Finite Elements in Analysis and Design 46 (2010) 783–791 Contents lists available at ScienceDirect Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel 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 a r t i c l e in fo abstract Article history: Received 24 May 2007 Received in revised form 8 October 2009 Accepted 24 April 2010 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., heatinduced 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. Keywords: Moisture and temperature Structural analysis FRP reinforced masonry Multi-layered permeable structure 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 contemporary 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 outstanding 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 Corresponding author. Tel.: + 1 310 658 3301. E-mail address: [email protected] (M. Khoshbakht). 0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ﬁnel.2010.04.002 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 multilayered porous structures. The most accepted governing equations capable of describing hygro-thermal problem for a porous solid were derived by Philip and De Vries  and Luikuv . Their mathematical models include two coupled partial differential equations for moisture content and temperature dependent variables, and are applicable to a single-material structure. 784 M. Khoshbakht, M.W. Lin / 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, nonpermeability 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-thermomechanical problem of a multi-layered structure constructed of distinctive permeable materials with general moisture/temperature 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  is augmented to develop a comprehensive 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 rc @T ¼ r ðlrTÞ þ hv r ðDyvap ryÞ, @t rc ð1Þ ð2Þ where y is the moisture content (kg/m3), T is the temperature (K), Dy is the moisture diffusion coefﬁcient (m2/s), DT is the thermal moisture diffusion coefﬁcient (kg/(m s K)), rc is the volumetric heat capacity (J/(m3 K)), l is the thermal conductivity (W/(m K)), hv is the latent heat of vaporization (J/kg), Dyvap is the isothermal vapor diffusivity (m2/s). @T dP ¼ r ðlrTÞ þ hv r rðhPs Þ , @t m ð4Þ where h is the relative humidity of the surrounding air (dimensionless), m is the vapor resistivity of the porous medium (dimensionless), dP is the vapor permeability of the air (kg/ (m s Pa)), Ps ¼ P0 exp½ðDH=RÞðð1=TÞð1=373:15ÞÞ is the saturation vapor pressure (Pa), in which P0 ¼101,325 (Pa) is the pressure of a standard atmosphere, and DH/R ¼5201 (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.  to substitute moisture content, y, with relative humidity, h, which resulted in the following form for the moisture diffusion equation: Ch @h ¼ r ðD h rhÞ þ r ðD T rTÞ, @t ð5Þ where Ch ¼ @y , @h ð6Þ Dh ¼ @y dP D þ Ps , @h h m ð7Þ DT ¼ dp DH m RT 2 hPs , ð8Þ in which Dh 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 rc The coupled moisture migration and heat transport problem in porous media was formulated by Philip and De Vries  in the form of the following system of differential equations: @y ¼ r ðDy ryÞ þ r ðDT rTÞ, @t The transport coefﬁcients are to be determined experimentally. Among these coefﬁcients, there is a lack of vigorous deﬁnition for the coefﬁcient of thermal moisture diffusion, DT. 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 . In the proposed mathematical model, the governing equations take the modiﬁed form as @y dP ¼ r ðDy ryÞ þ r ð3Þ rðhPs Þ , @t m @T ¼ r:ðlrTÞ, @t 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 . 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 ( ) hi ði ¼ 1,2, . . . , rÞ, with r being the total number of nodes Ti per element, 9 8 @hi > > ( ) > > = < hi @t þ½D e ¼ fQe g, ½Ce @T > > Ti i > > ; : @t ð10Þ M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 where ½D e ¼ R T ½D½B dV is the element humidity and heat 2 3 D hx 0 0 D Tx 0 0 6 7 D hy 0 0 D Ty 0 7 6 0 6 7 6 0 D hz 0 0 D Tz 7 0 6 7 diffusion matrix, ½D ¼ 6 7 is the 6 0 0 0 lx 0 0 7 7 6 6 0 ly 0 7 0 0 0 4 5 lz 0 0 0 0 0 V ½B material humidity and heat diffusion matrix, [B]¼LNT, in which ( )T N1 0 N2 0 Nr 0 is the element fNðx,y,zÞg ¼ 0 N1 0 N2 0 Nr shape function, @ @ @ @ @ @ the differential operator, @x @y @z @x @y @z " # 0 R Ch fNgfNgT dV is the element humidity and heat ½Ce ¼ V 0 rc R capacity matrix, fQe g ¼ S fNgq dS is the element boundary humidity and heat ﬂux, qn is the boundary humidity/heat ﬂux. fLg ¼ 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 deformations, 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 consideration 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 fsg ¼ ½Dfeelastic g ¼ ½Dfetotal eswelling ethermal g, ð11Þ where fsg ¼ fsx sy sz txy tyz txz gT is the stress vector, [D] is the stiffness matrix of the material, etotal is the total apparent strain, feelastic g ¼ fex ey ez gxy gyz gxz gTelastic is the elastic strain vector, eswelling is the moisture-induced strain, and ethermal 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 feswelling g ¼ n b1 ðyyref Þ b2 ðyyref Þ b3 ðyyref Þ 0 0 0 oT , ð12Þ fethermal g ¼ n a1 ðTTref Þ a2 ðTTref Þ a3 ðTTref Þ 0 0 0 oT , ð13Þ where bi, i¼ 1, 2, 3 are moisture swelling coefﬁcients in the principle material directions, ai, i¼1, 2, 3 are the thermal expansions coefﬁcients in the principle material directions, and yref and Tref are the reference moisture content and the reference temperature, at which the moisture and thermal induced strains are zero, respectively. Apparently, bi has the same magnitude in all directions for an isotropic material, such as masonry. Likewise 785 ai 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: eswelling ¼ b y ¼ b h, ð14Þ where bn 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, bn, has the form y b, b ¼ ð15Þ rh 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, bn, 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 bn becomes a nonlinear function of, h. The total strain vector can be related to displacements by the displacement–strain relation as 8 9 > <u> = T fetotal g ¼ fex ey ez gxy gyz gxzz g ¼ fL g v , ð16Þ > : > ; w where Ln is a 8 @ > > > > @x > > > < 0 L ¼ > > > > > > > :0 differential operator deﬁned as 9 @ @ >T > 0 0 0 > @y @z > > > > = @ @ @ 0 0 : @y @x @z > > > > @ @ @ > > > 0 0 ; @z @y @x ð17Þ Therefore, the ﬁnite element equation for the structural analysis incorporating the moisture and temperature strains results in the following form: ½KfUg ¼ fFB g þ fFS g þfFthermal g þ fFswelling g, ð18Þ T where fUg ¼ fu1 v1 w1 u2 v2 w2 . . . ur vr wr g is the nodal R displacement vector, ½K ¼ V ½B T ½D½B dV is the element stiffR T ness matrix, fFB g ¼ V ½N fFb g dV is the element body force vector, R FS ¼ S ½N T fFs g dS is the element force vector due to the surface R loading, fFthermal g ¼ V ½B T ½Dfethermal g dV is the element force R vector due to the thermal expansion, fFswelling g ¼ V ½B T ½Dfeswelling g dV is the element force vector due to the moisture swelling. T In the above equations ½B ¼ fLgfN g , in which Ln was deﬁned in Eq. (17) and 9T 8 0 N2 0 0 : : : Nr 0 0 > > = < N1 0 0 N2 0 : : : 0 Nr 0 0 N1 0 fN ðx,y,zÞg ¼ , > > : 0 0 N2 : : : 0 0 Nr ; 0 N1 0 is the shape function matrix. To implement the coupled hygrothermo-mechanical ﬁnite element analysis, the FEMLABs commercial ﬁnite element analysis package, which allows its user to 786 M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 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 . 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  was used with the default values of 0.01 and 0.001 for relative and absolute tolerances, respectively. Units: meter 0.2 0.15 0.1 0.05 0 0.4 0.35 -0.05 0.2 0.3 0.25 0.15 Z 0.2 0.1 0.15 0.1 0.05 Y X 0 0.05 0 Fig. 2. Schematic of the physical model with displacement boundary conditions. 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 mm3 partially covered with an FRP reinforcement 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 hin ¼50% and Tin ¼20 1C at FRP side, and hout ¼97% and Tout ¼10, 20, 40, 50, and 60 1C at the concrete side. The lateral, top, and bottom surfaces of the concrete block (normal to Z and X coordinates, 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 194 mm 396 mm 238 mm 194 mm Z Y X Fig. 1. Three-dimensional model of the FRP reinforced concrete block. displacement uz on all nodes on the Z¼ 0 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 Z¼ 0 is also restrained. The values or functional forms of the hygro-thermomechanical material properties adopted in the current analysis are given in the Appendix. 5. Results and discussion The resulting steady-state relative humidity and corresponding 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 distribution 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 distributions at various time instants along the centerline of the block for the case with Tin ¼ 20 1C and Tout ¼40 1C are shown in Fig. 6. Again, humidity boundary conditions were prescribed as hin ¼50% and hout 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 M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 787 Y (mm) -194 -155.2 -116.4 -77.6 -38.8 0 38.8 77.6 116.4 155.2 194 100 Relative Humidity (%) 90 T=60 C T=50 C 80 T=40 C T=30 C T=20 C 70 FRP Concrete 60 T=10 C 50 40 0 3.88 7.76 11.64 15.52 19.4 Y (mm) Fig. 3. Steady-state humidity distribution along the centerline (X ¼ 198 mm) for various outdoor temperatures. Y (mm) -77.6 -38.8 0 38.8 77.6 116.4 155.2 194 12 120 11 110 10 100 9 90 8 80 T=60 C 7 T=50 C 6 T=40 C 5 T=30 C 4 T=20 C 3 FRP Concrete T=10 C 2 1 70 60 50 40 30 20 10 0 0 0 3.88 7.76 11.64 15.52 19.4 23.28 27.16 31.04 34.92 3 -155.2 -116.4 Mositure Content in Concrete (kg/m ) Moisture Content for FRP (kg/m3) -194 38.8 Y (mm) Fig. 4. Steady-state moisture content distribution along the centerline (X ¼198 mm) for various outdoor temperatures. Y (mm) -194 -155.2 -116.4 -77.6 -38.8 0 38.8 77.6 116.4 155.2 194 70 T=60 C Temperature (°C) 60 T=50 C 50 T=40 C T=30 C 40 FRP Concrete T=20 C 30 T=10 C 20 10 0 0 3.88 7.76 11.64 15.52 19.4 23.28 27.16 31.04 34.92 38.8 Y (mm) Fig. 5. Steady-state temperature distributions along the centerline (X ¼198 mm) for various outdoor temperatures. 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 788 M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 Y (mm) Relative Humidity (%) -194 -155.2 -116.4 -77.6 -38.8 100 0 38.8 77.6 116.4 155.2 194 100 Steady State 10 months 90 90 80 80 70 70 60 60 1 month 50 5 days 5 months 3 months 50 FRP 40 0 3.88 Concrete 2 months 1 day 40 7.76 11.64 15.52 19.4 23.28 27.16 31.04 34.92 38.8 Y (mm) Fig. 6. Transient relative humidity distributions along the centerline (X¼ 198 mm) for Tin ¼20 1C and Tout ¼40 1C. 600 500 Tout = 60 °C τzy (kPa) 400 Tout = 40 °C 300 200 Tout = 20 °C 100 Tout = 10 °C 0 -100 40 60 80 100 120 140 160 180 200 Z (mm) Fig. 7. Interfacial shear stress, tyz distribution along the z-axis at X¼ 198 mm on the FRP laminate. gradient occurs at the interface, while this quantity has very small magnitudes at the locations away from the interface, a drastic increase in the relative humidity resulting from the change in temperature gradient appears only at the interface. The hygro-thermal induced interfacial shear stress tyz on the FRP laminate along the Z-axis for various outdoor temperatures is presented in Fig. 7 for the humidity/temperature loading obtained from the steady-state condition. It can be seen that the level of this stress, generally, increases with the increasing temperature gradient. Noting that the humidity gradient is the same for all these cases, it is apparent that this increase is caused by two factors. The ﬁrst factor is the increase in the thermal expansion due to higher temperature gradients. The second factor is the increase in the moisture swelling due to heat-induced moisture movement. It order to quantify this effect, two additional analyses were conducted. In the ﬁrst analysis, no moisture gradient was prescribed, i.e., hin ¼hout ¼50%, while a temperature gradient was given as Tout ¼60 1C and Tin ¼20 1C. In the second analysis, the temperature and humidity boundary conditions were kept the same, but the thermal expansion effect was eliminated by assigning zero thermal expansion coefﬁcients for concrete and FRP. It is apparent that in the second analysis the only induced strain is due to the heat-induced moisture movement. Fig. 8 shows the humidity distribution for the second analysis along the centerline of the block, where X ¼198 mm and Y changes from 19.4 to 194 mm. It is clearly shown that even in the absence of the humidity gradient; there exists an increase in the humidity close to the interface of the concrete and FRP laminate, which is caused by the heat-induced moisture movement. The comparison of the shear stress tyz proﬁle for these three analyses is shown in Fig. 9. It reveals that even with no moisture gradient and thermal expansion effect, there is still some level of the interfacial shear stress caused by heat-induced moisture movement. Noting the maximum value of this stress, it can be M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 789 90 Relative Humidity (%) 80 70 60 50 40 30 Concrete FRP 20 10 0 -20 0 20 40 60 80 100 120 140 160 180 200 Y (mm) Fig. 8. Relative humidity distribution at the centerline for the case with no relative humidity gradient. 600 500 Relative humidity gradient (97% / 50%) With thermal expansion effect τzy (kPa) 400 300 No relative humidity gradient With thermal expansion effect 200 100 No relative humidity gradient Without thermal expansion effect (Only heat-induced moisture movement) 0 -100 40 60 80 100 120 140 160 180 200 Z (mm) Fig. 9. Comparison of the interfacial shear stress, tyz distributions along the z-axis at X ¼ 198 mm on the FRP laminate. 2000 τxy (kPa) 1500 1000 Tout = 60 °C Tout = 40 °C 500 Tout = 20 °C Tout = 10°C 0 -500 198 208 218 228 238 248 258 268 278 288 298 308 318 X (mm) Fig. 10. Interfacial shear stress, txy distribution along the x-axis at Z ¼194 mm on the FRP laminate. concluded that the stress due to this phenomenon comprises about 25% of the total induced stress. The proﬁle of the interfacial shear stress, txy, for various outdoor temperatures along the x-axis at Z¼194 mm is shown in Fig. 10. Similar trend to that of tyz is also observed fortxy, although the stress level is less severe. The distribution proﬁle of the interfacial peeling stress, sy, for various outdoor temperatures along the x-axis at Z ¼194 mm in 790 M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 Sorption isotherm relation 1200 y ¼ 226:68h3 247:75h2 þ 123:45h þ:1076 ðkg=m3 Þ: 1000 ðA:1Þ Moisture diffusion coefﬁcient σy (kPa) 800 Dy ¼ 10:01942y10:5 ½m2 =s, 600 Tout = 60 °C 400 Tout = 40 °C kg for 0 o y o 200: m3 ðA:2Þ Thermal moisture diffusion coefﬁcient 200 Tout = 20 °C 5 4 3 DT ¼ r0 ð1:67 108 y 3:99 106 y þ 2:58 104 y 4:14 10 3 2 y þ:216y:035Þ10 13 ðkg=ðm s KÞÞ, ðA:3Þ 0 -200 Tout = 10 °C in which r ¼2,200 (kg/m3). Volumetric heat capacity rc ¼ r0 c0 þ 4187y ðJ=ðm3 KÞÞ, -400 198 208 218 228 238 248 258 268 278 288 298 308 318 X (mm) Fig. 11. Peeling stress, sy distribution along the x-axis at Z ¼194 mm on the FRP laminate. where c0 ¼940 (J/(kg K)) for dry material. Thermal conductivity l ¼ 2:74 þ :0032y ðWatt=ðm KÞÞ: 6. Conclusion A ﬁnite element modeling procedure for analyzing the hygrothermo-mechanical behavior of multi-layered structures consisted of distinct permeable materials was developed by incorporating the thermal and swelling strains in the coupled moisture/temperature ﬁnite element model based on the governing equations presented by Philip and De Vries. The developed model was then used to analyze the structural response of a concrete block reinforced with a unidirectional glass/epoxy FRP composite laminate partially covering one lateral surface of the block. The reinforced concrete block was subjected to humidity and temperature gradient loads resulted from the difference of indoor and outdoor environmental conditions. The results showed that signiﬁcant levels of interfacial shear stresses were induced by three factors: temperature gradient, moisture gradient and heat-induced moisture movement. The phenomenon of the heat induced moisture movement was clearly observed in the form of moisture accumulation at the interface of the concrete and FRP, even in the absence of a moisture gradient load. This moisture accumulation can cause up to 25% of the total induced stress under the general loading of temperature and humidity gradients. The effect of heat-induced moisture movement on interfacial stresses of multi-layered structures was shown in the presented study by incorporating the temperature gradient term in the ﬁnite element formulation of the moisture diffusion equation, which could not be detected otherwise. The developed ﬁnite element modeling procedure provides a general method for hygro-thermo-mechanical analysis of multilayered permeable structures. It can be used as an aid with the design of FRP/masonry structure or other similar structures for minimizing interfacial stresses due to the mismatch of moisture swelling and thermal expansion properties of the constituent materials. The functional forms or values of material parameters for concrete are ðA:6Þ Thermal expansion coefﬁcient a ¼ 9 106 ð1=KÞ ðA:7Þ The functional forms or values of material parameters for FRP laminate are Sorption isotherm relation y ¼ 7:53h4:3 ðkg=m3 Þ: ðA:8Þ Moisture diffusion coefﬁcient DH 1013 ½m2 =s, Dy ¼ 2659:7ð1:865y þ 49:125Þ Exp RT kg o 10, ðA:9Þ for 0 o y m3 where DH/R ¼2626.03 (K). Thermal moisture diffusion coefﬁcient dp h P ðkg=ðm s KÞ, DT ¼ m T s ðA:10Þ where m ¼vapor resistivity¼40,000, 0:81 T dP ¼ 1:83 1010 ðkg=ðm s PaÞÞ, 273:15 DH 1 1 ðPaÞ, Ps ¼ P0 exp R T 373:15 in which P0 ¼101,325 (Pa) is the pressure of a standard atmosphere, and DH/R¼5201(K) measured for liquid water. Volumetric heat capacity rc ¼ nf rx cx þ ð1nf Þre ce ðJ=ðm3 KÞÞ, ðA:11Þ where rxcx is the dry volumetric heat capacity of the ﬁber¼ 0.209 107 (J/(m3 1C)) for glass ﬁber, rece is the volumetric heat capacity of the epoxy matrix¼ r0c0 + 4187y, in which r0c0 ¼0.36 107 (J/(m3 1C)) for dry material, nf is the ﬁber volume fraction E0.6. Thermal conductivity Longitudinal direction lL ¼ nf lf þð1nf ÞlT ðW=ðm KÞÞ, Appendix ðA:5Þ Moisture swelling coefﬁcient b ¼ 0:00762: the FRP laminate is shown in Fig. 11. It can be seen that the maximum peeling stress occurs at the free edge, which is the resultant of the warping deformations of the concrete and FRP laminate about both x- and z-axis, as expected. ðA:4Þ Transverse direction 1 þ Znf lTrans ¼ lT , 1Znf ðA:12Þ ðA:13Þ M. Khoshbakht, M.W. Lin / Finite Elements in Analysis and Design 46 (2010) 783–791 where lf ¼1.05 (W/(m K)) is the thermal conductivity of a glass ﬁber, lT ¼ 9:522 1011 T 4 þ 1:111 107 bT 3 4:057 105 T 2 þ4:263 103 T þ :382 is the thermal conductivity of the epoxy matrix, Z ¼ ððlf =lT Þ1Þ=ððlf =lT Þ þ1Þ. Moisture swelling coefﬁcient b1 ¼ 0, b2 ¼ b3 ¼ 0:6 106 : ðA:14Þ Thermal expansion coefﬁcient a1 ¼ 6 106 ð1=KÞ, a2 ¼ a3 ¼ 21 106 ð1=KÞ ðA:15Þ References  H. Saadatmanesh, Extended service life of concrete and masonry structure with composites, Construction and Building Materials 11 (No. 5–6) (1997) 327–335.  T.C. 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