1-s2.0-S0168874X10000685-main

A finite element model for hygro-thermo-mechanical analysis of masonry
walls with FRP reinforcement
Mehran Khoshbakht , Mark W. Lin
University of Alabama in Huntsville, Department of Mechanical and Aerospace Engineering, Huntsville, AL 35899, USA
article info
Article history:
Received 24 May 2007
Received in revised form
8 October 2009
Accepted 24 April 2010
Keywords:
Moisture and temperature
Structural analysis
FRP reinforced masonry
Multi-layered permeable structure
abstract
Modeling the effects of humidity and temperature gradients on the structural behavior of masonry
walls reinforced with fiber reinforced polymer (FRP) composite is of great importance. Study of
interfacial stresses, in particular, is a key factor in predicting the durability of the bond between the
reinforcing FRP laminate and the host masonry. In this paper, a finite element modeling procedure for
analyzing the hygro-thermo-mechanical response of multi-layered structures constructed with
distinctive permeable materials was developed by incorporating structural stress analysis into the
coupled moisture/temperature finite element model based on the governing equations proposed by
Phillips and De Vries. The hygro-thermo-mechanical finite element model was used to analyze the
response of a concrete block reinforced with a unidirectional glass/epoxy FRP composite laminate. The
results demonstrated that the effect of temperature gradient on the moisture distribution, i.e., heat-
induced moisture movement, resulted in an accumulation of moisture at the interface, and induced
interfacial stresses even in the absence of a moisture gradient. The presented finite element modeling
procedure can be used to aid with the design of FRP/masonry structure or other similar structures for
minimizing interfacial stresses induced due to the mismatch of moisture swelling and thermal
expansion properties of the constituent materials.
&2010 Elsevier B.V. All rights reserved.
1. Introduction
Many of aging buildings were constructed using masonry
structures. Most of the existing masonry structures, except a few,
were not reinforced to meet modern engineering codes. Masonry
structures without any reinforcement have inadequate lateral
strength due to the presence of a matrix of adhesive joints of
mortar. Structures constructed with un-reinforced masonry,
therefore, are prone to failure when subject to lateral loading
such as high winds and earthquake events. Among the con-
temporary methods for strengthening un-reinforced masonry
structures, using fiber reinforced polymer (FRP) composites has
been shown to be an effective and efficient technique [1,2]. Fiber
reinforced polymer composites offer a combination of outstand-
ing properties that are very suitable for masonry wall upgrade
applications. They have low specific weight and excellent
mechanical stiffness and strength. They are also immune to
corrosion and have the ability to be formed in different shapes
and very long lengths.
Though many studies have demonstrated the effectiveness of
the FRP reinforcement for masonry structures [3,4], one important
issue that hinders its broad application is the understanding of
environmental temperature and moisture effects on the FRP
reinforcement and the FRP/masonry interface, in particular. The
day-to-day climate changes that these structures have to sustain
for several decades can result in substantial damaging effects on
the structural integrity, especially at the interface between the
masonry substrate and the FRP. The issue of thermo-mechanical
behavior of the building structures is well known. However, the
effect of moisture swelling mismatches between the masonry and
FRP materials, which can induce considerably high level of
interfacial stresses, has not been thoroughly investigated. This
effect may result in disintegration of the FRP/masonry interfacial
bond and damages in both masonry and FRP constituents [5,6].
Since the early 1990s, research on various aspects of using FRP
composites as upgrade materials on un-reinforced masonry
structures has received much attention [7–12]. To determine
the effect of the environment on the mechanical behavior of such
structures, it is necessary to derive a mathematical model for the
moisture diffusion and heat transfer phenomenon in multi-
layered porous structures. The most accepted governing equa-
tions capable of describing hygro-thermal problem for a porous
solid were derived by Philip and De Vries [13] and Luikuv [14].
Their mathematical models include two coupled partial differ-
ential equations for moisture content and temperature dependent
variables, and are applicable to a single-material structure.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/finel
Finite Elements in Analysis and Design
0168-874X/$ - see front matter &2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.finel.2010.04.002
Corresponding author. Tel.: [phone].
E-mail address: [email protected] (M. Khoshbakht).
Finite Elements in Analysis and Design 46 (2010) 783–791

These equations have been widely used as the basis for many
studies in this area with various simplifications according to the
specific physics and geometry of the problem investigated. Some
studies have been done to find the analytical solutions for the
coupled moisture/temperature problem in porous media with
regular geometries [15–17]. In all these studies, the material
transport properties were considered constant in order to obtain
closed-form analytical solutions, even for structures with simple
geometries. Other researchers have alternatively used various
numerical techniques to solve the coupled hygrothermal problem
[18–20]. Several published works are also available regarding the
analysis of the mechanical behavior of the porous structures
caused by the moisture variations through the medium [21–24].
These studies, however, have limitations such as specific simple
geometry, constant material properties, monolithic material, non-
permeability of part of the structure, or one-dimensional moisture
migration analysis.
In spite of these valuable studies, to date, there is little
research that addresses the three-dimensional hygro-thermo-
mechanical problem of a multi-layered structure constructed of
distinctive permeable materials with general moisture/tempera-
ture dependent material properties. Due to the complexity of the
governing equation for the hygro-thermo-mechanical problem as
well as the geometry of these structures, obtaining a closed-form
analytical solution is very difficult, if possible. Simplification of
the equations and the geometry would result in unrealistic
solutions with little or no practical value. On the other hand,
experimental study of the long-term environmental effects on
such structures would be very time consuming and expensive.
Using finite element technique to carry out comprehensive
studies for the long-term durability problem of FRP composites
reinforced masonry structures is a practical alternative.
In this paper, a finite element model, previously presented by
the authors to analyze the hygrothermal behavior of an FPR/
concrete wall structure [25] is augmented to develop a compre-
hensive model for hygro-thermo-mechanical analysis of this type
of structures. The structural analysis formulation is incorporated
into the hygrothermal model to determine the effects of moisture
swelling and thermal expansion on the mechanical response of
the wall structure. The developed hygro-thermo-mechanical
model is then implemented to analyze the behavior of a typical
concrete slab with FRP laminate reinforcement under various
temperature and humidity gradient loads. The phenomenon of
heat-induced moisture movement is particularly investigated to
evaluate its resulting interfacial stresses at the FRP/concrete
bonding surface.
2. Finite element implementation of the hygrothermal
problem
The coupled moisture migration and heat transport problem in
porous media was formulated by Philip and De Vries [13] in the
form of the following system of differential equations:
@ y
@t¼
r ðDy ry Þþ r ðD
T
r TÞ,ð1Þ
r c@T
@t¼
r ð lr TÞþh
v
r ðDy
vap
ry Þ,ð2Þ
where
y is the moisture content (kg/m
3
), Tis the temperature (K),
D
y
is the moisture diffusion coefficient (m
2
/s), D
T
is the thermal
moisture diffusion coefficient (kg/(m s K)),
r cis the volumetric
heat capacity (J/(m
3
K)), l is the thermal conductivity (W/(m K)),
h
v
is the latent heat of vaporization (J/kg), Dy
vap
is the isothermal
vapor diffusivity (m
2
/s).
The transport coefficients are to be determined experimen-
tally. Among these coefficients, there is a lack of vigorous
definition for the coefficient of thermal moisture diffusion, D
T
.
Therefore, American Society for Testing and Materials (ASTM) in
2001 suggested a modified version for this system of equations as
the standard tool for moisture analysis in building envelopes [26].
In the proposed mathematical model, the governing equations
take the modified form as
@ y
@t¼
r ðDy ry Þþ r
d
P
mr
ðhP
s
Þ ,ð3Þ
r c@T
@t¼
r ð lr TÞþh
v
r
d
P
mr
ðhP
s
Þ ,ð4Þ
where his the relative humidity of the surrounding air
(dimensionless),
m is the vapor resistivity of the porous medium
(dimensionless),
d
P
is the vapor permeability of the air (kg/
(m s Pa)), P
s
¼P
0
exp½ð D H=RÞðð1=TÞð1=373:15ÞÞ is the saturation
vapor pressure (Pa), in which P
0
¼101,325 (Pa) is the pressure of a
standard atmosphere, and
D H/R¼5201 (K) measured for liquid
water.
For a multi-layered structure, since the moisture content is
discontinuous at the interface of different layers of constituent
materials, a change of variable approach was taken by Lin et al.
[27] to substitute moisture content,
y , with relative humidity, h,
which resulted in the following form for the moisture diffusion
equation:
C
h
@h
@t¼
r ðD
h
r hÞþ r ðD
T
r TÞ,ð5Þ
where
C
h
¼@
y
@h,ð6Þ
D
h
¼@
y
@hD
h
þ
d
P
m
P
s
,ð7Þ
D
T
¼
d
p
m
D H
RT
2
hP
s
,ð8Þ
in which D
h
is the moisture diffusion coefficient of the material
expressed in terms of relative humidity using sorption isotherm
relation.
The second term in the heat transfer equation, Eq. (4), which is
associated with latent heat was shown to be insignificant for the
temperature range considered and, therefore, was omitted. As the
result, the change of variable did not affect this equation. Thus,
the heat equation yields a simple form as
r c@T
@t¼
r :ð lr TÞ,ð9Þ
The values or functional forms of the various coefficients in the
governing equations as well as sorption isotherm relation for
typical concrete and FRP materials have been experimentally
characterized and collectively summarized [28]. Applying the
Galerkin finite element formulation to the coupled humidity and
heat transport equations yields the following finite element
equation in terms of nodal humidty/temperature vector
h
i
T
i
()
ði¼1,2,...,rÞ, with rbeing the total number of nodes
per element,
½C
e
@h
i
@t
@T
i
@t
8
>
>
<
>
>
:
9
>
>
=
>
>
;
þ½D
e
h
i
T
i
()
¼fQ
e
g,ð10Þ
M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791784

where ½D
e
¼R
V
½B
T
½D½BdV is the element humidity and heat
diffusion matrix, ½D¼
D
hx
00D
Tx
00
0D
hy
00D
Ty
0
00
D
hz
00D
Tz
000 l
x
00
0000
l
y
0
00000
l
z
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
is the
material humidity and heat diffusion matrix, [B]¼LN
T
, in which
fNðx,y,zÞg ¼ N
1
0N
2
0N
r
0
0N
1
0N
2
 0N
r
()
T
is the element
shape function,
Lfg¼ @
@x
@
@y
@
@z
@
@x
@
@y
@
@z
the differential operator,
½C
e
¼R
V
C
h
0
0
r c
"#
fNgfNg
T
dV is the element humidity and heat
capacity matrix, fQ
e
g¼R
S
fNgqdS is the element boundary
humidity and heat flux, q
n
is the boundary humidity/heat flux.
3. Hygro-thermo-mechanical finite element model
To date, none of the commercial finite element software
packages has an available capability to model coupled structural/
moisture diffusion/heat transfer problem in a porous medium.
Therefore, in order to model the hygro-thermo-mechanical
problem using finite element technique, moisture-induced strain
and thermal strain need to be incorporated into the stress–strain
constitutive relation of the material in the finite element code.
Since hygro-thermal induced strains of most masonry materials
and FRP are relatively small and exhibit no permanent deforma-
tions, the linear elastic stress–strain relation is applicable. It is
also noted that stress-induced effects on thermal conductivity and
moisture diffusivity of the materials are not taken into considera-
tion in the current study, since such effects appear to be negligible
for the stress levels considered.
The linear elastic constitutive equation including the moisture
swelling and thermal strains can be expressed as
f
s g¼½Df e
elastic
g¼½Df e
total
e
swelling
e
thermal
g,ð11Þ
where f
s g¼f s
x
s
y
s
z
t
xy
t
yz
t
xz
g
T
is the stress vector, [D] is the
stiffness matrix of the material,
e
total
is the total apparent strain,
f
e
elastic
g¼f e
x
e
y
e
z
g
xy
g
yz
g
xz
g
T
elastic
is the elastic strain vector,
e
swelling
is the moisture-induced strain, and e
thermal
is the thermal
strain. For an orthotropic material, such as a unidirectional FRP
composite laminate, the moisture and thermal induced strain
vectors have the form
f
e
swelling
g¼
b
1
ð y y
ref
Þ b
2
ð y y
ref
Þ b
3
ð y y
ref
Þ000
no
T
,
ð12Þ
f
e
thermal
g¼
a
1
ðTT
ref
Þ a
2
ðTT
ref
Þ a
3
ðTT
ref
Þ000
no
T
,
ð13Þ
where
b
i
,i¼1, 2, 3 are moisture swelling coefficients in the
principle material directions,
a
i
,i¼1, 2, 3 are the thermal
expansions coefficients in the principle material directions, and
y
ref
and T
ref
are the reference moisture content and the reference
temperature, at which the moisture and thermal induced strains
are zero, respectively. Apparently,
b
i
has the same magnitude in
all directions for an isotropic material, such as masonry. Likewise
a
i
also has the same magnitude in all directions for isotropic
masonry materials.
The moisture swelling coefficient for a porous medium is
generally measured in terms of moisture content. Since the
moisture transport problem in a layered structure was converted
to a problem in terms of humidity potential, this coefficient also
needs to be modified accordingly. Due to the fact that the
moisture-induced strain calculated from the moisture content
state variable and from the humidity potential state variable must
yield the same induced strain, the following relation must hold:
e
swelling
¼ b y ¼ b
h,ð14Þ
where
b
n
is defined as the humidity swelling coefficient.
It is noted that in the case where
b is given in terms of specific
moisture content, the humidity swelling coefficient,
b
n
, has the
form
b
¼
y
r h
b ,ð15Þ
where
r is the mass density of the material.
In the present study, the swelling coefficients,
b , for both
concrete and FRP is treated as a constant, since only linear
response range is considered. The resulting humidity swelling
coefficient,
b
n
, is dimensionless and represents strain per unit
relative humidity. It should be noted that because the moisture
content,
y , is generally a nonlinear function of the relative
humidity, h, based on the characteristic sorption isotherm
property, the resulting swelling coefficient
b
n
becomes a non-
linear function of, h.
The total strain vector can be related to displacements by the
displacement–strain relation as
f
e
total
g¼f e
x
e
y
e
z
g
xy
g
yz
g
xzz
g
T
¼fLg
u
v
w
8
>
<
>
:
9
>
=
>
;
,ð16Þ
where L
n
is a differential operator defined as
L ¼
@
@x00@ @y0@ @z
0@
@y0@ @x
@
@z0
00@
@z0@ @y
@
@x
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
T
:ð17Þ
Therefore, the finite element equation for the structural
analysis incorporating the moisture and temperature strains
results in the following form:
½KfUg¼fF
B
gþfF
S
gþfF
thermal
gþfF
swelling
g,ð18Þ
where fUg¼fu
1
v
1
w
1
u
2
v
2
w
2
... u
r
v
r
w
r
g
T
is the nodal
displacement vector, ½K¼R
V
½B
T
½D½BdV is the element stiff-
ness matrix, fF
B
g¼R
V
½N
T
fF
b
gdV is the element body force vector,
F
S
¼R
S
½N
T
fF
s
gdS is the element force vector due to the surface
loading, fF
thermal
g¼R
V
½B
T
½Df e
thermal
gdV is the element force
vector due to the thermal expansion, fF
swelling
g¼R
V
½B
T
½Df e
swelling
gdV is the element force vector due to the moisture
swelling.
In the above equations ½B¼fLgfNg
T
, in which L
n
was defined
in Eq. (17) and
fNðx,y,zÞg ¼
N
1
00N
2
00:::N
r
00
0N
1
00N
2
0::: 0N
r
0
00N
1
00N
2
::: 00N
r
8
>
<
>
:
9
>
=
>
;
T
,
is the shape function matrix. To implement the coupled hygro-
thermo-mechanical finite element analysis, the FEMLAB
s
com-
mercial finite element analysis package, which allows its user to
M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 785

modify the predefined finite element formulation, was employed.
Considering the highly nonlinearly coupled nature of the problem,
the damped Newton method provided in the nonlinear solver of
FEMLAB software was used [29]. The software approximates the
nonlinear model with a linear model using initial values of
humidity, temperature, and displacement and calculates the
coefficients for the linearized system. The linear direct solver
(UMFPACK) was selected to solve the linearzied system. The
solution from the current linearized system is used to re-calculate
the coefficients and establish an updated system of linearized
equations. The process is repeated until a converged solution is
obtained. The linear direct solver uses the permutation of the
rows of the system of equations to achieve a more stable and
more accurate solution.
To solve the transient problem, the FEM discretization of the
time-dependent PDE problem, was carried out using the ‘‘method
of lines’’ technique. To solve the resulting system, DASPK solver
[30] was used with the default values of 0.01 and 0.001 for
relative and absolute tolerances, respectively.
4. Case study of a concrete slab with FRP laminate
reinforcement
A three-dimensional model of a concrete slab with dimensions
of 396 194 194 mm
3
partially covered with an FRP reinforce-
ment laminate was considered as a basis to carry out the coupled
hygro-thermo-mechanical analysis. The model configuration
along with the assigned coordinate system is shown in Fig. 1.
The thickness of the FRP laminate was assigned as 19.4 mm,
which corresponds to 10 layers of a typical woven glass/epoxy
composite lamina and is about 10% of the thickness of a typical
concrete masonry unit block used in building construction. The
length of FRP in X-direction was chosen as 60% of the length of the
concrete block, which is equal to 238 mm. Since applying FRP
laminates on the inner surface of a wall structure is a common
practice, outdoor and indoor boundary conditions were applied
on the concrete and FRP sides, respectively. The humidity and
temperature boundary conditions were considered as h
in
¼50%
and T
in
¼20 1C at FRP side, and h
out
¼97% and T
out
¼10, 20, 40, 50,
and 60 1C at the concrete side. The lateral, top, and bottom
surfaces of the concrete block (normal to Zand Xcoordinates,
respectively) and the top and bottom faces of the FRP laminate
were considered as insulated. Under the consideration of such
boundary conditions, the moisture and heat transport problem in
the model became essentially two-dimensional, whereas the
structural analysis was still a three-dimensional analysis.
The schematic of the model with prescribed displacement
constraint boundary conditions is shown in Fig. 2. The nodal
displacement u
z
on all nodes on the Z¼0 surface is constrained
based on the consideration of symmetry. The displacements in the
X-direction for the nodes on the central edges of the model with
Y¼194 and 194 mm are also restrained. Finally, to prevent rigid
body motion in the Y-direction, Y-displacement of the node at the
coordinates of X¼198 mm, Y¼194 mm, and Z¼0 is also
restrained. The values or functional forms of the hygro-thermo-
mechanical material properties adopted in the current analysis
are given in the Appendix.
5. Results and discussion
The resulting steady-state relative humidity and correspond-
ing moisture distributions along the centerline of the block in the
FRP and concrete, i.e., X¼198 mm, obtained for various outdoor
temperatures are shown in Figs. 3 and 4, respectively.
It is observed that increasing the outdoor temperature causes
the overall profile of the relative humidity/moisture content to
shift upward when the outdoor temperature is greater than the
indoor temperature, and to shift downward otherwise. This
behavior can be explained considering the temperature distribu-
tion shown in Fig. 5. An apparent abrupt change can be seen in the
slopes of the temperature distribution curves at the FRP/concrete
interface. For the cases where the outdoor temperature is greater
than the indoor temperature, the slope increases. The positive
second derivative of the temperature profile in these cases
behaves as a moisture source in the second term of Eq. (5),
which drives the relative humidity/moisture content to a higher
value. On the other hand, the negative second derivative of the
temperature profile, as in the case where the outdoor temperature
is lower than the indoor temperature, behaves as a moisture sink,
which drives the relative humidity/moisture content to a lower
value. This phenomenon is known as ‘‘heat-induced moisture
movement’’.
Transient analyses for various thermal gradient conditions
were also carried out. The resulting relative humidity distribu-
tions at various time instants along the centerline of the block for
the case with T
in
¼20 1C and T
out
¼40 1C are shown in Fig. 6. Again,
humidity boundary conditions were prescribed as h
in
¼50% and
h
out
¼97%.
At all time instants, apparent abrupt changes in the slopes of
the temperature distribution profile at the FRP/concrete interface
are present due to the distinctive coefficients of thermal
396 mm
194 mm
194 mm
238 mm
X
Z
Y
Fig. 1. Three-dimensional model of the FRP reinforced concrete block.
0.2
0.15
0.1
0.05
0
-0.05
0.2
0.15
0.1
0.05
0
0.2
0.15
0.1
0.05
0
0.25
0.3
0.35
0.4
Z
X
Y
Units: meter
Fig. 2. Schematic of the physical model with displacement boundary conditions.
M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791786

conductivity for the FRP and concrete materials. This trend is
similar to the temperature distribution profile shown in Fig. 5 for
the steady-state condition. As a result, the aforementioned effect
of thermal moisture flux on the moisture movement distribution
is also evident at all the time instants shown. Therefore, for the
case shown in Fig. 6, where the outdoor temperature is greater
than the indoor temperature, the positive second derivative of the
temperature profile has functioned as a moisture source, which
has driven the relative humidity/moisture content to higher levels
for all the curves. Since the major change in local temperature
0 3.88 7.76 11.64 15.52 19.4
Y (mm)
Relative Humidity (%)
-194 -155.2 -116.4 -77.6 -38.8 0 38.8 77.6 116.4 155.2 194
Y (mm)
T=60 C
T=50 C
T=40 C
T=30 C
T=20 C
T=10 C
FRP Concrete
40
50
60
70
80
90
100
Fig. 3. Steady-state humidity distribution along the centerline (X¼198 mm) for various outdoor temperatures.
0
1
2
3
4
5
6
7
8
9
10
11
12
0 3.88 7.76 11.64 15.52 19.4 23.28 27.16 31.04 34.92 38.8
Y (mm)
Moisture Content for FRP (kg/m3)
-194 -155.2 -116.4 -77.6 -38.8 0 38.8 77.6 116.4 155.2 194
Y (mm)
Mositure Content in Concrete
(kg/m
3)
T=60 C
T=50 C
T=40 C
T=30 C
T=20 C
T=10 C
FRP Concrete
0
10
20
30
40
50
60
70
80
90
100
110
120
Fig. 4. Steady-state moisture content distribution along the centerline (X¼198 mm) for various outdoor temperatures.
0
10
20
30
40
50
60
70
0 3.88 7.76 11.64 15.52 19.4 23.28 27.16 31.04 34.92 38.8
Y (mm)
Temperature (°C)
-194 -155.2 -116.4 -77.6 -38.8 0 38.8 77.6 116.4 155.2 194
Y (mm)
T=60 C
T=50 C
T=40 C
T=30 C
T=20 C
T=10 C
FRP Concrete
Fig. 5. Steady-state temperature distributions along the centerline (X¼198 mm) for various outdoor temperatures.
M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 787