Fracture Mechanics of Concrete and Concrete Structures High Performance, Fiber Reinforced Concrete, Special Loadings and Structural Applications- B. H. Oh, et al. (eds)
ⓒ 2010 Korea Concrete Institute, ISBN 978-89-5708-182-2
Cracking analysis of brick masonry arch bridge
J. M. Chandra Kishen & Ananth Ramaswamy
Dept. of Civil Engineering, Indian Institute of Science, Bangalore 560012, India
ABSTRACT: In this work, details of field measurements undertaken at a brick masonry arch bridge under
design train traffic and analytical work based on non-linear fracture mechanics are presented. A parametric
study is done to study the effects of tensile strength on the progress of cracking in the arch. Further, a stability
analysis to assess collapse of the arch due to lateral movement at the springing, in particular near the partially
filled land arches is done. The margin of safety with respect to cracking and stability failure is computed.
Conclusions are drawn on the overall safety of the bridge.
1 INTRODUCTION
Most of the railway bridges in the Indian Railway
system that have been built several decades ago have
deteriorated both in terms of strength and stiffness
due to a variety of reasons. These bridges have been
designed for live loads and service conditions that
have changed drastically with time. Increased axle
loads and traffic density have necessitated bridge
owners to get the bridge condition assessed in order
to determine their residual structural strength and
identify strengthening measures to be taken for safe
performance.
Condition assessment provides information regarding the intensity and extent of observed defects,
the cause for these defects and possible deterioration
processes that have strong impact on the safety and
service life of structures. Furthermore, this information forms the basis for estimating the residual structural capacity and possible remedial work that needs
to be undertaken.
The present study focuses on a brick masonry
arch bridge in the South Western Railway zone of
the Indian Railways. The bridge has been constructed with brick masonry with possibly a compacted granular soil infill and dates back to the
1870’s when it was part of a meter gauge line. Over
the years, the passenger and freight traffic have increased on this section and the section has been
transformed through a gauge conversion to broad
gauge traffic. The permitted axle load untill a few
years ago was classified as 18t axle load and has
subsequently undergone an upward revision to 22t
axle based on an in-house assessment undertaken by
the Indian Railways. At present, there has been a
growth in freight traffic in this section, in particular
for iron ore and coal movement, and the Indian
Railways is considering the possibility of further en-
hancing the permitted axle load to 25t immediately
with possible further upward revisions at a later date.
The condition assessment of the bridges in this
section has been initiated with a view to understand
the present state of the bridge and the ramifications
of increasing the payload on the bridge with or without any intervention for structural enhancement. In
this paper, details of field measurements undertaken
at the bridge under design train traffic and fracture
mechanics based finite element analysis are presented. Conclusions are drawn with some remarks
on the state of the bridge within the framework of
the information available and inferred.
2 DETAILS OF BRIDGE
The bridge considered in this study is a brick masonry arch bridge built in the 1870’s on a meter
gauge section. The bridge consists of two major
arches across a water fall with spans of 17.7m and
17.3m (Fig. 1) at springing level and a few smaller
arches of approximately 7.7m that are land arches
and partially closed. The carriage width of the bridge
is 8.8m and is slightly curved in plan and has a steep
rising gradient of 1 in 30. The railway line alignment
is eccentric with respect to the bridge centerline. The
arch has a central rise of about 4.5 meters. The rings
of the arch are about 0.93m in thickness across the
arch barrel, with a brick masonry facing on either
side rising up to the top of the bridge with a parapet
of a meter height on either side. The abutments and
piers appear to consist of brick masonry that has
been encased in reinforced concrete during an earlier
intervention. The piers and abutments rest on bed
rock. The pier width is about 3.82m at the base and
varies over the height. Figure 1 shows the schematic
sketch of the bridge.
J
= − D ( h , T ) ∇h
The proportionality coefficient D(h,T)
moisture permeability and it is a nonlinea
of the relative humidity h and temperature
& Najjar 1972). The moisture mass balanc
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J
Figure 1. Schematic sketch of brick masonry arch bridge.
(Left end: Kulem (K) end; Right end: Castle Rock (CR) end).
− ∂ = ∇•J
∂ in BFR wagon.
Figure 2. Spacing of axles
w
t
3 INSTRUMENTATION
A 64 channel data acquisition system (M/s
Dewetron, Austria) was used for acquiring the data
continuously from all the sensors. The sensors used
with this acquisition system consisted of linear variable differential transformer (LVDT) to measure
displacements, namely, vertical crown displacements and horizontal springing displacements; electrical resistance strain gauges to measure surface
strains in a particular direction, either along the
circumferential direction of the arch or the arch barrel at the crown, quarter and three quarter point and
springing levels and on the rails; vibrating wire
strain gauges mounted on the arch and parapet; uniaxial and tri-axial accelerometers to measure the acceleration levels in terms of ‘g’ levels at the track
level (sleepers) and corresponding locations on the
arch surface. A temperature sensor was fixed on the
bridge to measure the variations in the temperature.
A tilt sensor was placed on the vertical surface of the
pier. The LVDTs were located on the arch surface
and strain gauges were located on the arch intrados,
parapet and rail.
4 LOADING SCHEMES
A series of four major loading tests: static load tests;
quasi-static moving load tests; speed tests and longitudinal load tests were carried out over a four-day
period. In this study, only the first two tests are used
and hence are described in detail.
Static Load Test: Two BRN wagons whose axles
spacings are shown in Figure 2 and loaded with 200
(approx. 20.75 T axle load), 184, 168 and 152
prestressed concrete sleepers were placed on the
bridge structure at a fixed position. The two wagons
could apply a static load on both the spans of the
bridge. Displacements and strains were measured at
various locations on the bridge for each load level of
this test.
The water content w can be expressed a
of the evaporable water we (capillary wa
vapor, and adsorbed water) and the non-e
(chemically bound) water wn (Mil
Pantazopoulo & Mills 1995). It is reas
assume that the evaporable water is a fu
relative humidity, h, degree of hydration
degree of silica fume reaction, αs, i.e. we=w
= age-dependent sorption/desorption
(Norling Mjonell 1997). Under this assum
by substituting Equation 1 into Equati
obtains
Figure 3. Spacing of axles in BOXNEL wagon.
∂w
∂w ∂h
e
+ ∇ • ( D ∇h ) =
− e
h
∂α
∂h ∂t
∂w
α&c + e α&s + w
∂α
c
s
Quasi-static Moving Load Test : A train formation
consisting of two locomotive engines (WDG4), two
slopetwo
of the
sorption/
where ∂we/∂h isintheeach,
BFR wagons with 152 sleepers
goods
isotherm
(also
called
moisture
capac
wagon filled with 25 t axle load with four axles each
governing
equation
(Equation
3)
must
be
(BOXNEL whose axle spacing are shown in Figure
by
appropriate
boundary
and
initial
conditi
3), followed by two locomotive engines (WDG3A),
The relation between the amount of e
was positioned on the bridge at different prewater
and relative humidity is called ‘‘
determined locations and measurements were taken.
isotherm”
if measured with increasing
At the start of the test, the first wheel of WDG4
humidity
and
‘‘desorption isotherm” in th
wagon was placed on the left springing position of
case. Neglecting their difference (Xi et al.
span 1. The test progressed with subsequent wheels
the following, ‘‘sorption isotherm” will be
of the formation occupying this first reference point
reference to both sorption and desorption c
is succession. Since the formation had forty wheels,
By the way, if the hysteresis of the
the same number of measurements was taken. This
isotherm would be taken into account, two
test was performed to obtain the variations in deflecrelation, evaporable water vs relative humi
tions and strains for different positions of the load.
be used according to the sign of the varia
Such variations are typically determined in computarelativity humidity. The shape of the
tional models using the well known method of influisotherm for HPC is influenced by many p
ence lines.
especially those that influence extent and
chemical reactions and, in turn, determ
structure and pore size distribution (water5 RESULTS OF FIELD TESTS
ratio, cement chemical composition, SF
curing time and method, temperature, mix
In this section, the results of the various field tests
etc.). In the literature various formulatio
conducted on the arch bridge as indicated in Section
found to describe the sorption isotherm
4 have been presented.
concrete (Xi et al. 1994). However, in th
paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b
Proceedings of FraMCoS-7, May 23-28, 2010
J = −Results
D (h, T )∇ofh static tests
5.1
(1)
Table 1. Vertical deflections at the crown.
Load
1
Span 2 is called
The proportionalitySpan
coefficient
D(h,T)
mm
moisture permeability mm
and it is a nonlinear
function
168
sleepers
0.49
0.30 T (Bažant
of 184
the sleepers
relative humidity0.55
h and temperature
& Najjar
1972). The moisture
mass 0.60
balance requires
200 sleepers
0.57
0.60
that the variation in time of the water mass per unit
volume
of response
concrete in
(water
w) bedeflection
equal to the
The
static
termscontent
of vertical
at
divergence
of
the
moisture
flux
J
the crown central position is shown in Table 1. The
observed differences in the displacements of the two
spans
(2)
− ∂w =are
∇ • Jindicative of the mild asymmetric nature
that∂t exists in the arch geometry.
The water content w can be expressed as the sum
of the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable
(chemically bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
∂w the rail
∂w 4.∂hTangential strains along
Figure
e α ∂wcenterline.
eα w
+ ∇ • ( D ∇h ) =
− e
c
s
n
h
∂α
∂α
∂h ∂t
c
&+
s
&+ &
(3)
where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used with
reference to both sorption and desorption conditions.
By the
way, ifstrain
thealong
hysteresis
ofSpan
the1. moisture
Figure
5.Tangential
the barrel of
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of the
relativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of the
chemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,
etc.). In the literature various formulations can be
found to describe the sorption isotherm of normal
concrete (Xi et al. 1994). However, in the present
Figure
Transverse
strains at crown
of Span proposed
1 for different
paper 6.the
semi-empirical
expression
by
magnitude
of
loads.
Norling Mjornell (1997) is adopted because it
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly
accountsstrains
for the
evolution
hydration
The tangential
in span
1 alongofthe
centerreaction
SFarecontent.
This
sorption
line
of theand
track
shown in
Figure
4. Theisotherm
strains
readsminimum and almost zero near the crown and
are
maximum near the right springing position. Figure 5
shows the tangential strains
along the width
⎡
⎤ of the
arch at the crown position.
⎢ It may be1 mentioned
⎥
we (hone
, α , α ) = G (α , α ) 1 −
+ here
⎥
c micro
s
c corresponds
s ⎢ 10(g α to
that
strain
approximately
1
∞
2
⎢
c − α cin)h ⎥⎦the ma0.0018 MPa (0.018 kg/mm
e) of1 stress
⎣
(4)
sonry. It is seen that the strains are largest near the
⎡ 10(g α ∞ − α )h
⎤
eccentric position (far end
of1 the
barrel
of
c
c
⎢
K1 (αthe
− 1⎥ arch)
when compared to
c , αcentre
s )⎢e line of rail location.
⎥
⎣
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
w
0
K (α c α s ) =
,
1
⎡
⎢
−
⎢1 −
1
⎢
⎣
α s + 0.22α s G
− 0.188
c
s
⎛
10⎜
⎝
e
⎛
10⎜
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
g α c∞ − α c ⎞⎟h ⎤⎥
1
⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
Figure 7. Maximum strains measured in Span 1 and Span 2
2.2 Temperature evolution
during the quasi-static test.
Note that, at early age, since the chemical reactions
associated with cement hydration and SF reaction
6 shows
transversefield
strains
the
areFigure
exothermic,
the the
temperature
is notalong
uniform
crown
for differentsystems
magnitude
It is seen
for non-adiabatic
evenofif loading.
the environmental
temperature
is constant.
Heatareconduction
be
that
all the transverse
strains
tensile in can
nature.
The
maximum
value ofattransverse
tensile strainnot
is
described
in concrete,
least for temperature
about
17 microstrains
which corresponds
a tensile
exceeding
100°C (Bažant
& 2 Kaplan to1996),
by
stress
of about
0.00306
Fourier’s
law, which
readskg/mm . This value is less
than the codal permissible value [Code A, Rule A]
= − λ∇Tkg/mm2 in tension.
ofq 0.011
(7)
where q is the heat flux, T is the absolute
temperature, and λ is the heat conductivity; in this
5.2 Results of quasi-static moving load test
The influence line like study indicates that the maximum vertical displacement on the crown is about
0.88 mm and the maximum horizontal springing deflection is 0.1mm at abutment and 0.17 mm in the
central pier in span 1. In span 2 the maximum vertical deflection at the crown is 0.75 mm, while the
springing deflection in the abutment was 0.01mm
and -1.4mm / +1.2mm in the central pier due to a 25
t axle load. The negative value implies horizontal
movement of the central pier to the left (Kulem side,
Fig. 1) and the positive value represents horizontal
movement towards Castle Rock side (Fig. 1). It may
be mentioned here that the LVDT was mounted on
the repaired (encased) concrete at the springing
level. The large values of horizontal deflection suggests that delamination (separation) of the repaired
concrete from the parent masonry pier could have
occurred.
Figure 7 shows the maximum compressive and
tensile strains measured at different locations on the
bridge. The maximum value of strain is reckoned
across the measured responses taken for all forty
placements of the formation on the bridge.
6 FINITE ELEMENT ANALYSIS
In this section, details of finite element modeling of
the arch bridge under various loading conditions
have been presented. The analysis is done using the
finite element program ATENA (Cervenka & Pukl,
2007). The Atena software has been used to study
the bridge under monotonically increasing loads positioned as in rail load configurations with a view to
understand cracking in masonry and its propagation.
Furthermore, parametric studies have been undertaken by considering the tensile strength of brick
masonry and filler and the boundary conditions as
parameters.
The masonry arch bridge with a soil infill has
been idealized as composed of two isotropic homogeneous materials, namely, masonry and filler. A
two dimensional plane stress model of the bridge has
been used in this study. The finite element package
ATENA encompasses many material model formulations for quasi-brittle concrete like materials, such
as a bi-axial failure surface with different tension
and compression thresholds, post crack strain softening based on exponential and multi-linear softening,
specific fracture energy of the material, compression
softening in cracked concrete and other fracture
based parameters, such as crack interface shear
transfer.
A two-dimensional plane stress finite element
model for the masonry arch bridge is shown in Figure 8. The brick masonry has been assumed to have
= − D (and
h, T )∇ahPoisson’s ratio of 0.2.
a modulus of 1800J MPa
The soil infill has been idealized to have modulus of
800 MPa with a Poisson’s
ratio of 0.18. A
relatively D(h,T)
The proportionality
coefficient
small tensile strength
of
0.3
MPa
has
been
moisture permeability and assumed
it is a nonlinea
for the masonry with
similar value
of 0.3
MPa
for
of thea relative
humidity
h and
temperature
the infill. These values
are
obtained
through
an
itera& Najjar 1972). The moisture mass balanc
tive process of that
model
the calibration
variation inusing
time ofthethefield
water mas
measurements ofvolume
the static
load
–
deflection
andw) be eq
of concrete (water content
quasi-static moving
load studies.
boundary
divergence
of theThe
moisture
fluxconJ
dition on the vertical face of the abutment (left end
of span 2) is restrained
in the longitudinal traffic di− ∂w =of∇ the
• J abutments and central
rection and the base
∂t
pier have been constrained in the vertical direction.
The boundary at theThe
right
sidecontent
abutment
of be
span
1
water
w can
expressed
a
has been elastically
constrained
for
longitudinal
of the evaporable water we (capillary wa
movement using vapor,
linear and
springs
(the water)
value of
adsorbed
andthis
the non-e
spring constant is(chemically
reported later).bound) water w (Mil
n
Pantazopoulo & Mills 1995). It is reas
The following assume
studies have
undertaken:water is a fu
that been
the evaporable
relative humidity, h, degree of hydration
1.
Simulation
of static
load
– deflection
degree
of silica
fume
reaction, αtest
s, i.e. we=w
(BFR wagons
carrying
PSC
posi= age-dependent sleepers
sorption/desorption
tioned as(Norling
in the field
studies).
Mjonell
1997). Under this assum
2.
Simulation
of
quasi-static
moving load
test Equati
by substituting Equation
1 into
(25 t axle
load BOXNEL wagons arranged
obtains
3.
at critical position as in the field studies)
Parametric studies varying tensile strength
∂w
∂w
∂w ∂h
e α&
e α& + w
e +and
of brick −masonry
bound∇ • ( filler
= the
Dh ∇h)and
c +
s
∂
α
∂
α
∂
∂
h
t
ary condition of the abutment of span
c 1 ons
the Castle Rock end. It may be recalled
that a few
smaller
land
arches
of of
approxiispresent
the
slope
the sorption/
where
∂we/∂h
mately 7.7m
span
are
on
the
Castle capac
isotherm
(also
called
moisture
Rock end
of Span
1 that(Equation
are partially
governing
equation
3) must be
closed. by
Hence,
this
boundary
is and
simulated
appropriate
boundary
initial
conditi
by providing
springs
and
also
by
restrainThe relation between the amount of e
ing it completely.
water and relative humidity is called ‘‘
isotherm” if measured with increasing
and test
‘‘desorption isotherm” in th
6.1 Simulation humidity
of static load
case.
Neglecting their
difference
(Xi et al.
In this study, the the
twofollowing,
BFR wagons
with PSC
sleep-will be
‘‘sorption
isotherm”
ers (as detailed above)
are placed
on
the bridge
with
reference
to both
sorption
and
desorption
c
loads coinciding with
the
axle
position
as
in
the
field
By
the
way,
if
the
hysteresis
of
the
test.
isotherm would
be taken
into account,
two
The vertical deflection
obtained atwater
the crown
of humi
relation,
evaporable
vs
relative
span 1 is 0.55 mm
as against
a measured
value
of varia
be with
used
according
to theThe
sign
of the
0.57 mm for BFR
200
sleepers.
correrelativity
humidity.
The vertical
shape of the
sponding value of
computed
and
measured
isotherm
for HPC
is0.54
influenced
by
many p
deflection at the crown
of span
2 isthat
mm andextent
0.6 and
especially
those
influence
mm. This indicates
that the
materialand,
properties
of determ
chemical
reactions
inforturn,
modulus of elasticity
and
Poisson’s
ratio
used
the
structure
and analysis
pore sizeare
distribution
(watermasonry and the ratio,
infill in
the
in
agreecement chemical
composition,
SF
ment with the curing
deformations
observed
at field, mix
time
and
method,
temperature,
thereby calibrating
the material
model. various formulatio
etc.).
In the literature
found to describe the sorption isotherm
(Xi etmoving
al. 1994).
in th
6.2 Simulation concrete
of quasi-static
load However,
test
paper
the
semi-empirical
expression
pro
In this study, the Norling
entire rake
used for(1997)
this testisas adopted
deMjornell
b
scribed above is used for applying the load on the
Proceedings of FraMCoS-7, May 23-28, 2010
J = − D (in
h, Ta )sequential
∇h
bridge
way. One particular position
(1)
of the rake on the bridge which causes maximum effects
been simulated.coefficient
Figures 8 D(h,T)
and 9, show
the
Thehasproportionality
is called
finite
element
model
and
load
positions,
respecmoisture permeability and it is a nonlinear function
tively.
of the relative humidity h and temperature T (Bažant
& Najjar 1972). The moisture mass balance requires
that the variation in time of the water mass per unit
volume of concrete (water content w) be equal to the
divergence of the moisture flux J
− ∂ = ∇•J
∂
(2)
w
t
The water content w can be expressed as the sum
of the evaporable water we (capillary water, water
vapor,8.and
water)
the non-evaporable
Figure
Finiteadsorbed
element model
withand
boundary
conditions.
(chemically bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
Figure
obtains9. Loads on the finite element model.
∂w
∂w ∂h
e α of∂wthis
e α simulation(3)
e +2∇shows
Table
at
) = results
• ( D ∇hthe
c
s
n
h
∂
α
∂
α
∂
∂
h
t
various points together with the field results for the
−
c
&+
s
& + w&
loading configuration as in Figure 9. This loading
configuration
onesorption/desorption
of an episode of
is the slope ofto the
where ∂we/∂h corresponds
the
quasi-static
load
test
that
results
in the highest
isotherm (also called moisture capacity).
The
crown
vertical
displacement.
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
Table
2. Results
of finite
element
The
relation
between
theanalysis.
amount of evaporable
Description
Span 1
2
water and relative humidity is calledSpan
‘‘adsorption
isotherm”
if measured with increasing relativity
Crown displacements
humidity
and
‘‘desorption0.64
isotherm” in0.3the opposite
Computed (mm)
0.68
Measured
(mm)
case. Neglecting their difference (Xi et0.2al. 1994), in
strains ‘‘sorption isotherm” will be used with
theCrown
following,
1.26conditions.
Computed
reference
to(microstrain)
both sorption 10.7
and desorption
8.35
Measured (microstrain)
By the way, if the hysteresis of Not
the measmoisture
ured
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
6.3
Parametric
be used
accordingstudy
to the sign of the variation of the
relativity
humidity.
Thetheshape
of configuration
the sorption
In the parametric study,
loading
isotherm
forquasi-static
HPC is influenced
manyinparameters,
used
in the
test and by
shown
Figure 4.3
especially
those
that
influence
extent
the
is considered. The tensile strength of and
the rate
brickofmachemical
reactions
and,
in
turn,
determine
pore
sonry and the filler and the boundary condition at the
structure and
pore1size
abutment
of span
has distribution
been varied (water-to-cement
to study their inratio,
cement
chemical
composition,
SFmode.
content,
fluence on the crack widths and the failure
curing time and method, temperature, mix additives,
etc.).
In the Analysis
literature various formulations can be
No
Cracking
found to describe the sorption isotherm of normal
concrete
et al. analyses
1994). However,
present
The
finite(Xi
element
are doneinbythe
assuming
paper
the
semi-empirical
expression
proposed
by
higher values of tensile strength of the brick maNorling
Mjornell
(1997)
is
adopted
because
sonry and the filler material in such a way that noit
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly
the the
evolution
of due
hydration
cracks
are accounts
developedfor
when
axle load
to the
reaction and
SF to
content.
Thist. sorption
BOXNEL
reaches
about 105
The load isotherm
configureads used is the same as shown in Figure 9. In the
ration
analysis the axle loads on the BOXNEL wagons are
increased in steps of 5 t from
an initial value⎤ of 25 t.
⎡
The corresponding loads⎢ of the 1681 sleepers⎥ loaded
(α , α ) 1 −
we (h, αwagons
+ The
⎥
c , α s ) = Gare
cscaled
s ⎢ up10(proportionately.
BRN
1
∞
g1α c − α c )h ⎥
⎢
boundary of the abutment
1 on the
e span
⎣ of
⎦ Castle
(4)
Rock end (Fig. 1) is assumed to be on springs. The
⎡ 10(g α ∞ − α )h
⎤
spring constant (1300 N/mm)
is calibrated
such
c
1 c
⎢
K1 (α25
− 1⎥ that
for an axle load of
the
c , αt son)⎢ethe BOXNEL wagons,
⎥
⎦
deflection at the crown of⎣ spans 1 and 2 match
with
the field measurements.
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
Figure
Maximum
Principal pores
Stress and
(without
self weight)
fill all 10.
pores
(both capillary
gel pores),
one
versus Axle Load at Crown of Span 1.
can calculate K1 as one obtains
w
0
K (α c α s ) =
,
1
⎡
⎢
−
⎢1 −
1
⎢
⎣
α s + 0.22α s G
− 0.188
c
s
⎛
10⎜
⎝
e
⎛
10⎜
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
g α c∞ − α c ⎞⎟h ⎤⎥
1
⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that,
at early
age, since
the of
chemical
reactions
Figure
11. Vertical
Deflection
at Crown
Span 1 versus
Axle
associated with cement hydration and SF reaction
Load.
are exothermic, the temperature field is not uniform
forFigure
non-adiabatic
systems
even
the environmental
10 shows
the plot
of ifmaximum
principal
temperature
is constant.
conduction
be
stress
at the crown
of SpanHeat
1 with
respect to can
the apdescribed
concrete,
at least
not
plied
axle in
load.
From this
figure,forit temperature
is seen that the
exceeding
100°C (Bažant
& stress
Kaplanis only
1996),about
by
rise
in the maximum
principal
Fourier’s
law,anwhich
readsof 100 t axle load which is
0.1
MPa for
increase
quite small. Furthermore, at 25 t axle load and withq = the
− λ∇Tself weight, the maximum principal stress
out
(7)
near the crown is 0.0245 MPa (0.16 MPa with self
weight).
was observed
load
where qSince
is no
the cracking
heat flux,
T is theat this
absolute
in
the
field
studies,
it
is
expected
that
the
minimum
temperature, and λ is the heat conductivity; in this
tensile strength of the masonry arch is over
0.16MPa.
Figure 11 shows the vertical deflection at the
crown of span 1 with the self weight and increasing
live load. The live load corresponding to a relative
displacement of 1.2 mm (Code A) caused by live
load alone is 45 t. This corresponds to a margin of
safety at 25 t equal to 1.8. This margin of safety is
over and above the margin already provided by the
code. For completeness, the vertical crown deflection of Span 2 are also computed and are found to be
lower than those in Span 1 since this part of the span
is subjected to loads lower than those in Span 1.
J = −The
D (h, Thorizontal
) ∇h
by the code itself.
deflection observed at other springing (Figs. 13 – 15) is less than
0.2 mm for an axle load
105 t.
The ofproportionality
coefficient D(h,T)
moisture permeability and it is a nonlinea
of the relative humidity h and temperature
& Najjar 1972). The moisture mass balanc
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J
− ∂ = ∇•J
∂
w
t
The water content w can be expressed a
of the evaporable water we (capillary wa
vapor, and adsorbed water) and the non-e
(chemically bound) water wn (Mil
Pantazopoulo & Mills 1995). It is reas
Figure 14. Horizontal Deflection at Springing of Span 2 (CRassume that the evaporable water is a fu
end) versus Axle Load.
relative humidity, h, degree of hydration
degree of silica fume reaction, αs, i.e. we=w
= age-dependent sorption/desorption
(Norling Mjonell 1997). Under this assum
by substituting Equation 1 into Equati
obtains
−
Figure 12. Horizontal Deflection at Springing of Span 1
(CR-end) versus Axle Load.
Figure 13. Horizontal Deflection at Springing of Span 1
(K-end) versus Axle Load.
Figures 12 to 15 show the horizontal deflection at
the springing of abutment and piers of span 1 and
span 2 respectively with increasing axle load.
From Figure 12 it is seen that for a relative displacement (i.e., due to live load alone) of 0.2 mm
(Code A) at the springing the corresponding live
load is 65 t. This offers a margin of safety at 25 t
axle load of 2.6 over and above the margin offered
∂w ∂h
e + ∇ • ( D ∇h) = ∂we
h
∂α
∂h ∂t
∂w
α&c + e α&s + w
∂α
c
s
where ∂we/∂h is the slope of the sorption/
isotherm (also called moisture capac
governing equation (Equation 3) must be
by appropriate boundary and initial conditi
The relation between the amount of e
water
and relative
humidity
called ‘‘
Figure 15. Horizontal Deflection
at Springing
of Span is
2 (Kisotherm”
if
measured
with
increasing
end) versus Axle Load.
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
Cracking analysisthe following, ‘‘sorption isotherm” will be
reference to both sorption and desorption c
By isthe
way,outifto the
hysteresis
A cracking analysis
carried
determine
if theof the
isotherm
be taken
account, two
failure of the arch
bridge would
takes place
due into
to excesevaporable
water
vs mode.
relative humi
sive cracking nearrelation,
the crown
or by any
other
be of
used
according
signMPa
of the
The tensile strength
brick
masonrytoofthe
0.20
is varia
relativity
humidity.
The
shape
used and the analysis is carried out by applying the of the
for HPC
is influenced
by many p
self weight and isotherm
incrementally
increasing
the axle
especially
thoseinthat
influence
loads on the BOXNEL
wagons
steps
of 5 t. extent
The and
chemical
reactions
in BRN
turn, determ
corresponding loads
of the 168
sleepersand,
loaded
structure
and
pore
size
distribution
wagons are scaled up proportionately. It was found (waterratio,to crack
cementat an
chemical
composition,
SF
that the crown starts
axle load
of 15 t.
curingcontinues
time andand
method,
The cracking process
at antemperature,
axle load mix
In the
the entire
literature
variousof formulatio
of 40 t the crack etc.).
reaches
thickness
the
found
to
describe
the
sorption
isotherm
brick arch. The crack widths from this incremental
(Xiaxle
et al.load
1994).
However,
analysis is plottedconcrete
against the
in Figure
16. in th
semi-empirical
pro
It is seen that thepaper
crackthe
widths
increase at expression
a faster
Norling
Mjornell
is adopted
b
rate after 40t axle
load that
is when(1997)
the crack
has
Proceedings of FraMCoS-7, May 23-28, 2010
J = − D ( h , T )∇
h
propagated
through
the entire thickness of the brick
(1)
arch. Thereafter, on increasing the axle load the
crack
increases without
the crack
propagating
Thewidth
proportionality
coefficient
D(h,T)
is called
further.
The
arch
softens
considerable
duefunction
to inmoisture permeability and it is a nonlinear
creased
opening
of the crack.
The final failure
may
of the relative
humidity
h and temperature
T (Bažant
take
place
due
to
cracking
at
the
crown
of
the
arch.
& Najjar 1972). The moisture mass balance requires
that the variation in time of the water mass per unit
volume
100 of concrete (water content w) be equal to the
divergence
of the moisture flux J
90
80
70
w
60
t 50
40
30
20
10
0
Load (t)
− ∂ = ∇•J
∂
(2)
The water content w can be expressed as the sum
of the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable
(Mills
(chemically
w0.5
n
0
0.1 bound)
0.2
0.3water
0.4
0.6
0.71966,
Pantazopoulo & Mills
1995).
It
is
reasonable
to
Crack Width (mm)
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
Figure
axle load
crown
we=ofwSpan
degree16.ofCrack
silicawidth
fumeversus
reaction,
αs,ati.e.
e(h,α1.
c,αs)
= age-dependent sorption/desorption isotherm
(Norling
Mjonell
6.4
Analysis
with 1997).
tractiveUnder
forcesthis
due assumption
to braking and
by substituting Equation 1 into Equation 2 one
A
longitudinal tractive force analysis is carried out
obtains
by including horizontal forces at the rail level at
wheel
equal to ∂about
on an aver∂w position
we 35∂twbased
e α& + w&a tensile
e ∂h +measured
age
For
this
analysis,
&
α
+
) =
∇ • ( D ∇hvalue.
− field
(3)
c
s
n
h is∂αassumed
∂αfor both
∂h ∂t of 0.2 MPa
strength
the brick
c
s
masonry and the filler material. From this analysis, it
was
observed
at the crown of
is thecracking
slope ofinitiated
the sorption/desorption
where
∂we/∂h that
Span
1
at
a
live
load
of
65
t,
offering
a marginThe
of
isotherm (also called moisture capacity).
safety
of
2.6
with
respect
to
a
25
t
axle
load.
Furgoverning equation (Equation 3) must be completed
thermore,
no cracks
wereand
observed
at the springing
by appropriate
boundary
initial conditions.
of the
and abutments,
suggesting
thatevaporable
the hingThepier
relation
between the
amount of
ing
mode
of
failure
is
unlikely.
It
may
be noted
water and relative humidity is called ‘‘adsorption
here
that
the
failure
of
the
arch
through
the
formaisotherm” if measured with increasing relativity
tion
of
hinge
near
the
crown
can
take
place
if
there
is
humidity and ‘‘desorption isotherm” in the opposite
excessive
horizontal
case. Neglecting
theirmovement
difference at(Xitheet springing
al. 1994), of
in
the
abutment.
the following, ‘‘sorption isotherm” will be used with
reference to both sorption and desorption conditions.
By the way, if the hysteresis of the moisture
7isotherm
ESTIMATION
FATIGUE
LIFE two different
would beOF
taken
into account,
relation, evaporable water vs relative humidity, must
The
bridge
structure
to repeated
be used
according
to isthesubjected
sign of the
variationfluctuof the
ating
loads
due
to
the
passage
of
trains.
Thesorption
loss of
relativity humidity. The shape of the
mechanical
of the structure
dueparameters,
to this reisotherm forintegrity
HPC is influenced
by many
peated
fluctuating
load
is
known
as
fatigue.
especially those that influence extent and rateA ofcomthe
ponent
fails atand,
a high
load, maypore
fail
chemicalwhich
reactions
in constant
turn, determine
under
a substantially
fatigue(water-to-cement
load. The procstructure
and pore sizesmaller
distribution
ess
of fatigue
the masonry
arch bridge SF
maycontent,
lead to
ratio,
cementinchemical
composition,
excessive
deformations,
formation
of
tension
zones
curing time and method, temperature, mix additives,
at
the
crown
region
of
the
arch
and
cracking
of
etc.). In the literature various formulations canmorbe
tar
in the
found
to masonry.
describe the sorption isotherm of normal
It may(Xi
be etmentioned
that a insmall
tensile
concrete
al. 1994).here
However,
the present
stress
is
observed
in
the
static
analysis
of
the
paper the semi-empirical expression proposedarch
by
bridge.
may not(1997)
be the is
caseadopted
when thebecause
bridge isit
NorlingThis
Mjornell
newly constructed and is at its beginning of service
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly
the evolution
of hydration
life.
Due toaccounts
repeatedfor
loading
over a period
of time
reaction120and
SF stiffness
content. degradation
This sorption
(about
years),
and isotherm
damage
reads
accumulation
could have taken place and hence the
major principal stress near the crown region is inferred to be tensile.
⎡
⎤
1
Empirical relations are
especially
⎢ developed,
⎥
we (h, α cthat
, α ) = G (α , α ) 1 −
+ for
⎢
⎥value of
s relate
cthe sapplied
metals,
stress
(peak
1
∞
10(g α
− α c )h ⎥
1 c of cycles
the fluctuating load) and⎢⎣ theenumber
re⎦ N (4)
quired to cause the failure. This relation is generally
⎡ 10(g α ∞ − α )h
⎤
known as the S – N curve.
The
majorc drawback
of
1 c
⎢
K (α c , α s ) eis that it does not
− 1⎥explore
the S – N curve 1approach
⎢
⎥
⎦
the mechanisms of failure⎣ and it does not distinguish
between crack initiation life and crack propagation
whereonly
thethefirst
term fatigue
(gel isotherm)
representsFurthe
life,
overall
life is considered.
physicallyinbound
(adsorbed)there
water
and theanysecond
thermore,
this approach,
is hardly
conterm (capillary
represents
the capillary
sideration
on sizeisotherm)
effects: that
is, data generated
on
water.
This
expression
is
valid
only
for
low
content
small size specimen in a laboratory is directly apthe data
amount
of SF.onThe
G1 represents
plied
largecoefficient
size components.
Also, the
on S of
–
water
per has
unit avolume
held insuggesting
the gel pores
N
curve
large scatter
that at
the100%
forrelative humidity,
it canrigorous.
be expressed
(Norling
mulation
needs to and
be more
To overcome
Mjornell
1997)
as
these drawbacks of the S – N curve approach, more
sophisticated models are developed using the concepts
fracture
c αmechanics.
s α s In this theory of fracture
G1 (α cof
,α ) = k
(5)
s a vgcrack
c c +isk vgassumed
s
mechanics,
to exist at the position where maximum tensile stress occurs. The rate
of
propagation
withparameters.
respect to the
numksvgthis
arecrack
material
From
the
where
kcvg and of
ber
of
cycles
of
fatigue
load
is
computed
and
this
demaximum amount of water per unit volume that can
fines
fatigue
of the component
/ structure.
fill allthepores
(bothlifecapillary
pores and gel
pores), one
can calculate K1 as one obtains
The simplest crack propagation law is the Paris law
which is defined by [Kumar 1999]
⎡
⎛
⎞ ⎤
∞
⎜ g α − α ⎟h ⎥
⎢
c
c
⎝
⎠
w −
α s + α s −G ⎢ −e
⎥
c
s
⎥
⎢
da
m
⎦ (6)
⎣
(∆ K )
(1)
)=
KdN(α c =α C
s
⎛
⎞
∞
−
α
α
g
h
⎜
⎟
e ⎝ c c⎠ −
where a is the crack length, N is the number of cyclesThe
of fatigue
(∆K) is the
factor
and ksintensity
materialload,
parameters
kcvgstress
vg and g
1 can
range,
C and mbyare
material
constants.data relevant to
be calibrated
fitting
experimental
10
0
0.188
0.22
1
1
1
,
1
10
1
1
free (evaporable) water content in concrete at
The fatigue
estimation
for 2009b).
the arch bridge is
various
ages (DilifeLuzio
& Cusatis
done using the above Paris law. The material parameters for mortar used in the present case for ma2.2 Temperature
sonry
are assumedevolution
as C = 1.70E-03 m/cycle and m =
2.1.
Note that, at early age, since the chemical reactions
associated with cement hydration and SF reaction
stress intensity
factor range
as
areThe
exothermic,
the temperature
fieldisiscomputed
not uniform
[Kumar
1999]
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
described
at least for temperature (2)
not
∆K = f ( β )in
∆σconcrete,
πa
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s∆law,
where
the reads
stress amplitude range =
σ iswhich
σ max − σ min . This is considered to be equal to 0.06
q = − λ ∇T
(7)
MPa
which is the difference of the maximum and
minimum principal stress as obtained in the analysis
β ) isabsolute
the geusing
wheretheq 25is t axle
the load.
heat The
flux,factor
T is f (the
ometry
factor and
= 1.3λ for
circular
An initialincrack
temperature,
is the
heat arch.
conductivity;
this
size of 0.1 mm is assumed at the crown of the arch.
Numerical integration is performed on the Paris
law and the number of cycles required for an initial
crack of size 0.1 mm to propagate until the crack
grows to a size of 10 mm is obtained as 242.6E+03
cycles. Assuming each cycle of loading to be a passage of train (54 Wagons + 7 WDG4) loaded to 25 t
per axle, and assuming ten trains running on this
bridge per day, it takes about 66 years for a crack of
size 0.1mm to reach a size of 10 mm.
It may be noted that a crack initiated at the crown
of the arch may propagate until half the ring thickness of the brick masonry which is 465 mm. Thereafter, this crack does not propagate further as due to
stress redistribution, the nature of stress would be
compressive. A compressive stress near a crack tip
tends to close a crack and prevent further propagation. Hence, it is unlikely that the arch ring may collapse due to crack propagation. The cracking process
may only result in excessive displacements and so
these cracks have to be sealed during the regular
maintenance program.
8 CONCLUSIONS
Based on the investigations, the following conclusions are made:
(1)
From Field Measurements
• Under known loads (load deformation and quasi
static load tests) the structural load deformational behavior is within the elastic regime based
on the full recovery of the strains observed in the
test. The stiffness properties used in the analytical simulations have been obtained from these
tests.
• In the field measurements, it was observed that
there were no visible cracks observed in the
brick masonry under the applied axle load of 25t.
• The maximum stresses in the static and quasistatic tests are within the code permissible limits.
• The maximum displacements observed in the
quasi-static tests indicate that they are within
code permissible limits except for the springing
displacement of the pier in Span 2. This may be
due to the delamination caused between the outer
concrete repair and the inner masonry pier.
(2)
•
•
From Finite Element Analysis:
Based on the IRS codal provisions on the displacements for Arch bridges, a margin of safety
of 2.6 on crown displacements for a 25 t axle
load is obtained over and above the margin
provided by the code.
Based on the IRS codal provisions on the displacements for Arch bridges, a margin of safety
•
(3)
•
J = − D ( h, T
) ∇h
of 2.6 on springing
displacements
for a 25 t
axle load is obtained over and above the margin
provided by theThe
code.proportionality coefficient D(h,T)
The above margin
safety valuesand
drop
moistureof permeability
it to
is a2.6nonlinea
for 25 t axleofload
over and
above hthe
the relative
humidity
andmargin
temperature
provided by &
theNajjar
code under
the
presence
of
lon- balanc
1972). The moisture mass
gitudinal tractive
forces
(42
t)
due
to
braking.
that the variation in time of the water mas
volume of concrete (water content w) be eq
From Fatigue
Analysis:of the moisture flux J
divergence
The estimated ∂number
of years for a crack at
w
=0.1
∇ •mm
J to grow to 10 mm asthe crown of−size
∂t
suming ten trains (54 wagons+7 WDG4) per
day loaded to 25
t /axle
on thew bridge
66
The
waterpass
content
can beisexpressed
a
years.
of the evaporable water w (capillary wa
e
vapor, and adsorbed water) and the non-e
(chemically bound) water wn (Mil
9 ACKNOWLEDGEMENT
Pantazopoulo & Mills 1995). It is reas
assume that the evaporable water is a fu
The financial support
provided
by theh,South
Western
relative
humidity,
degree
of hydration
Railway of the degree
Indian of
Railways
is gratefully
, i.e. we=w
silica fume
reaction, αsacknowledged.
= age-dependent sorption/desorption
(Norling Mjonell 1997). Under this assum
by substituting Equation 1 into Equati
REFERENCES obtains
Cervenka V. & Pukl R., 2007, ATENA program Documenta∂w
∂w ∂hCzech Republic, Praha.
tion, Cervenka Consulting,
e α ∂we α w
( D ∇and
) = construction
∇ •design
− e for +the
h
Code A: Code of practice
of s
c
h
∂α
∂α
∂h ∂t
masonry and plain concrete arch bridges (Arch
c Bridge
s
Code), RDSO, Lucknow, India.
Kumar P, 1999.Elements
of Fracture
Wheeler
is the slope
of thePubsorption/
where
∂we/∂h Mechanics,
lishing, New Delhi.
isotherm
(also
called
moisture
Rule A: Rules specifying the loads for design of superstructure capac
governing
equation
3) must be
and sub-structure
of bridges
(Bridge(Equation
Rules), RDSO,
Lucknow, India. by appropriate boundary and initial conditi
&+
&+
The relation between the amount of e
water and relative humidity is called ‘‘
isotherm” if measured with increasing
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
the following, ‘‘sorption isotherm” will be
reference to both sorption and desorption c
By the way, if the hysteresis of the
isotherm would be taken into account, two
relation, evaporable water vs relative humi
be used according to the sign of the varia
relativity humidity. The shape of the
isotherm for HPC is influenced by many p
especially those that influence extent and
chemical reactions and, in turn, determ
structure and pore size distribution (waterratio, cement chemical composition, SF
curing time and method, temperature, mix
etc.). In the literature various formulatio
found to describe the sorption isotherm
concrete (Xi et al. 1994). However, in th
paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b
Proceedings of FraMCoS-7, May 23-28, 2010