Creep in concrete beams strengthened with composite materials

European Journal of Mechanics A/Solids 29 (2010) 951e965

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Creep in concrete beams strengthened with composite materials Ehab Hamed*, Mark A. Bradford Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, UNSW Sydney, NSW 2052, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 October 2009 Accepted 31 May 2010 Available online 9 June 2010

This paper investigates the creep behaviour of concrete beams strengthened with externally bonded composite materials. The challenges associated with the creep modelling of the different materials involved are discussed and a theoretical model is developed. The model derived in the paper accounts for the viscoelasticity of the materials using differential-type constitutive relations that are based on the linear Boltzman’s principle of superposition. The model also accounts for the deformability of the adhesive layer in shear and through its thickness, and for its ability to resist stresses in these directions. These aspects are not fully accounted for in the existing models. An incremental formulation of the field equations is conducted via the variational principle of virtual work, which considers the variation of the internal stresses in time and their effect on the creep response. A numerical study that examines the capabilities of the model and quantifies the response of the strengthened beam to sustained loads is presented, with special focus on the edge stresses that develop at the adhesive interfaces and which initiate debonding failures. The effect of flexural cracking of the concrete is also considered through an enhancement of the model, along with a numerical example that describes the variation with time of the forces and stresses in the concrete beam, the internal steel reinforcement, and the FRP strip at the cracked section. Ó 2010 Elsevier Masson SAS. All rights reserved.

Keywords: Composite materials Concrete Creep FRP Strengthening Viscoelasticity

1. Introduction Externally bonded composite materials in the form of fibre reinforced polymers (FRP) are being used for the strengthening of concrete and masonry structures, with their effect on the strength, stiffness, ductility, durability, and dynamic performance of strengthened structures being widely investigated (Buyukozturk and Hearing, 1998; Rabinovitch and Frostig, 2000; Bakis et al., 2002; Rizkalla et al., 2003; Hamed and Rabinovitch, 2005, 2007a, 2008). The ongoing research in this field points in favour of using these materials, but some difficulties need to be overcome and some aspects of the behaviour of strengthened members need to be further clarified and studied. One of the main aspects that hitherto has not been investigated properly is the creep behaviour of strengthened members, when strengthening or upgrading is required to resist additional sustained loads or when strengthening is applied to a structure that creeps continuously. The current paper focuses on this aspect in strengthened concrete beams. The short-term response of concrete beams strengthened with FRP has been widely studied and reported. One of the main characteristics of strengthened beams is the debonding failure mechanism

* Corresponding author. Tel.: þ61 2 93859765; fax: þ61 2 93859747. E-mail address: [email protected] (E. Hamed). 0997-7538/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2010.05.007

of the FRP at the edges and near cracks, which results from relatively high shear and vertical normal stress at the adhesive interfaces. Under sustained loads, these stresses become dependent upon the creep characteristics of the concrete, as well as those of the adhesive and the FRP. Since composite and polymer materials exhibit different creep characteristics to concrete, the creep behaviour of the strengthened member may lead to stress redistributions. This may eventually lead to debonding failures over time, although the structure may be subjected to sustained loads that are less than its short-term failure loads. In order to shed light on this effect, as well as, on the stresses redistribution at cracked sections and on the creep behaviour of strengthened members in general, an understanding of the effect of creep on the deformations, internal forces and stresses is required, bearing in mind the dependence of the creep strains on the variable stress level and their interaction with environmental effects (temperature, humidity, etc.), which make predicting the behaviour of strengthened members a challenging and difficult task. Plevris and Triantafillou (1994) presented theoretical and experimental studies of the time-dependent behaviour of strengthened beams. The study revealed the positive effect of the bonded FRPs on the long-term deflection, but their effects on the localized stresses at the adhesive interfaces were not addressed. Also, the creep of the adhesive layer, which may significantly affect the stress concentrations, was not accounted for. Masia et al. (2005) studied the problem based on the strain compatibility approach

952

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and ignored the creep of the FRP and the adhesive. The prediction of the long-term deflections was based on straightforward formulae provided by design codes (ACI Committee-209, 1982; CEB-FIP, 1990). Tan and Saha (2006) used the effective modulus method (EMM) for the creep modelling of the concrete material, while the effect of creep of the adhesive was again ignored. As in Plevris and Triantafillou (1994), a power law was used for modelling the FRP material, which yielded an equivalent time-dependent modulus. However, the time-dependent moduli approach, which is a straightforward procedure that does not account for the stresses variation in time, is more applicable for simple, homogeneous, and statically determinate structures for which creep deformations do not affect the stress distribution. Nevertheless, creep of one of the components of the strengthened beam may modify the interfacial stresses, as well as the entire stresses distribution in the structure as with any reinforced concrete member (Gilbert, 1988). The accuracy of using the EMM is therefore questionable for a more detailed analysis, and because the creep behaviour of concrete and FRP is different, use of the strain compatibility approach is problematic because it does not account for the different strain differentials through the cross-section. An incremental formulation that is based on modelling the concrete material by a generalized Maxwell chain (Bazant and Wu, 1974) was developed by Muller et al. (2007), which accounts for the variation of the internal stresses with time. However, the creep of the FRP and the adhesive layer was not considered and the developed theoretical model does not address the local effects and stress concentrations. Al Chami et al. (2009) observed in their experimental study that the use of CFRP laminates has a negligible effect on the long-term deflections of strengthened beams. This could be a combination of the creep of the concrete material and the creep of the adhesive and the FRP, for which the latter can significantly decrease the efficiency of the strengthening system. Studying this aspect requires a detailed modelling approach and analysis that accounts for creep in all structural components. Benyoucef et al. (2007a,b) developed a theoretical model for studying the effect of creep on the adhesive stresses in strengthened beams considering the creep in the concrete only by the approximate age-adjusted EMM. However, the model assumes that the stresses in the adhesive layer do not change through its thickness based on a layered structure modelling approach. This assumption is not compatible in terms of material point level (continuum) equilibrium in the adhesive. Based on the findings of Rabinovitch and Frostig (2000) and Hamed and Rabinovitch (2005), the vertical normal stresses vary through the height of the adhesive layer, attain different values and different distributions at the upper and lower interfaces, and in many cases, even have different signs (tension/compression) at the different interfaces. Furthermore, in order to quantify the edge stress concentrations, the shear stress free boundary condition in the adhesive must be fulfilled, along with modelling the interaction between the shear and the vertical normal stresses stemming from the continuum level equilibrium condition. A study of the creep behaviour of the FRPeconcrete interface was conducted by Wu and Diab (2007) and Diab and Wu (2007) in concrete beams strengthened with prestressed FRP. Generalized Maxwell constitutive models were developed for the interface and implemented into a finite element (FE) code that considers the interface as a cohesive layer with zero thickness. This modelling is suitable for predicting failures in pure shear, but it does not account for the vertical normal stresses and their distribution through the thickness, which play important roles in the debonding mechanisms (Rabinovitch, 2008). In addition, their works focused on the shear slip mechanisms of the concreteeFRP interface, without describing the behaviour of flexural members and their stress and deformation fields. The creep of the adhesive at the concreteeFRP

interface in a double-lap shear setup was also studied by Choi et al. (2007) and by Meshgin et al. (2009), which revealed the stress redistribution caused by the differences in the retardation times of the Epoxy adhesive and concrete. The creep behaviour of other strengthened concrete members was investigated in Stierwalt and Hamilton (2005) and Savoia et al. (2005), which showed that external FRP bonding has beneficial effects on the creep behaviour and that different adhesives lead to desirably different behaviour. The aim of this paper is to provide insight into the creep behaviour of concrete beams strengthened with externally bonded composite materials and to contribute to the establishment of a foundation of knowledge required for the creep analysis of general strengthened structures. For this, a theoretical model that considers the strengthened beam as a layered structure that consists of the reinforced concrete (RC) beam, the adhesive layer and the FRP, and accounts for the creep of each component through Boltzmann’s principle of superposition (Bazant, 1988) is developed. Although the strengthened beam and its components are characterized by many nonlinear effects of material behaviour, interfacial debonding, temperature dependence, and others, one must know and understand the effect of creep on the stress transfer mechanism and the stress concentrations at the adhesive interfaces before turning to the more complicated studying of the modes of failure under sustained loads. This paper therefore focuses on the behaviour of the structure in a framework of linear viscoelasticity in its first part assuming perfect bonding between the structural components. In its second part (Section 5), the effect of flexural cracking of concrete is accounted for assuming linear viscoelastic behaviour in compression. The model accounts for the deformability of the adhesive layer in shear and through its thickness by a high-order modelling approach (Rabinovitch and Frostig, 2000; Hamed and Rabinovitch, 2005) that was first developed for the analysis of sandwich structures (Frostig et al., 1992), and which allows for quantitative evaluation of the edge stresses. This modelling approach was validated by Rabinovitch and Frostig (2000, 2003) and Hamed and Rabinovitch (2007a,b, in press) through comparison with experimental and finite element results obtained from static loading of concrete beams and masonry walls. Differential-type constitutive relations are used for modeling the creep in each component, which are interpreted into incremental quasi-elastic relations that involve an incremental time analysis and which accounts for the stress histories in the structure (Bazant and Wu, 1974). The various assumptions made in the creep modelling are included in the mathematical formulation. 2. Mathematical formulation The model uses lamination theory with first order shear deformation for the modelling of the FRP laminates because in general their shear modulus is significantly smaller than their normal elastic modulus and in many cases they exhibit different creep behaviour in shear than in tension or compression (Scott et al., 1995). For consistency, the RC beam is modelled as a first order shear deformable Timoshenko’s beam (Hamed and Rabinovitch, 2007a). Following Frostig et al. (1992), Rabinovitch and Frostig (2000), and Hamed and Rabinovitch (2005), the adhesive layer is modelled as two dimensional (2D) continuum with shear and vertical normal rigidities, while its in-plane longitudinal rigidity is neglected with respect to that of the RC beam and the FRP. The assembly of the structural components into the whole structure is achieved through compatibility and continuity requirements. For each component, it is assumed that the stress and deformation fields are uniform through the width. The sign conventions for the coordinates, deformations, loads, stresses and stress resultants of a strengthened beam are shown in Fig.1. Note that in most practical applications, it is not possible to extend the FRP to the supports and an ordinary beam

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

modelling is required for the unstrengthened region. For brevity however, the formulation focuses on the strengthened region whereas the modelling for the unstrengthened region is obtained by degenerating the model (Hamed and Rabinovitch, 2005). The variational principle of virtual work (dU þ dW ¼ 0, d is the variational operator) is used to derive the incremental equilibrium equations along with the boundary conditions. Thus, the time of concern t is subdivided into nt discrete times with Dtr ¼ tr  tr1 (r ¼ 1,2,.nt). The virtual work of the internal stresses dU at time tr þ Dtr is (Wunderlich and Pilkey, 2003):

Z

dU ¼ 





Z h

Z

(3b)

where wa and ua are the vertical and in-plane displacements of the adhesive, respectively. The virtual work of the loads that are assumed to be applied to the RC beam only at the time tr þ Dtr is: xZ¼ L

dW ¼

sfrp xx

þ

Dsfrp xx



d3frp xx

þ



sfrp xz

þ

Dsfrp xz









þ ðmx þ Dmx ÞdDfc dx



dgfrp xz



ð1Þ

where the superscripts c, frp and a refer to the concrete beam, FRP strip, and the adhesive layer, respectively; sixx and 3ixx (i ¼ c, frp) are the in-plane normal stress and strain in the RC beam and the FRP strip; sixz and gixz (i ¼ c, frp, a) are the shear stresses and shear strains; sazz and 3azz are the vertical normal stress and strain in the adhesive layer; D is the incremental time operator; and Vi represents the region occupied by each component. All stresses and strains are functions of the independent coordinates x and z (Fig. 1) and time tr, which for brevity, are omitted here. Note that Eq. (1) does not include the virtual work at the time of first loading t ¼ to. Nevertheless, the instantaneous response is obtained by choosing a fairly small first time step, i.e., Dt1 ¼109 day. The incremental kinematic relations for the RC beam and the FRP strip are assumed to follow first order shear deformation theory and the assumptions of small displacements as follows:



D3ixx x; zi ; tr ¼ Duoi;x ðx; tr Þ  zi Dfi;x ðx; tr Þ

(2a)

Dgixz ðx; zi ; tr Þ ¼ Dwi;x ðx; tr Þ  Dfi ðx; tr Þ

(2b)

where wi, uoi and fi (i ¼ c, frp) are the vertical displacement, inplane displacement and rotation of the cross-section, and (),x denotes a derivative with respect to x. The kinematic relations for the adhesive which are based on 2D linear elasticity are

ua za ,wa

qz

c.g

uofrp zfrp ,wfrp

(4)

where qz, nx, and mx are external distributed loads and bending moments, respectively (Fig. 1). The compatibility conditions at the adhesive interfaces assuming perfect bonding are

i dVfrp

saxz þ Dsaxz dgaxz þ sazz þ Dsazz d3azz dVa

x,uoc z c ,wc

½ðqz þ Dqz ÞdDwc þ ðnx þ Dnx ÞdDuoc x¼0

Va



Dgaxz ðx; za ; tr Þ ¼ Dua;z ðx; za ; tr Þ þ Dwa;x ðx; za ; tr Þ



Vfrp



(3a)

scxx þ Dscxx d3cxx þ scxz þ Dscxz dgcxz dVc

Vc







953

  D3azz x; za ; tr ¼ Dwa;z ðx; za ; tr Þ





Dwa x; za ¼ 0; tr ¼ Dwc ðx; tr Þ

(5a)

Dua ðx; za ¼ 0; tr Þ ¼ Duoc ðx; tr Þ  Yc Dfc ðx; tr Þ

(5b)

  Dwa x; za ¼ da ; tr ¼ Dwfrp ðx; tr Þ

(6a)

Dua ðx; za ¼ da ; tr Þ ¼ Duofrp ðx; tr Þ þ dfrp =2Dffrp ðx; tr Þ

(6b)

where Yc is the centroid of the RC beam, and dfrp and da are the depths of the FRP, and the adhesive layer respectively (Fig. 1).

2.1. Equilibrium equations The incremental quasi-elastic equilibrium equations of the strengthened beam are formulated using the variational principle (Eqs. (1) and (4)), along with the kinematic relations (Eqs. (2) and (3)), and the compatibility requirements (Eqs. (5) and (6)), and they read c DNxx;x þ bDsaxz ðza ¼ 0Þ ¼ Dnx

(7)

frp DNxx;x  bDsaxz ðza ¼ da Þ ¼ 0

(8)

c DVxx;x þ bDsazz ðza ¼ 0Þ ¼ Dqz

(9)

frp DVxx;x  bDsazz ðza ¼ da Þ ¼ 0

(10)

c

mx nx

M xx RC Beam

L

Yc da dfrp

b

c Nxx

a (x,0) τxz

σzza (x,0)

zs

c Vxx

Adhesive Layer

a (x,da) τxz

σzza (x,da)

frp

FRP Strip frp

Vxx

a

Mfrp xx

Nxx

b

Fig. 1. Geometry, loads, sign conventions, and stress resultants: (a) Geometry, coordinate systems, and loads; (b) Stresses and stress resultants.

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E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

c c DMxx;x  DVxx þ bYc Dsaxz ðza ¼ 0Þ ¼ Dmx

(11)

frp frp DMxx;x  DVxx þ bDsaxz ðza ¼ da Þdfrp =2 ¼ 0

(12)

Dsaxz;x þ Dsazz;z ¼ 0

(13)

Dsaxz;z ¼ 0

(14)

i , V i , and M i where Nxx xx xx

(i ¼ c, frp) are the axial force, shear force, and bending moment respectively, and b is the width of the FRP strip. 2.2. Constitutive relations In order to clarify the creep effects, the thermo-mechanical material behaviour of the structural components is not accounted for (Sullivan et al., 1993; Scott et al., 1995; Ferrier and Hamelin, 2002). Thus, the viscoelastic characteristics are assumed to be independent of temperature and other environmental effects. This is only an assumption that is made to simplify the formulation and to highlight the creep effects. The case of uncracked beam is considered first in order to clarify the general creep behaviour and the time variation of the local edge stresses, which are not significantly affected by flexural cracking close to mid-span. For this, the effect of the internal steel reinforcement is neglected. The case of cracked beam that also accounts for the contribution of the internal steel reinforcement is discussed in Section 5. Boltzmann’s principle of superposition is adopted for describing the time-dependent stressestrain relations of the different materials (Bazant, 1988). For example, it is given by the following history integral-type for the stresses in the adhesive layer:

Zt

sazz ðtÞ ¼ to

Razz ðt; t 0 Þd3azz ðt 0 Þ;

saxz ðtÞ ¼

Zt

Raxz ðt; t 0 Þdgaxz ðt 0 Þ

(15)

to

where Razz ðt; t 0 Þ and Raxz ðt; t 0 Þ are the normal and shear relaxation functions or relaxation moduli, respectively, which represent the stress at time t caused by a unit constant strain imposed at time t0 . However, this creep law requires the storage of the complete strain histories and consequently huge computational resources that are impractical for large structures, or structures with many degrees of freedom such as the one investigated here. Therefore, a differential-type form of Boltzmann’s principle, which was presented and validated in detail in Bazant and Wu (1974) for the analysis of concrete structures, is adopted here for modelling the different materials, along with its incremental form that is effective for step-by-step analysis and takes into account the effect of the variation of stresses in time (see also Wu and Diab, 2007; Jurkiewiez et al., 1999; Borchert and Zilch, 2005). For this, the relaxation functions of each material are expanded into Dirichlet series, and the incremental differential-type constitutive relations lead to a generalized Maxwell model for each material. In general, studying the creep behaviour of concrete structures is combined with studying its shrinkage behaviour, as these two effects interact, and both affect the long-term behaviour. However, because strengthening is usually required for relatively old structures with small active shrinkage straining, the effect of shrinkage is assumed to be negligible, and for brevity it is not included in the mathematical formulation. For the same reason, the effect of aging of the concrete material in terms of increasing its strength and stiffness with time (hydration progress effects) and its effect on the creep characteristics are ignored here as these effects are more prominent at the first 15e20 years of service. The adhesive layer also shrinks during curing. However, in order to focus on the creep effects, the stresses induced due to shrinkage of the adhesive are

neglected. This simplification can be justified because most adhesives that are used for concrete strengthening are based on Epoxy, which shrinks much less than other Polyester resins (Kaw, 1997), and its shrinkage generally terminates after 18e24 h of curing when the structure is still unloaded (Fischbei, 1966). The aging effects in the adhesive are also ignored as most Epoxy adhesives develop 90% of their characteristic mechanical properties after less than 24 h (Choi et al., 2007; Meshgin et al., 2009). Similarly, aging in the FRP is also ignored as the time scale for which aging activates is relatively small (Sullivan et al., 1993, 1995), and it may terminate when the FRP is still not applied on site. The viscoelastic constitutive relations of each material are presented in the following. For the lack of available data, it is assumed that the material properties of the concrete, adhesive and FRP do not change once they are in a composite layered structure. Also, with the absence of aging, the creep parameters are no longer depend upon t0 , and therefore the time coordinate t0 is omitted hereafter, while the time t is measured from the time of first loading. 2.2.1. Adhesive The relaxation moduli of the adhesive are expressed in a Dirichlet series as follows (Bazant and Wu, 1974): man X

Razz ðtÞ ¼

m¼1

a Ema et=Tm þ EN ; Raxz ðtÞ ¼ an

mas X

m¼1

Gam et=Tm þ GaN as

(16)

where Ema and Gam are the moduli of the mth Hookean spring in the Maxwell chain for the modelling in the normal and shear directions respectively, man and mas are the number of units in each direction, Tman and Tmas are the relaxation times of the mth unit in each direction, a and Ga are the moduli of the m þ 1 and m þ 1 Hookean and EN an as N spring respectively that are not coupled to any dashpot. The relaxation times are given by:

Tman ¼ ham =Ema ;

(17a)

a

Tmas ¼ zm =Gam

(17b) a

where ham and zm are the viscosities of the mth dashpot in each direction. The relaxation times are chosen in advance considering the time of interest (t  to). A suitable choice is given by Bazant and Wu (1974) as:

Tman ¼ T1an am1 ; Tmas ¼ T1as b T1an ,

(18a)

m1

(18b)

T1as ,

where a and b are chosen parameters that control the time range. The moduli of the springs in the Maxwell chain are determined by the least squares or other method to fit a test data or a known expression of Razz ðtÞ and Raxz ðtÞ. The relaxation moduli of the adhesive material as well as the other materials are discussed in the sequel in Section 3. The incremental constitutive relations are based on a trapezoidal numerical integration and take the following form (see Bazant and Wu (1974) for more details):

D3azz ðtr Þ ¼ Dsazz ðtr Þ=Ea00 ðtr Þ þ D300a ðtr Þ

(19a)

Dgaxz ðtr Þ ¼ Dsaxz ðtr Þ=G00a ðtr Þ þ Dg00a ðtr Þ

(19b)

Ea00 ðtr Þ ¼

man  X

m¼1

1  eDtr =Tm

an

T an m

Dtr

a Ema þ EN

(20a)

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965 mas  X

G00a ðtr Þ ¼

1  eDtr =Tm

as

 T as

m¼1

D300a ðtr Þ ¼

Dg00a ðtr Þ ¼

m

Dtr

man  X

Gam þ GaN  an

1 1  eDtr =Tm Ea00 ðtr Þ m ¼ 1 mas  X

1

G00a ðtr Þ m ¼ 1

(20b)

(21a)

 as 1  eDtr =Tm sam ðtr1 Þ

(21b)

After the solution at time tr is achieved, the stresses at each unit are determined as follows:



an



Dtr =Tmas a

sm ðtr Þ ¼ e a

an

Dtr =Tmas

sm ðtr1 Þþ 1e

T an m

Dtr T as m

Dtr



a

Gm Dgaxz ðtr Þ





c DMxx ðtr Þ ¼ Ec00 ðtr ÞIc Dfc;x ðtr Þ  Dc00c ðtr Þ

  Dwc;x ðtr Þ  Dfc ðtr ÞÞ

(22)

(23)

(24)





c DVxx ðtr Þ ¼ kc G00c ðtr ÞAc Dwc;x ðtr Þ  Dfc ðtr Þ  Dg00c ðtr Þ

(25) 

(26)

(32)

2.2.3. FRP strip Following the Maxwell model for the RC beam and the lamination and viscoelastic theories (Wilson and Vinson, 1984; Kim and Hong, 1988; Jones, 1999; Hamed and Rabinovitch, 2007c), the constitutive relations of the FRP laminate are:

h

Ema Deazz ðtr Þ

2.2.2. Concrete beam The constitutive relations of the concrete follow the procedure outlined above and take the form of Eqs. (16)e(23) with subscript/ superscript c instead of a, and xx instead of zz. The relations in the cross-section level are determined using the classical definition of the stress resultants and by substitution of the kinematic relations (Eq. (2)) into Eq. (19) (with c instead of a): c DNxx ðtr Þ ¼ Ec00 ðtr ÞAc Duoc;x ðtr Þ  D300oc ðtr Þ

 T cs  cs m c c Dtr =Tmcs c Vxx Vxxm ðtr1 Þ  1  eDtr =Tm G k A m ðtr Þ ¼ e Dt m c c r

sam ðtr1 Þ;

sam ðtr Þ ¼ eDtr =Tm sam ðtr1 Þþ 1eDtr =Tm

955



frp DNxx ðtr Þ ¼ b A0011 ðtr Þ Duofrp;x ðtr Þ  D300ofrp ðtr Þ



 i  B0011 ðtr Þ Dffrp;x ðtr Þ  Dc00frp ðtr Þ

(33)

h   frp DMxx ðtr Þ ¼ b B0011 ðtr Þ Duofrp;x ðtr Þ  D300ofrp ðtr Þ  i  D0011 ðtr Þ Dffrp;x ðtr Þ  Dc00frp ðtr Þ 

frp DVxx ðtr Þ ¼ bA0055 ðtr Þ Dwfrp;x ðtr Þ  Dffrp ðtr Þ  Dg00frp ðtr Þ

(34) 

(35)

where A0011 , B0011 , D0011 and A0055 are the extensional, extensionalbending, flexural, and shear viscoelastic rigidities of the FRP laminate in the x direction, which are defined by:

2

3

A0011 4 B00 5 11 D0011

2

3

ðz  k  zk1 Þ. NK NK X X k 4 z2  z2 k 00 25 k1 ¼ R11  k R55 ðzk  zk1 Þ . ; A55 ¼ k¼1 k¼1 3 z3k  z3k1 (36) k R11

k R55

D300oc ðtr Þ ¼

mcn   cn 1 X c 1  eDtr =Tm Nxx m ðtr1 Þ=Ac Ec00 ðtr Þ m ¼ 1

(27)

Dc00c ðtr Þ ¼

mcn   1 X Dtr =Tmcn c 1  e Mxx m ðtr1 Þ=Ic Ec00 ðtr Þ m ¼ 1

(28)

where and are the longitudinal and shear transformed reduced relaxation stiffnesses of the kth lamina within the composite laminate, NK is the number of layers in the laminate, and zk and zk1 are the distances to the bottom and top of the kth layer respectively (Jones, 1999). For brevity, the formulation here focuses on unidirectional and single-ply laminates, which are commonly used for flexk ural strengthening of concrete beams. For this, B0011 vanishes and R11 k 00 00 and R55 are replaced by Efrp and Gfrp respectively, which correspond to the quasi-elastic normal and shear moduli of the FRP laminate respectively. Based on this, Eqs. (24)e(32) hold also for the FRP with the subscript/superscript frp instead of c, and the viscoelastic properties of the FRP are discussed in Section 3. For the general case described in Eqs. (33)e(36), the viscoelastic properties can be determined based on the classical lamination theory (Lee, 1989).

(29)

2.3. Stress and displacement fields in the adhesive layer

where Ac and Ic are the cross-section area and geometrical moment of inertia of the concrete beam, kc is the shear correction factor, Ec00 and G00c are the quasi-elastic normal and shear moduli (given by Eq. (20) with subscript c instead of a), and D300oc ðtr Þ, Dc00c ðtr Þ and Dg00c ðtr Þ are the incremental prescribed creep membrane strain, change of curvature, and shear strain, respectively, which follow Eq. (21) with D300c ðtr Þ ¼ D300oc ðtr Þ  zc Dc00c ðtr Þ, and are given by:

Dg00c ðtr Þ ¼

1

mcs  X

G00c ðtr Þ m ¼ 1

 cs c 1  eDtr =Tm Vxx m ðtr1 Þ=ðkc Ac Þ

where Tmcn and Tmas are the relaxation times of the mth unit under normal and shear tractions respectively, and mcn and mcs are the corresponding number of Maxwell units. The forces and bending moments in each unit are defined as follows:

 cn cn Tm c Dtr =Tmcn c Nxx Nxxm ðtr1 Þ þ 1  eDtr =Tm Ec A Du ðt Þ m ðtr Þ ¼ e Dtr m c oc;x r (30) 

c Dtr =Tm Mxx m ðtr Þ ¼ e

cn

T cn  m c c Dtr =Tmcn Mxx ðt Þ  1  e E I Df ðt Þ r1 m Dtr m c c;x r (31)

Based on Eq. (14), the shear stresses are uniform through the depth of the layer and read:

Dsaxz ðx; za ; tr Þ ¼ Dsaxz ðx; tr Þ ¼ Dsa ðx; tr Þ

(37)

The vertical normal stresses are determined by integration of Eq. (13) as follows:

Dsazz ðx; za ; tr Þ ¼ Dsa;x ðx; tr Þza þ Cs ðx; tr Þ

(38)

where Cs(x,tr) is an unknown function of x and t only. The vertical deformation is determined through the use of the kinematic relation (Eq. (3a)) and the constitutive relation (Eq. (19a)), and integration through the depth of the adhesive layer:

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E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

 1 Dsa;x ðx; tr Þz2a ðx; t Þz þ C s r a 2 Ea00 ðtr Þ z a Z D300a ðx; z; tr Þdz þ Cw ðx; tr Þ þ

Dwa ðx; za ; tr Þ ¼

stated in terms of the unknown deformations and rotations, (wc, wfrp, uoc, uofrp, 4c, 4frp), and the unknown shear stress (sa), as follows (for brevity, the notation of the independent variables is omitted):

(39)

0

where Cw(x,tr) is a second unknown function, that together with Cs(x,tr) are determined using the compatibility conditions of the vertical deformations at the adhesive interfaces (Eqs. (5a) and (6a)), and D300a is a known function that is given by Eq. (21a). Thus, the distributions of the vertical deformation and stress through the thickness of the adhesive become

Dwa ðx; za ; tr Þ ¼  Dsa;x ðx; tr Þ 

 2  za  da za 2Ea00 ðtr Þ



Dwfrp ðx; tr Þ  Dwc ðx; tr Þ za

þ

da za da



Zda

D300a ðx; z; tr Þdz þ

0

Zza

þ Dwc ðx; tr Þ

D300a ðx; z; tr Þdz

ð40Þ

0

ð2za  da Þ 2

Dsazz ðx; za ; tr Þ ¼ Dsa;x ðx; tr Þ þ 

  Ea00 ðtr Þ Dwfrp ðx; tr Þ  Dwc ðx; tr Þ Zda

D300a ðx; z; tr Þdz

(41)

0

The distribution of the longitudinal deformation is determined using the kinematic relation (Eq. (3b)) and the corresponding constitutive relation (Eq. (19b)). Integration through the depth of the adhesive layer then yields

Zza

Dua ðx; za ; tr Þ ¼

Dsa ðx; tr Þ G00a ðtr Þ

0

Zza þ

Zza

Dwa;x ðx; z; tr Þdz

dz  0

Dg00a ðx; z; tr Þdz þ Cu ðx; tr Þ

(42)

0

where Cu(x,tr) is determined through the compatibility of the longitudinal deformations at the concreteeadhesive interface (Eq. (5b)). The final form of the longitudinal deformation field is determined by substituting Eq. (40) into Eq. (42) and integration, and it is given by:



Dua ðx;za ;tr Þ ¼

Dsa ðx;tr Þza Dsa;xx ðx;tr Þ z3a þ

G00a



Dwfrp;x ðx;tr Þz2a 2da

2Ea00

3

 da

z2a 2



 z2  Dwc;x ðx;tr Þ za  a 2da

þ Duoc ðx;tr Þ Yc Dfc ðx;tr Þ þza Dg00a ðx;tr Þ z2 þ a 2da

Zda

D300a;x ðx;z;tr Þdz 

0

Zza Zza







(46) 

00 Ifrp Dffrp;x  Dc00frp ;x þkfrp G00frp Afrp Dwfrp;x  Dffrp  Dg00frp Efrp

þ bDsa dfrp =2 ¼ 0

ð47Þ

   bda bE00  Dsa;x þ a Dwfrp  Dwc G00c Ac Dwc;x  Dfc  Dg00c ;x þ 2 da da Z bE00 D300a dz ¼ Dqz  a ð48Þ da 0



 bda Dsa;x G00frp Afrp Dwfrp;x  Dffrp  Dg00frp ;x þ 2  bE00 Zda bE00  D300a dz ¼ 0  a Dwfrp  Dwc þ a da da

ð49Þ

0

da Ea00 ðtr Þ da

  Ec00 Ac Duoc;x  D300oc ;x þbDsa ¼ Dnx (44)   00 Afrp Duofrp;x  D300ofrp ;x bDsa ¼ 0 (45) Efrp     00 00 00 00 Ec Ic Dfc;x  Dcc ;x þkc Gc Ac Dwc;x  Dfc  Dgc bYc Dsa ¼ Dmx

D300a;x ðx;z;tr Þdz2 ð43Þ

0 0

2.4. Governing equations The governing equations are derived using the equilibrium equations (Eqs. (7)e(14)), the constitutive relations (Eqs. (24)e (26)), the compatibility requirement (Eq. (6b)), and the stress and deformation fields of the adhesive layers (Eqs. (40)e(43)). They are

 Ds d Ds d3 d d  Duoc  Duofrp  a Dwfrp;x þ Dwc;x þ a00 a  a;xx00 a  frp Dffrp 2 Ga 12Ea 2 Yc Dfc þ

d2a 2

Zda

D300a;x dz 

0

Zda Zza 0

D300a;x dz2 ¼ 0

ð50Þ

0

The boundary conditions that also result from the variational calculus are: i zDNxx ¼ wDNk or Duoi ¼ Duoi

(51)

i zDMxx ¼ wDMk or Dfi ¼ Dfi

(52)

i zDVxx ¼ wDPk or Dwi ¼ Dwi

(53)

Dsa ¼ 0 or Dwa ðza Þ ¼ Dwa ðza Þ

(54)

where Pk, Nk and Mk are external loads and bending moments at x ¼ 0 or x ¼ L; the over-bar designates prescribed deformations; z ¼ 1 for x ¼ L; z ¼ 1 for x ¼ 0; w ¼ 1 for the boundary conditions of the RC beam; and w ¼ 0 for the FRP. The solution of Eqs. (44)e(50) is achieved by the numerical multiple shooting method (Stoer and Bulirsch, 1993) at each time increment.

3. Viscoelastic properties The incremental mathematical model developed in Section 2 is valid for any desired creep models, but the specific viscoelastic properties that are adopted for the numerical study are discussed here. Relaxation moduli or relaxation test data of the different materials are required for the determination of the spring moduli of the rheological Maxwell models. However, because of the limited experimental data available to calibrate the relaxation modulus owing to the fact that relaxation tests are more difficult to conduct than creep tests (Bazant, 1988), the relaxation modulus is generated here based on the compliance function (J), which corresponds to the strain caused by a unit sustained stress and is obtained by creep tests.

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

In general, this is achieved through the following expression (Findley et al., 1976) in the normal direction of the concrete for example:

Zt c c Jxx ðt  xÞRcxx ðxÞdx ¼ t for t > 0; Jxx ð0ÞRcxx ð0Þ ¼ 1 for t ¼ 0

3.3. FRP laminate

Eq. (55) presents a Volterra integral that is solved numerically or analytically by the aid of a Laplace transformation. 3.1. Concrete Many theoretical models which aim to predict the timedependent behaviour of concrete are available (Bazant, 1988; Gilbert, 1988). Here, the influence of creep in the normal direction (tension or compression) is introduced based on the recommendations of ACI Committee-209 (1982), which have been widely investigated and validated through comparison with test results:

1 þ 4ðtÞ Ec

(56)

where Ec is the modulus of elasticity and 4(t) is the creep coefficient that are both evaluated at the age of loading to:

"

4ðtÞ ¼

# tl 4u ðto Þ; c þ tl

4u ðto Þ ¼ 1:25  toð0:118Þ 4ðN; 7Þ

(57)

in which l and c are parameters that control the creep coefficient, 4 (N,7) is the final creep coefficient (infinity) for a load applied at age 7 days, and the times t and to are measured in days. When aging is not considered (see Section 2.2), 4u actually remains constant with time. Knowing the compliance function of the concrete, Bazant and Kim (1979) suggested the use of an approximated expression for obtaining the relaxation modulus, which provides a very good estimate to Eq. (55), and takes the following form when aging is not considered:

Rcxx ðtÞ ¼

1  do c ðtÞ Jxx

for t > 0; Rcxx ð0Þ ¼ Ec

for t ¼ 0

frp

Rxx ðtÞ ¼

3.2. Adhesive The compliance functions of the adhesive follow the well known Findley’s power law (Findley et al., 1976) and take the following form in the normal direction:

1 þ Szza t n Ea

(59)

where Szza and n are determined to best fit a test data (Wilson and Vinson, 1984; Maksimov and Plume, 2001) as shown in the numerical study in the subsequent. The relaxation modulus in this case is given by Nielsen (2005):

Razz ðtÞ ¼ Ea

N X ð1Þj t jn ðSzza Ea Gð1 þ nÞÞj j¼0

Gð1 þ jnÞ

Many of the fibres (Glass, Carbon) used for civil engineering applications are linear elastic, but the overall behaviour of the laminate may exhibit some level of viscoelasticity due to the matrix. Aramid fibres, on the other hand, exhibit significant creep which together with the creep of the matrix and the adhesive, may radically affect the efficiency of the strengthening system over time. Fibres are also characterized by a phenomenon called creep or stress rupture, for which the strength of the material significantly decreases over time, especially for Glass and Aramid fibres. For this, the existing codes and design guidelines for structural strengthening introduce limits on the stress and strain of the FRP in the serviceability limit state (CEB-FIB fib Bulletin 14, 2001; ACI Committee 440.2R-02, 2002; Ascione et al., 2008). Nevertheless, the creep rupture effects are out of the scope of this paper, and the behaviour of the FRP is assumed linear viscoelastic. Due to the lack of data on the viscoelastic creep properties of FRP laminates, a micromechanical modelling is used here (Wilson and Vinson, 1984). For this, a perfect bonding is assumed between the fibres and the matrix, and the stresses arising from the differences in Poisson’s ratios are assumed to be negligible (Maksimov and Plume, 2001). In many cases, adhesives and fibres do not exhibit the same material properties in a joint as they do in bulk (Brinson, 1982). However, following Lee (1989) and for the seek of simplicity and brevity, it is assumed that the mechanical properties of each constituent do not change. The TsaieHahn semi-empirical role of mixture, which is a modification of the classical role of mixture is used (Sullivan et al., 1995; Wen et al., 1997), and the relaxation modulus of the laminate is given by:

(58)

in which do z 0.008. Eq. (58) is then used for the determination of the spring moduli in the normal direction in the concrete beam via Eq. (16a) (with c instead of a and xx instead of zz). Due to the lack of data on the creep behaviour of concrete in shear, it is assumed that Poisson’s ratio nc is constant with time (Bazant, 1988), and so, the spring moduli in shear are determined as Gcm ¼ Emc =ð2ð1 þ nc ÞÞ, while the relaxation times and number of units are assumed as the ones in the normal direction (i.e. Tmcs ¼ Tmcn , mcs ¼ mcn), and the shear dashpot viscosities are determined through Eq. (17b), with c instead of a.

a ðtÞ ¼ Jzz

rheological modelling in shear is derived from that in the normal direction or vice versa. Once Razz or Raxz are known, they are expressed by a Dirichlet series using Eq. (16).

(55)

0

c Jxx ðtÞ ¼

957

(60)

where G is the gamma function. The Poisson’s ratio na is assumed constant with time (Findley et al., 1976; Wen et al., 1997), and the

h   i 1   Vf Rfxx þ j 1  Vf Rm xx Vf þ j 1  Vf

(61)

f

where Rxx and Rm xx are the relaxation moduli of the fibres and the matrix respectively, Vf is the fibre volume fraction, and j is obtained by setting Eq. (61) to be equal to a measured or provided modulus of elasticity of the laminate. Assuming a creep power law for the behaviour of the fibres and the matrix (Findley et al., 1976; Scott et al., 1995), their compliance and relaxation moduli follow Eqs. (59) and (60) with subscript/superscript xx instead of zz, and f instead of a for the fibres and m instead of a for the matrix. Note that Sxxf ¼ 0 for the common case of elastic fibres. The relaxation modulus in shear is also obtained using the same procedure outlined above. For simplicity and for the lack of data on the shear behaviour of the fibres, the ratio of the shear relaxation modulus to the normal relaxation modulus of the laminate is assumed to be independent of time and equals to the ratio of the shear modulus to the normal elastic modulus of the FRP laminate. 4. Numerical study The geometry, basic material properties, and loading of the investigated beam appear in Fig. 2. The beam is assumed to be strengthened at the age of 20 years with extra load applied 2 weeks after strengthening. For this, the modulus of elasticity of the concrete (N32) at that age is taken as Ec ¼ 31 GPa (AS3600, 2001) with nc ¼ 0.17. The time variation of the normalized relaxation modulus of the concrete is shown in Fig. 3a (with Rcxx ð0Þ ¼ Ec ), along with its Dirichlet series approximation (Eq. (16)). The

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Ec=31GPa

280

20 kN/m

2φ16

2500 3000 Ea= 2.6GPa Efrp=160GPa d a= 3.0mm dfrp=1.2mm

150 200

Fig. 2. Geometry, material properties and load.

following values are adopted for the parameters that control the creep behaviour of concrete (Eq. (57), ACI Committee 209, 1982): l ¼ 0.6, c ¼ 10 days, 4(N,7) ¼ 2.35; and for clarity, the creep coefficient after 5 years since loading is evaluated as 0.93, which can also be obtained from Fig. 3a. The properties of the adhesive are calibrated based on test data reported in Maksimov and Plume (2001) for pure EDT-10 Epoxy specimens under tension. The results in Maksimov and Plume (2001) reveal that the creep behaviour depends slightly on the stress level. For this, the data reported on an average stress of 13.6 MPa, which appear in Fig. 3b, is used for calibrating the material properties and they are assumed constant with time. The following values are obtained (Eq. (59)): Ea ¼ 2.6 GPa, Szza ¼ 0.048 GPa1(day)n, n ¼ 0.4, which also provide a good approximation to the data reported in Maksimov and Plume (2001) for a stress level of 6.8 MPa. The predicted and the reported strains from which the compliance function is evaluated (J ¼ 3/13.6) appear in Fig. 3b. The normalized relaxation modulus of the adhesive and its Dirichlet approximation appear in Fig. 3a, and na is taken as 0.3. The FRP laminate is based on carbon fibres with Efrp ¼ 160 GPa, Gfrp ¼ Efrp/20, Vf ¼ 68%. The fibres are linear elastic with Ef ¼ 234 GPa, and the properties of the matrix are assumed to follow those of the adhesive. The relaxation modulus is obtained via Eq. (61) with j ¼ 1 and is described in Fig. 3a, which shows that the relaxation modulus of the CFRP laminate is almost constant in time. Six Maxwell chain units are used for the modelling of the different materials with T1cn ¼ T1an ¼ T1frpn ¼ 1 and a ¼ 5. The Dirichlet series approximations of the relaxation moduli, which appear in Fig. 3a, show very good agreement with the given moduli. For reference, the distributions of the deflection, axial force, and the shear and vertical normal stresses at the adhesive interfaces that develop under instantaneous loading (i.e. t ¼ 0) are shown in Fig. 4. Fig. 4c shows the shear stress concentration at the edge of the adhesive layer, which reaches a peak value of 0.603 MPa at a distance of about 4.0 mm away from the edge, and drops to zero at the edge and actually fulfils the stress free boundary condition (Eq. (54a)), also see Rabinovitch and Frostig (2000). Fig. 4d shows

the vertical normal stress concentrations at the edges, which have different signs at the different interfaces. These stresses decay rapidly to almost zero at a distance of about 5 times the thickness of the adhesive layer away from the edge. The combined traction of relatively high tensile normal stresses at the concreteeadhesive interface and the shear stress concentrations, which are transferred to the concrete beam via a thin concrete reinforcement cover, may lead to debonding failure mechanisms that have been observed in many experimental studies (Bakis et al., 2002; Rizkalla et al., 2003). The variations of the peak deflection, axial force, shear and vertical normal stresses with time appear in Fig. 5. The results are normalized with respect to the instantaneous response appearing in Fig. 4. In order to clarify and understand the creep behaviour, the creep effects are gradually studied and introduced here by three steps. In the first step, only the creep of the concrete beam is accounted for, which simulates a case of strengthening with elastic adhesive and elastic FRP materials. In the second step, the creep in the adhesive is also accounted for in addition to the concrete creep and in the third step, the creep of all components is included. In all cases the material properties follow those appear in Figs. 2 and 3. The results in Fig. 5 show an increase with time in the vertical deflection and the axial force in the FRP as predicted by all three cases with very small differences. However, the time variation of the interfacial edge stresses in the adhesive layer exhibits significantly different behaviour according to the three cases studied. As expected from Fig. 3a, the minor creep behaviour of the CFRP strip has a negligible influence on the response of the beam compared to the case with creep in the adhesive and the concrete. In the case of creep in the concrete beam only, the increase in the deformations is associated with an increase in the axial force and in the interfacial shear and vertical normal stresses as well. This is because once the concrete beam undergoes creep deformations, the strengthening system tends to restrain its deflections, which is interpreted by an increase in the composite action (the moment carried in terms of tension in the FRP and compression in the beam) and an increase in the axial force and the adhesive edge stresses. On the other hand, once the adhesive material is

Fig. 3. Creep properties: (a) Normalized relaxation modulus (d) and Dirichlet approximation (- - -); (b) Experimental (Maksimov and Plume, 2001) (* normal creep strains of the adhesive material based on a creep power law.

* *

) and theoretical (d)

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

959

Fig. 4. Instantaneous response: (a) Vertical deflection; (b) FRP axial force; (c) Shear stresses at the edge of the adhesive layer; (d) Vertical normal stresses at the adhesive interfaces.

a viscoelastic one that does creep, the creep deformations of the concrete beam and the increase in the axial force are no longer associated with an increase in the edge stresses after 5 months from first loading. During the first 5 months where the rate of creep of the concrete is high (Fig. 3a), there is an increase in the edge stresses. Following this, the rate of creep of the concrete becomes relatively small while that of the adhesive is still high (Fig. 3a), which leads to a relaxation of the edge stresses although there is an ongoing increase in the axial force.

To further clarify this physical behaviour, consider the distribution of the axial forces along the beam and the distribution of the adhesive shear stresses close to the edge at three different times, as shown in Fig. 6. It can be seen that the axial forces and peak shear stresses at t ¼ 151 days are higher than those that develop at t ¼ 10 days, while the response at t ¼ 700 days is associated with an increase in the axial force and a decrease in the peak shear stresses compared to the response at t ¼ 151 days. However, the decrease in the peak shear stresses is associated with a slight change in their distribution along

Fig. 5. Normalized creep response: (a) Peak vertical deflection; (b) Peak FRP axial force; (c) Peak shear stress at the adhesive layer; (d) Peak vertical normal stresses at the adhesiveeconcrete interface (Legend: d Creep in all components; . . . . Creep in concrete and adhesive; - - - Creep in concrete only).

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E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

a

b

Fig. 6. Response at three different times: (a) Axial force in the FRP; (b) Shear stresses close to the edge of the adhesive layer.

modulus of the AFRP is Efrp ¼ 73.6 GPa, and the shear modulus is taken as Gfrp ¼ Efrp/20. The geometric properties of the concrete beam and the strengthening system are the same as those which appear in Fig. 2. For reference, the peak deflection, peak axial force in the FRP, and peak interfacial stresses at the adhesive layer in this case are: frp wc ðx ¼ L=2Þ ¼ 1:9 mm, Nxx ðx ¼ L=2Þ ¼ 3:7 kN, sa ðx ¼ 253Þ ¼ a 0:38 MPa, szz ðx ¼ 250; za ¼ 0Þ ¼ 1:58 MPa, sazz ðx ¼ 250; za ¼ 3Þ ¼ 0:48 MPa, which as expected, reveal larger deformations and smaller internal forces and stresses compared to the case with a stiffer CFRP laminate (see Fig. 4). The time variations of the normalized deflection and internal forces and stresses appear in Fig. 8 for the case where creep in all components is considered. The results show that contrary to the case with CFRP laminate (Fig. 5), the increase in the axial force is much smaller. The peak axial force due to creep is only 1.1 times the instantaneous one (Fig. 8b), while that observed in Fig. 5b for the CFRP laminate is about 1.8 times the instantaneous force. This is due to the significant creep characteristics of the AFRP laminate, which also lead to the development of smaller interfacial adhesive stresses and to a sharper decrease of these stresses with time. Because of the high rate of creep of the AFRP laminate short time after loading (Fig. 7), the magnified plots in Fig. 8 between t ¼ 0 and t ¼ 100 days show a decrease in the axial force and the interfacial stresses immediately after loading up to about t ¼ 2.5 days. Following this, a slight increase is observed, which is followed by a gradual decrease of the forces and stresses, where the latter are sharper due to the creep of the adhesive material itself. The decrease in the axial force over time when AFRP laminates are used was also indicated in the study of Plevris and Triantafillou (1994). As can also be expected, the creep of the FRP laminate leads to an increase in the normalized deflection of the beam. Fig. 8a shows that the deflection 5 years after loading is about 1.88 times the instantaneous one, which is higher than that observed in Fig. 5a (1.81).

the beam and is associated with an increase of the stresses 15 mm away from the edge towards mid-span, so that that integrated effect of the shear stresses still increases the axial forces (see Eq. (8)). A similar trend of behaviour was observed in the experimental study of Diab and Wu (2007) who performed double-lap shear tests to investigate the response of the FRPeconcrete interface. Note that the normalized creep deflections after 5 years are 1.81 times the instantaneous deflection (0.81 creep) as illustrated in Fig. 5a. This deflection is slightly smaller than what is expected based on the estimated creep coefficient at this time (0.93), due to the heterogeneity of the strengthened RC beam that consist of different viscoelastic and elastic materials. In order to illustrate and understand the effect of creep of the bonded FRP, a viscoelastic strengthening laminate made of Aramid fibres (AFRP) is examined. Thus, although AFRP laminates are not likely to be used in flexural strengthening applications, the understanding of the effect of creep of the bonded FRP, which can be due to the use of viscoelastic fibres or due to the use of strongly viscoelastic matrix with high volume ratio, is important for the design of strengthened members. The properties of the fibres are calibrated based on test data reported in Maksimov and Plume (2001) for Aramid fibres under a tension stress of 1000 MPa. A creep power law is assumed for the modelling and the following values are obtained (Eq. (59) with subscript/superscript f instead of a and xx instead of zz): Ef ¼ 107 GPa, Sxxf ¼ 0.193  102 GPa1(day)n, n ¼ 0.18. The predicted and reported strains from which the compliance function of the fibres is evaluated (J ¼ 3/1000) are plotted in Fig. 7a and reveal good agreement. The relaxation modulus of the fibres is obtained via Eq. (60) with subscript/superscript f instead of a and xx instead of zz. The matrix of the laminate and the adhesive are assumed as the adhesive material previously studied, i.e. EDT-10 Epoxy, and their creep characteristics follow Fig. 3. Taking Vf ¼ 68%, the relaxation modulus of the AFRP laminate is obtained via Eq. (61) with j ¼ 1 and is described in Fig. 7b. Eq. (61) actually reveals that the elastic

a 1.8

b

1.4 1 0.8 0.6 0.4 0.2 0 0

AFRP

1.0 0.8

1.2 R(t)/R(0)

Normal Strain [%]

1.6

0.6 0.4

0.2 1

t [years] 2 3

4

5 * * *

0

0

1

t [years] 2 3

4

5

Fig. 7. Creep properties of AFRP: (a) Experimental (Maksimov and Plume, 2001) ( ) and theoretical (d) normal creep strains of Aramid fibres based on a creep power law; (b) Normalized relaxation modulus (d) and Dirichlet approximation (- - -) of AFRP laminate.

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

961

Fig. 8. Normalized creep response with AFRP strengthening: (a) Peak vertical deflection; (b) Peak FRP axial force; (c) Peak shear stress at the adhesive layer; (d) Peak vertical normal stresses at the adhesiveeconcrete interface.

5. Cracking effect In the mathematical formulation and numerical study presented above, a focus was made on the general creep behaviour of FRP strengthened beams and particularly on the time variation of the edge stresses, but without the consideration of cracking of the RC beam. In this section, the effect of flexural cracking and the contribution of the internal steel reinforcement to the structural response, which is more prominent in cracked sections, are investigated. A smeared simplified cracking model is adopted with linear viscoelastic behaviour in compression and brittle behaviour in tension for the concrete. The steel reinforcement is assumed linear elastic. The effect of cracking is introduced into the model through equivalent time-dependent rigidities (Gilbert, 1988; Rabinovitch and Frostig, 2001), with the same Maxwell chain model outlined above being used for the creep modelling. While the use of rheological models for the creep analysis of linear viscoelastic materials is physically justified and mathematically proven once the relaxation function is expanded into Dirichlet series (Bazant and Wu, 1974; Bazant, 1988), such models are also commonly used for the viscoelastic modelling of different nonlinear materials with very good degree of accuracy (Sathikumar et al., 1998; Van Zijl et al., 2001). Papa et al. (1998) used Maxwell chain for the nonlinear modelling of concrete by the introduction of a damage variable that reduces the magnitudes of the chain spring constants with time. Nevertheless, here the Maxwell coefficients are kept constant with time because the stresses at the uncracked zone are assumed linear.

5.1. Mathematical modelling of cracked beams The model distinguishes between cracked and uncracked regions along the beam with appropriate continuation conditions between them. For simplicity, the length of the cracked region is assumed constant with time. The variational principle, equilibrium

equations, constitutive relations of the adhesive and FRP strip, and the stress and deformation fields of the adhesive layer follow Section 2, whereas the effect of cracking on the constitutive relations of the RC beam is discussed here. The model assumes uniform time-dependent rigidities along the cracked region that are determined based on the strains at the critical section. Following the same incremental approach outlined earlier, it is actually assumed that the equivalent rigidities are constant at each time increment. The constitutive relations in the cross-section level are determined using the classical definition of the stress resultants and Eq. (19) (with subscript/superscript c instead of a and xx instead of zz):

Zzact

c DNxx ðtr Þ

  bc Ec00 ðtr Þ D3cxx ðtr Þ  D300c ðtr Þ dzc þ Es D3s As

¼

(62)

dc =2

c DMxx ðtr Þ ¼

Zzact

  bc Ec00 ðtr Þ D3cxx ðtr Þ  D300c ðtr Þ zc dzc þ Es D3s As zs

dc =2

(63) Zzact c DVxx ðx; tr Þ ¼ k

  bc G00c ðtr Þ Dgcxz ðtr Þ  Dg00c ðtr Þ dzc

(64)

dc =2

where bc and dc are the width and depth of the RC beam respectively; zact is the distance of the lower tip of the active (uncracked) zone in the case of positive bending moment with respect to the reference line of the beam, which is measured positive downwards, and equals to dc/2 in the uncracked region while it should be iteratively determined in the cracked region as shown next; Es, D3s and As are the modulus of elasticity, incremental strain, and area of the internal steel reinforcement, respectively; and zs is the distance of the steel reinforcement from the reference line of the beam (Fig. 1). Eq. (64) actually assumes that only the active zone of the concrete contributes to the shear rigidity. The incremental strain of the steel

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reinforcement can be obtained from (Eq. (2)) with zc ¼ zs. Substitution of the kinematic relations (Eq. (2)) into Eqs. (62)e(64) leads to the following constitutive relations:

8 9 c < DNxx = c DMxx ¼ : c ; DVxx

2

00

3

8 9 8 9 0 Duoc;x = < DNcr = < Dfc;x  DMcr 0 5 : 00 Dwc;x  Dfc ; : DVcr ; kA55

00

A11 4 B00 11 0

B11 00 D11 0

and deformation fields of the adhesive layer (Eqs. (40)e(43)). For brevity they are not presented here. However, the resulting equations are associated with nonlinearity due to their implicit 00 00 00 00 dependence on the viscoelastic rigidities A11 , B11 , D11 and A55 of the cracked section, which is solved using the following iterative procedure (Rabinovitch and Frostig, 2001): Step 1: Initial guess: The beam is assumed uncracked through its entire length. Step 2: Analysis of the structure: Using the rigidities calculated in the initial guess or in the previous iteration, the governing equations become linear ordinary differential equations with constant coefficients, which can be solved analytically or numerically. Step 3: Analysis of the cracked section: Based on the solution obtained in step 2, the distance of the lower tip of the active zone from the reference line is determined based on Eq. (2) and ignoring the concrete capacity to undergo tensile strains without cracking as follows:

(65) 00

00

00

00

where A11 , B11 , D11 and A55 are the extensional, extensionalbending, flexural, and shear viscoelastic rigidities of the RC beam, and DNcr, DMcr and DVcr are incremental effective forces that result from creep. The viscoelastic rigidities are:

Zzact

00

A11 ðtr Þ ¼

bc Ec00 ðtr Þdz þ Es As ;

(66)

bc Ec00 ðtr Þzdz þ Es zs As

(67)

dc =2 00 B11 ðtr Þ

Zzact ¼

zact ¼

dc =2

Zzact

00

bc Ec00 ðtr Þz2 dz þ Es z2s As ;

D11 ðtr Þ ¼

uoc;x ðx ¼ L=2Þ fc;x ðx ¼ L=2Þ

(73)

(68)

Once zact is determined, the rigidities are evaluated using Eqs. (66)e(69).

(69)

Step 4: Convergence criterion. If the norm of the relative difference between the magnitudes of the equivalent rigidities in two successive iterations is sufficiently small, the iterative procedure is stopped. Otherwise, the procedure returns to step 2 with the updated rigidities determined in step 3.

dc =2 00 A55 ðtr Þ

Zzact ¼

bc G00c ðtr Þdz

dc =2

Substitution of the incremental creep strains (Eq. (21) with subscript/superscript c instead of a) into Eqs. (62)e(64) leads to the following effective creep forces:

Zzact

DNcr ðtr Þ ¼

mcn  X

bc Zzact bc dc =2

mcn  X

1  eDtr =Tm

mcs  X

bc dc =2

c

sm ðtr1 Þdz

(70)

cn



scm ðtr1 Þzdz

(71)

 cs 1  eDtr =Tm scm ðtr1 Þdz

(72)

m¼1

Zzact

DVcr ðtr Þ ¼ k

1e



m¼1

dc =2

DMcr ðtr Þ ¼

Dtr =Tmcn

m¼1

The governing equations in this case can be obtained using the equilibrium equations (Eqs. (7)e(14)), the constitutive relations (Eq. (65)), the compatibility requirement (Eq. (6b)), and the stress

a

5.2. Numerical example of cracked beam The beam appears in Fig. 2 is investigated here. It is assumed that the beam is symmetrically cracked at the middle between x ¼ L/3 and x ¼ 2L/3. Fig. 9 shows the distribution of the instantaneous normal stresses and strains at x ¼ L/2, which reveals that zact ¼ 60.3 mm with respect to the reference line. It can also be seen that the peak stress in the concrete is within the range of linear stresses as assumed by the model. In order to clarify the effect of cracking, the adhesive layer is assumed linear elastic in this case. The time variation of the deflection, internal forces, bending moment, distance of active zone from the reference line, and peak concrete normal stresses and strains appear in Fig. 10. It is seen in Fig. 10a that the vertical deflection increases from 3.48 mm up to 5.54 mm. Thus, an increase factor of 1.59, which is less than the factor obtained in the uncracked beam in Fig. 5a (1.81). This is because the deflection of the cracked RC beam is significantly restrained by the tensioning of the elastic internal steel

b

Fig. 9. Distribution of strains (a) and stresses (b) in the cracked RC beam at x ¼ L/2.

E. Hamed, M.A. Bradford / European Journal of Mechanics A/Solids 29 (2010) 951e965

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Fig. 10. Creep response of cracked beam at x ¼ L/2: (a) Deflection; (b) Axial forces; (c) Bending moment in RC beam; (d) Distance of active zone with respect to reference line; (e) Peak compressive strain in the RC beam; (f) Peak compressive stress in the RC beam.

reinforcement in addition to that of the elastic bonded FRP strip. Fig. 10b shows an increase in the absolute axial forces in the structural constituents. Note that the axial force in the RC beam as c Þ is actually the total force carried by defined in the formulation ðNxx the concrete beam and the internal steel reinforcement. The force in the steel can be calculated in a post-processing type of procedure c E 3 A . as Es3sAs, while that in the concrete is determined by Nxx s s s This increase in the internal forces along with the decrease in the bending moment shown in Fig. 10c reveals that the moment carried in terms of tension in the FRP and compression in the RC beam increases with time. Fig. 10d shows that the lower tip of the active zone shifts downwards with time towards the reference line revealing an increase in the depth of the active zone. This observation is in agreement with Gilbert (1988) and Plevris and Triantafillou (1994). Fig. 10e shows that the peak total compressive strains in the concrete are more than doubled with time, while the peak compressive stress is slightly reduced as shown in Fig. 10f due to the increase in the active depth of the concrete and the change in the stress transfer mechanism with time. 6. Conclusions The challenges associated with the creep analysis and modelling of concrete structures strengthened with composite materials have

been discussed and a theoretical model has been developed. The proposed model takes into account many aspects of the strengthened structure that the existing models do not consider. Among these are the creep characteristics of all components, the variation of stresses in time and their creep effect, the layered structure modelling, and the deformability of the adhesive layer in shear and through its thickness. These aspects have been considered through a mathematical modelling of the structure and its constitutive laws, which have yielded a set of incremental differential governing equations. The results have shown that the creep behaviour of FRP strengthened beams is associated with physical phenomena that some of which cannot be observed and accurately predicted using existing models. It has been shown that creep of the concrete beam leads to a significant increase in the axial force in the FRP laminate, as well as an increase in the shear and vertical normal stresses in the adhesive layer at the edges. These stresses initiate debonding failures, and their increase in time due to creep is of critical importance for the design of strengthened beams. It has also been shown that creep of the adhesive layer leads to a reduction in the interfacial adhesive edge stresses with small effect upon the deformations and the axial forces. Thus, in some cases, viscoelastic adhesives may have a favourable effect on the behaviour of strengthened beams. FRP laminates that exhibit significant creep (like AFRP) however, increase the deformations and decrease the efficiency of the strengthening system.

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Due to the lack of a detailed experimental study and because the proposed model does not account for some material and other interfacial nonlinearities, a comparison between the theoretical predictions and test results has not been made. Yet, the results reported in this paper are in qualitative agreement with some experimental and theoretical observations obtained in other studies. This agreement and the use of a validated high-order modelling approach, as well as, well known and experimentally calibrated constitutive relations provide some level of verification to the proposed model. In conclusion, the analytical model developed here sets a theoretical platform for the creep modelling and analysis of concrete structures strengthened with composite materials. It also clarifies some aspects of the structural behaviour and sheds light on the creep characteristics of FRP strengthened structures. Further aspects of the structural behaviour that include the material nonlinearity, tension stiffening, creep rupture, development of debonded regions, shear cracking, and temperature effects, need to be investigated, which can be built on the modelling concepts and outcomes of this paper. Acknowledgment The work reported in this paper was supported by the Australian Research Council (ARC) through a Discovery Project (DP0987939) awarded to the authors. References ACI Committee-209, 1982. Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures. American Concrete Institute (ACI), Detroit, Michigan. ACI Committee 440.2R-02, 2002. Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures. American Concrete Institute (ACI), Farmington Hills, Michigan. Ascione, F., Berardi, V.P., Feo, L., Giordano, A., 2008. An experimental study on the long-term behaviour of CFRP pultruded laminates suitable to concrete structures rehabilitation. Composites, Part B: Engineering 39 (7e8), 1147e1150. AS3600, 2001. Australian Standard: Concrete Structures. Standard Association of Australia, Sydney. Al Chami, G., Therialt, M., Neal, K.W., 2009. Creep behaviour of CFRP-strengthened reinforced concrete beams. Construction and Building Materials 23 (4), 1640e1652. Bakis, C.E., Bank, L.C., Brow, V.L., Cosenza, E., Davalos, J.F., Lesk, J.J., Machida, A., Rizkalla, S.H., Triantafillou, T.C., 2002. Fiber-reinforced polymer composites for construction e state-of-the-art review. Journal of Composites for Construction (ASCE) 6 (2), 73e87. Bazant, Z.P., 1988. Mathematical Modelling of Creep and Shrinkage of Concrete. Wiley, New York. Bazant, Z.P., Kim, S.S., 1979. Approximate relaxation function for concrete. Journal of the Structural Division (ASCE) 105 (ST12), 2695e2705. Bazant, Z.P., Wu, S.T., 1974. Rate-type creep law of aging concrete based on Maxwell chain. Materials and Structures 7 (1), 45e60. Benyoucef, S., Tounsi, A., Adda Bedia, E.A., Meftah, S.A., 2007a. Creep and shrinkage effect on adhesive stresses in RC beams strengthened with composite laminates. Composites Science and Technology 67 (6), 933e942. Benyoucef, S., Tounsi, A., Benrahou, K.H., Adda Bedia, E.A., 2007b. Time-dependent behavior of RC beams strengthened with externally bonded FRP plates: interfacial shear analysis. Mechanics of Time-Dependent Materials 11, 234e248. Borchert, K., Zilch, K., 2005. Time depending thermo mechanical bond behaviour of epoxy bonded pre-stresses FRP-reinforcement. In: Shield, C.K., Busel, J.P., Walkup, S.L., Gremel, D.D. (Eds.), Proceedings of the 7th International Symposium on Fiber Reinforced Polymer (FRP) Reinforcement for Concrete Structure (FRRRCS 7), Kansas City, MO, pp. 671e683. Brinson, H.F., 1982. The viscoelastic constitutive modelling of adhesive. Composites 13 (4), 377e382. Buyukozturk, O., Hearing, B., 1998. Failure behavior of precracked concrete beams strengthened with FRP. Journal of Composites for Construction (ASCE) 2 (3),138e144. CEB-FIP, 1990. Design Code. Comite Euro-International du Beton, Thomas Telford House, London. CEB-FIB fib Bulletin 14, 2001. Externally Bonded FRP Reinforcement for RC Structures. International Federation for Structural Concrete (fib), Lausanne, Switzerland. Choi, K.K., Meshgin, P., Taha, M.M.R., 2007. Shear creep of epoxy at the concreteeFRP interfaces. Composites, Part B: Engineering 38 (5e6), 772e780.

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