.16
22 kN/ m3
4,000 MPa
Fig. 10. 1:5 scale model of a multiring arch [cm]
Fig. 13. Load-deflection response of the Bridgemill arch
Fig. 11. Load-displacement response of the arch of Fig. 10.
Displacement below the load.
Fig. 14. Geometry for the shallow sample arch [cm]
Fig. 12. (a) Geometry of the Bridgemill Bridge [cm] and (b) finite
element mesh and load distribution
Fig. 15. Load-displacement
Displacement below the load.
curves
for
the
sample
arch.
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Table 4. Main Parameters of the Sample Arches
Property
Span s
Ring thickness d
Fill (crown) f
Arch geometry—Shallow arch
Rise r
Arch geometry—Deep arch
Rise r
Density and loads
Arch density
Fill density
Masonry
Young’s modulus
Value
1,500 cm
75 cm
75 cm
Property
Width
d/s
f /d
Value
100 cm
1 / 20
1/1
375 cm
r/s
1 / 4 = 0.25
525 cm
r/s
7 / 20= 0.35
22 kN/ m3
24.1 kN/ m3
Load diffusion
Loaded length
40° + 40°
Knife-type
15,000 MPa
Compressive strength
5 / ⬁ MPa
In the following the width of the arch is assumed to be 100 cm
and all the main geometric and mechanical parameters are summarized in Table 4. In order to compare the results to the rise-tospan and barrel factors of the MEXE-MOT method (DOT 1993),
two sample arches have been considered, a shallow and a deep
one, differing in the rise/span ratio, whose main parameters are
summarized in Table 4.
A concentrated load was located in different positions along
the bridge assuming for masonry, first, the perfectly NTR model,
a perfectly elastoplastic model with compressive strength ␴c of 5
MPa and no limits to ductility and then setting a limit to the
allowed ductility. The procedure has been implemented in both
the ANSYS 5.6 (ANSYS 1999) and LUSAS (FEA 2001) FE
codes, assuming a beam length of approximately two brick units.
Fig. 15 shows the load-displacement response of the barrel, the
displacement being computed just below the load, and assuming
both a NTR model and a 5 MPa compressive strength material.
For both the arches the activation of the collapse mechanism, i.e.,
the sudden drop of the arch stiffness, takes place far after the
compressive strength is first reached, circles, and also far after the
maximum allowed ductility is reached, squares. This means that
these two geometries collapse because of the collapse of some
sections which takes place at rather limited displacements. This
shows that a mechanical model for the material which does not
set any limit to inelastic strain would overestimate the effective
load carrying capacity.
The diagrams of Fig. 16 represent the load carrying capacity of
the sample arches for masonry models and as a function of the
load position. The lowest load is found in the position nearby
1/4–1/3 of the arch span, specifically for x / l = 0.3. It can be observed that the models setting no limits to inelastic strains again
dramatically overestimate the load carrying capacity. Once an ultimate ductility is defined, the load carrying capacity is found to
have little variations as the load changes positions. The increment
of the limit load due to a limited ductility 共␦ = 1.5– 2兲, compared
to the perfectly brittle model 共␦ = 1兲, is some 10%, showing that,
for some geometries, a perfectly brittle model is acceptable. Besides, it can be seen that the effect of the limit on the maximum
allowable ductility leads to a limit load that is almost half of the
load that would be obtained by other models, such as the mechanism one.
Discussion
Fig. 16. Limit load versus loading position and masonry strength for
(a) r / s = 0.25 and (b) r / s = 0.35
The comparison of the discussed model with analytical data
shows an encouraging efficiency of the procedure. Comparison
with experimental data generally provides reasonably good approximation at the expenses of low computational effort provided
that the load bearing structure is the arch barrel, such as in the
case of the sample single span arch. When other “nonstructural”
elements, such as spandrel walls, collaborate with the barrel increasing the limit load, i.e., in the case of the Bridgemill test, the
comparison is less satisfactory. Nevertheless, the procedure is still
acceptable being conservative. As already pointed out by the ex-
588 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2004
perimental work by Melbourne et al. (1995), the effect of spandrel
wall on the global strength of a bridge can be as much as 30% or
more of the global collapse load. The comparison with the experimental data on the Bridgemill arch seems to corroborate this conclusion.
The parametric study of the load-displacement response and of
the limit loads for a shallow 共r / s = 0.25兲 and a deep 共r / s = 0.35兲
arch showed that the activation of the collapse mechanism would
require much higher values of ductility than what masonry is able
to exhibit. In fact, a reasonable value for ductility 共␦
= 1.4 to 1.5兲 would lead to a load carrying capacity that is about
60% of the previsions by other methods not taking ductility into
account, such as in the mechanism approach. Besides, the collapse would be caused by crushing of the critical section rather
than by the activation of a mechanism, as assumed by many assessment procedures.
As the Bridgemill arch showed, in some cases also a limited
value for the ultimate ductility of masonry may significantly increase the limit load. Since this is not a general feature, as the
sample arches showed, the effect of masonry ductility on the load
carrying capacity should be carefully analyzed for each bridge.
The barrel factor introduced by the MEXE method (DOT
1993) is related to the quality of masonry and ranges from 0.7 to
1.2 for brickwork arches, i.e., a maximum reduction of 43% due
to masonry strength. The reduction of the load carrying capacity
due to the compressive strength in the sample arches has been
found to be as much as 50%, in good agreement with the MEXE
factor.
If a quasi-brittle 共␦ = 1.4兲 model is given to the masonry of the
sample arches, the limit load for the deep one 共r / s = 0.35兲 is found
to be 55% of the load carrying capacity of the shallow 共r / s
= 0.25兲 arch. This is in contrast with the MEXE approach which
assumes, given all the other parameters, the same load carrying
capacity for all the arches with a rise/span ratio above 0.25.
Conclusions
An iterative procedure for the elastoplastic analysis of masonry
arch structures has been discussed. It proved to be user-friendly
since it can be developed simply making use of the programming
facilities of commercial FE codes and, in spite of some approximations, it proved to give good precision. The procedure can be
considered a reliable tool for the simplified analysis of masonry
arch bridges, giving conservative estimates of the limit load since
many beneficial effects are neglected, such as the contribution of
the spandrel walls and the arch barrel–fill interaction.
The discussed approach may be an alternative to the standard
nonlinear facilities of commercial FEM codes, which are generally related to concretelike materials. An example is the WillamWarnke limit surface, developed for concrete, that is usually used
for brickwork, but only few experimental data corroborate this
choice. Besides, the discussed procedure does not need to compute the stress field by means of numerical integration, leading to
more accurate results, for low-cost analyses of masonry arch
bridges, than standard FEM codes.
The FEM model of a multispan bridge is often a challenge to
the computational capacities of modern PCs, requiring often
many hours of computing time. The discussed procedure can be
applied to multispan bridges at very low computational costs.
Examples of analyses of multispan models and of a real bridge
are presented in Part II of the work.
Acknowledgments
This research has been carried out by partial financial support
from National Earthquake Defence Group (GNDT), VIA research
project “Seismic vulnerability reduction of infrastructural systems
and environment,” part of the Biennial 2000–2002 Research
Project, and PRIN 2002/2003 “Safety and control of masonry
bridges.”
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