Theoretical analysis on long-term deflection of GFRP-concrete hybrid structures with partial interaction

Composite Structures 216 (2019) 1–11

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Theoretical analysis on long-term deflection of GFRP-concrete hybrid structures with partial interaction

T

Shiqian Zhanga,b, W.C. Xueb, , Xiandong Liaoa ⁎

a b

China Construction Eighth Engineering Division Co. Ltd, Shanghai, China Department of Structural Engineering, Tongji University, Shanghai, China

ARTICLE INFO

ABSTRACT

Keywords: GFRP-concrete hybrid structure Creep Shrinkage Long-term deflection Partial interaction Calculation model

The creep and shrinkage of concrete and the creep of glass-fiber reinforced polymer (GFRP) may lead to the increase of long-term deflection of GFRP-concrete hybrid structures under service loading, due to the change of internal forces and stresses. This paper presents a calculation model to consider the influence of the aforementioned changes in analyzing the long-term deflection of GFRP-concrete hybrid structures with partial interaction under service loading, based on the age-adjusted effective modulus method (AEMM). The calculated results from the proposed calculation model agree well with the available experimental results. A further discussion is also made about the effects of the concrete creep and shrinkage and the GFRP creep, as well as the shear stiffness of shear connectors on long-term behavior of GFRP-concrete hybrid structures. Especially, the significant influence of concrete shrinkage on long-term interface slip, interface shear and long-term deflection is found. What’s more, the long-term deflection may increase with the augment of horizontal shear stiffness of shear connectors.

1. Introduction Glass fiber-reinforced polymer (GFRP)-concrete hybrid structures consist of GFRP pultruded profiles in tension zone and concrete slabs in compression zone, which are connected by shear connectors to work together. It may be in favor of decreasing not only the instantaneous deflections but also in diminishing the long-term deflections due to the combination of GFRP and concrete slab. The remarkable advantages of GFRP-concrete hybrid structure have been discovered, leading to a large growth in the construction and bridge projects in recent three decades [1–5]. However, there is a factor that prevents GFRP pultruded profiles from being used more widely, which is the knowledge deficiency of the coupling effects of concrete and GFRP. Especially, the sensitivity to creep represents one of the major restrictions of using GFRP materials in construction under sustained loading. GFRP creep has a great influence on the long-term deflection of GFRP-concrete hybrid structural systems, which should be taken into account at the serviceability limit states [6–8]. The creep and the shrinkage of concrete, as well as the creep of GFRP exert a large influence on the long-term behavior of GFRP-concrete hybrid structures. Under sustained loading, they will induce timedependent performance changes in stress, deflection, and interface slip between concrete and GFRP material. At present, there’s a mounting ⁎

number of researchers carrying out the studies on short-term behavior of GFRP–concrete hybrid structural systems, which mainly focus on evaluation of stiffness, strength, deflection, and theoretical analysis [8–16]. But the lack of attention paid on the time-dependent effects brings unreasonable prediction of the behavior of the hybrid structures in the serviceability limit state, in especial when the concrete shrinkage and partial interaction are neglected. For the moment, the long-term behavior of GFRP–concrete hybrid structures has been investigated by several researchers, e.g. Deskovic et al. [17] Mendes et al. [18] Gonilha et al. [19], Xue et al. [20] and Alachek et al. [21]. Their research results revealed that the concrete creep and the GFRP creep significantly affected the long-term behavior of the hybrid structures. In addition, the long-term slips at the interface were also investigated by Xue et al. [20]. However, experiments on the effect of shrinkage have not been performed and theoretical studies have also been inadequate. Simplified methods based on equivalent section flexural stiffness or Timoshenko’s beam model (TBM) are used to predict the long-term deflection in the studies [18,19]. These methods are indeed simple to use, but the results obtained may be inaccurate since the shrinkage effect and the interface slip for partial interaction in the hybrid structures under sustained loading have been ignored. Moreover, the available methods cannot reveal the law of long-term internal forces as well, especially for partial interaction. Therefore, these time-dependent behaviors of GFRP-

Corresponding author at: Department of Structural Engineering, Tongji University, Shanghai, China. E-mail addresses: [email protected] (S. Zhang), [email protected] (W.C. Xue).

https://doi.org/10.1016/j.compstruct.2019.02.072 Received 31 August 2018; Received in revised form 28 December 2018; Accepted 18 February 2019 Available online 20 February 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature α

a modifying index, the average value of a is 1.33 × 10–2

A

= KL

Ac Af Ap Aweb A0 =

A01 =

1 E (t , t0) A

+

height of concrete slab height of GFRP profiles the inertia moment of concrete the inertia moment of GFRP profiles Ic / n (t , t 0 ) moment of inertia of hybrid girders with no interaction I02 = E (t , t0 ) I f + c (t , t0) Ic / n (t , t 0 ) time-dependent moment of inertia k connectors shear stiffness KL horizontal shear stiffness per unit length L span of hybrid structures M (t 0 ) derivative of cross section moment Mc (x , t ) moment in concrete slab under long-term loading Mf (x , t ) moment in GFRP profiles under long-term loading n (t , t 0) = Ef (t , t 0)/Ec (t , t 0 ) modular ratio ne, ng flexural creep rate exponents and shear creep rate exponents p longitudinal spacing of connectors q uniform load applied to hybrid beam s slip s (x , t ) long-term slip t0 the age of concrete t time d u = EIc V (x , t ) horizontal shear force or axial force at concrete slab yoct(t) distance from the top fiber of concrete slab to the composite centroid yofb(t) distance from the bottom fiber of GFRP profiles to the composite centroid ze, zg, we, wg correlation coefficients c, b (x , t ) strain at the concrete slab bottom c 0, b (x , t 0 ) instantaneous strain at the bottom of the concrete slab f 0 , t (x , t 0 ) instantaneous strain at top of GFRP profiles f , t (x , t ) strain at the GFRP profile top slip (x , t ) slip strain shrinkage strain of concrete sh (t , t 0 ) ( t ) creep coefficients of GFRP due to longitudinal deflections E creep coefficients of GFRP due to shear deflections G (t ) creep coefficient of concrete c (t , t 0 ) creep coefficient of GFRP E (t , t 0 ) ( x , t ) additional curvature s hc hf Ic If I01 = If +

dc 2 E (t , t0) I

area of the concrete slab area of GFRP profile the concrete wedge area in GFRP perfobond shear connectors the web sectional area A01 2 transformed area

I01 + A01 dc Ac Af

equivalent area

Ac + n (t , t0) Af

B = EI + EA · dc 2 equivalent stiffness BEI = BS 2 b2 dc

EA dc d2

1

= KL ( EIc + EA ) distance between concrete slab centroid and GFRP profiles centroid Ec(t0) elastic modulus of concrete at t0 E (t ) Ec (t , t 0) = [1 + c (0t , t )] the age-adjusted effective modulus concrete 0 c at t Ef(t0) longitudinal tensile modulus of GFRP at t0 Ef (t ) the time-dependent elasticity modulus Eft creep moduli for the elasticity in flexure Ef (t0)

the age-adjusted effective longitudinal tensile modulus of GFRP at t EI = Ec (t 0) Ic + Ef (t 0 ) I f stiffness of hybrid girders with no interaction 1 1 1 = E (t ) A + E (t ) A

Ef (t , t 0 ) =

EA

fe0 f shear f pflex

f shear f pflex f f flex Gf (t 0 ) Gf (t ) Gweb(t0) Gweb (t , t0 ) Gft

1 + E (t , t 0 )

c 0

c

f

0

f

total instantaneous deflection shear deflection flexural deflection

long-term shear deflections long-term flexural deflections with partial interaction total long-term deflection long-term flexural deflections with full interaction the instantaneous shear modulus the time-dependent shear modulus the web shear modulus at t0 the age-adjusted effective shear modulus of the web at t shear modulus in flexure

concrete hybrid structures should be studied well, which will help designers to deal with these factors effectively in practice. The effect of concrete shrinkage on long-term behavior of steelconcrete hybrid beams was assessed by Fan et al. [22]. In their study, after 1,100 days, the deflection caused by the concrete shrinkage was about 3.8 times of the deflection caused by the creep. In 1992, Tarantino and Dezi [23] explored the effect of the slip between the concrete slab and the steel beam on the time-dependent behavior of steelconcrete hybrid beams based on the step-by-step method, and the distribution of shear force at the steel-concrete interface. Based on the analytical results, significant effects of creep and shrinkage of concrete on long-term internal forces were investigated, especially for shear force on the connectors. The step-by-step method including the creep compliance and the shrinkage function was always used to analyze the long-term behavior of steel-concrete hybrid beams [24,25]. However, this method needs a complex computational procedure in usual. To simplify the analysis of the long-term behavior, Bradford et al developed the one-step method called the age-adjusted effective modulus method (AEMM) [26], and used it to analyze the long-term behavior of a two-span steel-concrete hybrid beam. Thus, the theoretical foundation has been established to analyze the long-term behavior of GFRP-

concrete hybrid structures on the basis of these efforts. Above all, the long-term behavior of the GFRP-concrete hybrid structures is usually more complex than that of traditional steel-concrete hybrid structures, which is attributed to the coupling effects of the shrinkage and the creep of concrete, the creep of FRP, as well as the interface ship between the two materials. Neverthless, these coupling effects were not considered by the available predictive models. Moreover, very few theoretical analyses on long-term deflection of GFRP-concrete hybrid structures with interface slip have been reflected in the literature so far. On account of the aforementioned problems in previous studies, this paper presents a theoretical analysis based on the age-adjusted effective modulus method (AEMM) to predict the longterm deflection of GFRP-concrete hybrid structure, considering the effects of the shrinkage and the creep of concrete, the creep of GFRP and the interface slip. 2. Creep models 2.1. GFRP profiles creep models GFRP pultruded profiles are susceptible to creep under compression, 2

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S. Zhang, et al.

tension, bending moment and shear [7]. And under different load forms, there are obvious differences in the creep laws. A few studies have indicated that creep has much more marked effect on GFRP materials when under compression than in tension [7,18,19], which is mainly caused by the sensitivity to creep when under compression. Findley’s power law which is an empirical model based on experimental results is often used to predict the GFRP creep behavior [8,19]. But Gonilha, J.A. et al. [19] considered this method was not applicable to the long-term behavior of the GFRP–concrete hybrid structures, as a result of the inconsistency with their experiment tests. Afterwards, three representative GFRP creep models were come up to predict the creep behavior of GFRP pultruded profiles by Bank [27], the EuroComp [28] and Italian Guidelines [29], respectively. Bank [27] derived a new creep model of GFRP pultruded profiles based on Findley’s power law, in which the coefficients needed to be further determined by test results of GFRP pultruded profiles subjected to compression and bending moment; the EuroComp [28] provided the long-term moduli of unidirectional FRP materials in tension and pure shear based on graphical data [19]; the Italian Guidelines [29] suggested discrete coefficients of timedependent elasticity modulus and shear modulus (similar to creep coefficient of concrete). All these three creep models have considered the effect of the shear deflection on the long-term deflection of GFRP pultruded profiles. Meanwhile, appropriate creep model should be selected according to the actual stress distribution on the cross-section of the hybrid structure. For easy interpretation, the three creep models of GFRP profiles can be rewritten into a uniform format:

Ef (t ) =

Gf (t ) =

between the above two models. 2.2. Concrete creep and shrinkage models Until now, ACI 209(92) Model, CEB-FIP MC90 Model, B3 Model and GL2000 Model are widely used to predict the long-term response of concrete due to creep and shrinkage [30,31]. Among them, CEB-FIP MC90 Model could relatively accurately calculate the creep coefficient and shrinkage strain of any age [32]. But CEB-FIP MC90 Model fails to consider the effect of the basic creep and the basic shrinkage on the long-term behavior of concrete because of the concrete loading time, the strength grade of cement and the maturity of concrete etc. So, on the basis of CEB-FIP MC90 Model, CEB-FIP MC 2010 [33] presents an improved creep and shrinkage model of concrete to predict the longterm behavior of concrete, including basic creep, dry creep, basic shrinkage and dry shrinkage. The improved model, namely CEB-FIP MC2010 Model takes into consideration many factors, such as the loading time of concrete, the compressive strength of concrete, the strength grade of cement, the maturity of concrete and the environmental conditions. In addition, different from the CEB-FIP MC90 Model, creep calculation formula in the form of logarithm in CEB-FIP MC2010 Model has no limit value as time goes by, which means creep can develop all the time. Therefore, in this paper, the creep and shrinkage behavior of the concrete is estimated according to the CEB-FIP MC2010 Model. 3. Theoretical analysis on long-term deflection of GFRP-concrete hybrid structure

Ef (t 0 ) 1+

(1)

E (t )

In this section, the theoretical formula for GFRP-concrete hybrid structures is deduced as follows. At first, internal force equilibrium equations of cross-section are given considering the concrete creep and shrinkage, the GFRP creep, as well as the partial interaction at the slab/ GFRP profile interface. Then, through the cross-section analysis, the additional curvature equation is determined according to the age-adjusted effective modulus method (AEMM) [26,31]. At last, the timedependent additional curvature is deduced according to the cross-section analysis by combining the above equations, and the long-term deflection is derived through the integral of curvature along the span. And In order to understand the formula derivation process more clearly, the flow chart of formula derivation is shown in Fig. 2.

Gf (t 0 ) 1+

(2)

G (t )

where Ef (t ) and Gf (t ) are the time-dependent elasticity modulus and shear modulus, respectively. Ef (t 0 ) and Gf (t 0 ) are the instantaneous elasticity modulus and shear modulus, respectively. E (t ) and G (t ) are the coefficients of creep due to longitudinal deflection and shear deflection, respectively. From the models of Bank [27], E (t ) and G (t ) are determined from Eq. (3) and Eq. (4), respectively; according to the EuroComp [28] and reference [19], E (t ) and G (t ) are re-written as Eq. (5) and Eq. (6), respectively; the Italian Guideline [29] presents the discrete coefficients of creep E (t ) and G (t ) over time as shown in Table 1. E (t )

=

G (t )

=

E (t )

=

G (t )

=

Ef (t 0) Eft Gf (t0) Gft

t ne

3.1. Basic assumptions

(3)

t ng

The proposed creep model is used to analyze the long-term behavior of GFRP-concrete hybrid structures under sustained loading, which is based on the following assumptions:

(4)

(z e

1 we ln t )

1

(z g

1 wg ln t )

1

(1) Generally, the compressive stress of concrete induced by loading should be lower than 40% of its compressive strength. The constitutive relationship of GFRP material is linear-elastic. (2) The connectors between GFRP profiles and concrete slabs are assumed to be linear-elastic under service loading. (3) The connectors are fixed continuously in the hybrid structure. And equal spacing is required.

(5) (6)

where t is time; are known as creep modulus for the elasticity and shear modulus in flexure proposed by Bank [27]; ne and ng are the creep rate exponents, 0.3 suggested by Bank [27]; ze, zg, we and wg are the coefficients obtained from reference [19]. The evolution coefficients of creep with time based on the abovementioned models are shown in Fig. 1. The main conclusions are as follows: the creep for the elasticity modulus proposed by Bank [27] subjected to bending moment is much more likely to happen than that of EuroComp models in tension; the creep for the shear modulus suggested by the EuroComp [28] is more sensitive to pure shear than that of Bank models in bending; the models regardless of elasticity modulus and shear modulus in the Italian Guidelines [29] are completely

Eft

and Gft

Table 1 Coefficients of creep at different times in the Italian guidelines. t (years) 1 5 10 30 50

3

E (t )

0.26 0.42 0.50 0.60 0.66

G (t )

0.57 0.98 1.23 1.76 2.09

Composite Structures 216 (2019) 1–11

S. Zhang, et al.

Fig. 1. Evolution of the different coefficients of creep of the GFRP material.

(4) Assume that the two elements meet the plane cross-section hypothesis, respectively, with the similar curvature. (5) There is no vertical separation between the slab and the beam. (6) Superposition principle is adopted to calculate the deflection due to bending moment and shear.

1 1 1 = + EA Ec (t0 ) A c Ef (t 0 ) Af where s is slip; KL is horizontal shear stiffness per unit length; p is longitudinal spacing of connector; k is stiffness of connector; M (t 0 ) is the derivative of cross section moment; Ec(t0) and Ef(t0) are elastic modulus of concrete and longitudinal tensile modulus of GFRP, respectively; Ac and Af are area of concrete slab and GFRP profile, respectively; Ic and If are the inertia moment of concrete about concrete slab centroid and GFRP profiles centroid, respectively; dc is distance from the concrete slab centroid to the GFRP profiles centroid. The solutions of the slip and deflection, subjected to a uniform loading q, are shown in Eq. (8) and Eq. (9), respectively.

3.2. Short-term analysis According to reference [34], the slip s is calculated as follows:

s KL =

d2

KL ( EIc +

1 EA

)s =

k p

dc M EI

(t 0 ) (7)

and define

s=

EI = Ec (t 0) Ic + Ef (t 0 ) I f

uq sinh[b (l/2 b3 cosh(bl/2)

Fig. 2. Flow chart of formula derivation. 4

x )] +

uq (2x 2b 2

l)

(8)

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S. Zhang, et al.

f pflex =

qx (2lx 2 24B

x3

l3)

qx (l 2Bs b2

q cosh[b (x Bs b4 cosh(bl/2)

x) +

l/2)]

q Bs b4

s (x , t ) =

B = EI + EA · dc 2 dc2 1 + ) EI EA

slip (x ,

BEI BS = EA dc2

c , b (x ,

V (x , t ) = x2

c , b (x ,

t)

f , t (x ,

t)

t) =

c (t , t 0 ) c 0, b (x , t 0 )

+

sh (t , t 0 )

+

Mc (x , t ) hc · Ec (t , t 0 ) Ic 2

V (x , t ) Ec (t , t 0) Ac (14a)

f , t (x ,

(10)

t) =

E (t , t 0 ) f 0 , t (x , t 0 )

in which

where Gweb(t0) is the web shear modulus of the GFRP profiles; Aweb is the web sectional area. The total instantaneous deflection (fe0) of the hybrid structure (beam/slab) is therefore equal to the sum of the flexural deflection ( f pflex ) of the hybrid section and the shear deflection ( f shear ) of the GFRP profile (Eq. (11)):

fe0 = f pflex + f shear

2

1 KL

According to the basic assumptions, the strains in the concrete slab and GFRP profiles are calculated based on the superposition principle. The strains in bottom of the concrete slab and top of the GFRP profiles are:

where l is span of hybrid structures. The GFRP profile has a low ratio of shear modulus to longitudinal modulus, therefore the shear deflection of the hybrid structure must be taken into account. Under uniform loading, the shear deflection ( f shear ) was usually calculated according to Timoshenko’s beam theory [35] (Eq. (10)):

f shear

s (x , t ) = x

t) =

(13)

d u= c EI

ql2 8Aweb Gweb (t0 )

(12)

where s (x , t ) is slip between the GFRP profile and the concrete slab; V (x , t ) is horizontal shear force at the interface. The slip strain slip (x , t ) should be equal to the difference between the interfacial strains in the concrete slab c, b (x , t ) and that in the GFRP profile f , t (x , t ) (Eq. (13)) [23]:

(9)

in which

b2 = KL (

1 d V (x , t ) KL dx

Ec (t , t 0) =

[1 +

Ec (t0) c (t , t0)]

Mf (x , t ) hf V (x , t ) · + Ef (t , t 0 ) I f 2 Ef (t , t 0) Af

and Ef (t , t 0 ) =

(14b)

Ef (t0) 1 + E (t , t 0 )

where Ec (t , t 0) and Ef (t , t 0) are the age-adjusted effective moduli of concrete and GFRP at time t, respectively; χ is the aging factor of concrete; c (t , t 0 ) and E (t , t 0 ) are the creep coefficients of concrete and GFRP from time t0 to t, respectively; c 0, b (x , t 0) and f 0 , t (x , t 0) are instantaneous strains in the bottom of concrete slab and the top of GFRP profiles, respectively; sh (t , t 0 ) is the shrinkage strain of concrete; hc and hf are height of concrete slab and GFRP profile, respectively. According to the equilibrium of cross section moments (M), the equations are as follows:

(11)

3.3. Long-term analysis

(15)

Mc (x , t ) + Mf (x , t ) + V (x , t ) dc = 0

Available analytical models for predicting the long-term deflection of GFRP-concrete hybrid structures were presented based on Timoshenko’s beam model (TBM) [19] in which the instantaneous moduli were replaced by time-dependent moduli. However, it is imprecise to use TBM to calculate the long-term deflection of GFRP-concrete hybrid structures based on equivalent section flexural stiffness, because the effects of the long-term internal force, the concrete shrinkage and the interface slip have been ignored. Therefore, the deflections herein need to be re-calculated with new calculation model. When the beam is subjected to sustained loading, the cross-section mechanical analysis model from time t0 to t can be listed in Fig. 3. According to basic assumptions (2) and (3), the relationship between the long-term slip s (x , t ) and the horizontal shear force V (x , t ) is shown in Eq. (12):

where Mc (x , t ) is moment in concrete slab under long-term loading; Mf (x , t ) is moment in GFRP profiles under long-term loading. According to the AEMM and linear creep theory, it is assumed that there is an identical additional curvature ( s (x , t ) ) in the concrete slab and the GFRP profile (Eq. (16)): s (x ,

t) =

c (t , t0 )

(x , t 0 ) +

Mc (x , t ) = Ec (t , t0) Ic

E (t , t0 )

(x , t 0 ) +

Mf (x , t ) Ef (t , t0 ) I f (16)

where ϕ(x,t0) is the initial curvature at time t0. By combining Eq. (15) and Eq. (16), Mf (x , t ) and Mc (x , t ) can be computed by

Fig. 3. Analysis of the cross-section. 5

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S. Zhang, et al.

[ c (t , t0 )

Mf (x , t ) =

E (t , t 0 )] Ec (t , t 0 ) Ic Ec (t , t0) Ic + Ef (t , t0) I f

(x , t 0 )

(17)

V (x , t ) d c + [ c ( t , t 0 )

E (t , t 0 )] Ef Ef (t , t0) I f

Mc (x , t ) =

Ec (t , t0) Ic

where Gweb (t , t0 ) the age-adjusted effective shear modulus of GFRP at t. So the total mid-span long-term deflection can be computed by Eq. (27)

V (x , t ) d c

1 ( t , t 0 ) I f (x , t 0 )

f = f pflex + f shear

+1

(18)

4. Examples, prediction and analysis with full interaction

Replace Eqs. (14a) and (14b) with Eq. (13) to eliminate c, b (x , t ) and f , t (x , t ) , and replace Eqs. (17) and (18) with Eqs. (14a) and (14b) to eliminate Mf (x , t ) and Mc (x , t ) , thus Eq. (19) can be derived: 1 2 V (x , t ) KL x2

1 E (t , t 0 ) A

=

+

dc 2 E (t , t0) I

V (x , t )

sh (t , t0 ) c (t , t 0 ) c 0, b (x , t 0 ) Ec (t , t0) Ic hf / 2 Ef (t , t0) I f h c / 2 [ c (t , t 0 ) E (t , t0) I

Considering A = KL

C = KL

{

E (x ,

t)

Ec (t , t0) Ic hf / 2

1 E (t , t 0 ) A

+

f 0 , t (x , t 0 )

E (t , t 0 ) I

[ c (t , t 0 )

sh (t , t 0 )

E (t , t 0 )]

1 E (t , t 0 ) A

=

Eq. (19) can be rewritten as Eq. (20)

V (x , t ) x2

(19)

dc2 E (t , t 0 ) I

fine E (t , t 0 ) I = Ec (t , t 0 ) Ic + Ef (t , t 0 ) I f ; 2

(x , t 0 )

c (t , t 0 ) c 0, b (x , t0 )

Ef (t , t0) I f h c / 2

For the hybrid structures with full interaction, the shear stiffness of the connectors, namely KL theoretically tends to be infinite (KL → ∞). And replace Eq. (28) with Eq. (25) to eliminate f 0 , t (x , t 0) and c 0, b (x , t 0 ) , thus the Eq. (25) can be rewritten as Eq. (29) with full interaction:

E (t , t0 ) f 0, t (x , t 0 )

E (t , t0 )]

}

+

Ax

(20)

Ax

+ C2 e

(21)

V (0, t ) = 0 at x = 0 V (l, t ) = 0 at x = l

1 KL

s (x , t ) =

Ax

+e

+

1+e

Al

V (x , t ) = dx

A01 =

(23)

A (e

Ax

e

1+e

(24)

Al

A01

I01 + A01 dc2 Ac Af Ac + n (t , t0) Af

E (t , t0 ) I f

+

c (t , t0 ) Ic / n (t , t 0 )

n (t , t 0) = Ef (t , t 0)/Ec (t , t 0 ) So the long-term deflection at mid-span due to flexure with effect of interface slip can be calculated by

f pflex =

s (x ,

¯ t ) Mdx =

I02 f + A0 dc (1 I01 e0

2e

A l /2

1+e

Al

)

C ¯ Mdx KL (25)

In addition, the long-term shear deflection provided only by the web of the GFRP profiles should be considered by Eq. (26) [19,27].

f shear

ql2 8Aweb Gweb (t , t 0 )

Af +

)+

Ac hc 2n (t , t0)

Ac n (t , t0)

; yofb (t ) = h

yoct (t )

c (t , t 0 ) yoct (t 0 )

E (t , t 0 ) yofb (t0 )

I02 f I01 + A01 dc2 e 0

+ A 0 dc

1

(1

hc 2yoct (t0) hf

2yofb (t0)

l2 sh (t , t 0 ) 8

)

fe0 (29)

(30)

In order to consider the effect of concrete shrinkage and internal force on long-term deflection of hybrid structures, the Eq. (30) proposed by the authors is selected to calculate the long-term deflections of these simply supported hybrid structures with full interaction subjected to a long-term loading in the references. However these effects are ignored in TBM. Six selected examples PROT_125 from Mendes et al. [18], Test 1 and Test 2 from Gonilha et al. [19], as well as PCB2, PCB4 and PCB5 reported by Alachek et al. [21], have been calculated by the proposed formula. PROT_125, Test 1 and Test 2 were all made up of GFRP pultruded profiles and thin steel fiber reinforced self-compacting concrete (SFRSCC), while PCB2, PCB4 and PCB5 all consisted of GFRP pultruded profiles and normal concrete slab. It should be noted that for SFRSCC structures, in particular, if the volume of steel fibers is lower than 1.0%, the effect of the fibers on the creep behavior will be neglected [19]. SFRSCC used in the footbridge deck has a fiber volume content of 1.9% for PROT_125, and 2.5% for Test 1 and Test 2. However, because of the relatively low stress levels in the tests and the compression state(in longitudinal direction) of the slabs, there is no crack emerging on the slabs. So the fiber reinforcement is not effective for the structures [36] at these load levels. Therefore, the creep and the shrinkage model presented by fib model code for concrete structure 2010 for regular concrete elements could be used. The temperature and relative humidity during measurement and the age of concrete have a significant influence on creep and shrinkage of concrete as is known and should be taken into account in the

A (l x ) )

I01 = If + Ic / n (t , t 0 ) I02 =

2

f = f flex + f shear

Combine Eqs. (16)–(18) and Eq. (23), and define

A0 =

hf

4.1. Calculation examples

1 1 C KL A

(28)

Mid-span long-term deflection is

(22)

A (l x )

Af (hc +

+

Expressions for the shear force V (x , t ) and slip s (x , t ) throughout the beams can be obtained from Eqs. (23) and (24):

C e A

c 0, t (x , t 0 )

f flex = A 0 dc

C A

C1 and C2 are constants which can be determined based on the boundary conditions as follows:

V (x , t ) =

f 0, b (x , t 0 )

= h c (x , t 0 )

where f 0, b (x , t 0) and c 0, t (x , t 0) are instantaneous strains in the bottom of GFRP profiles and the top of the concrete slab, respectively; yoct(t) and yofb(t) are the distances respectively from the top fiber of concrete slab and the bottom fiber of GFRP profiles to the hybrid centroid.

A general solution of Eq. (20) is given in Eq. (21):

V (x , t ) = C1 e

= h f (x , t 0 )

c 0, b (x , t0 )

yoct (t ) =

1 Ef (t , t0) Af

A V (x , t ) = C

f 0 , t (x , t 0 )

and define

and de-

(x , t 0 )

1 Ec (t , t0) Ac

(27)

ql2 8Aweb Gweb (t 0 )

(26) 6

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calculation formula. During the test period, average temperature and average relative humidity of PROT_125 are 32 °C and 78% [18], respectively, and they are 23 °C and 60% for Test 1 and Test 2 [19]. In the creep test of Alachek et al. [21], average temperature and average relative humidity of PCB2, PCB4 and PCB5 are 18.4 °C and 65%, 12.3 °C and 80%, 12.3 °C and 40%, respectively. The age of concrete of PROT_125, Test 1, Test 2, PCB2, PCB4 and PCB5 are 90 days, 562 days, 254 days, 129 days, 174 days and 174 days, respectively [18,19,21]. Initial deflections of the six specimens PROT_125, Test 1, Test 2, PCB2, PCB4 and PCB5 which are calculated by TBM are 12.55 mm, 25.8 mm, 15.13 mm, 13.7 mm, 13.55 mm and 13.55 mm, respectively, with a 17% maximum errors compared with their test results. After the calculation, the heights of the neutral axis of PROT_125, Test 1, Test 2, PCB2, PCB4 and PCB5 are 41 mm, 29.7 mm, 30.9 mm, 34.88 mm, 35.22 mm and 35.22 mm, respectively. Except for PROT_125, the height of the neutral axis of other specimens is located in the concrete slab. Considering the actual cross-section stress distribution of the GFRPconcrete hybrid structures, Bank’s model is applied to calculate the long-term deflection of PROT_125, since the GFRP profiles are in bending. Besides, the mixed model, combined by EuroComp’s tension

4.2. Prediction of long-term deflection Fig. 5 compares the long-term predictions using the proposed 6

10

PROT_125-experimental Proposed formula Induced by shrinkage only

8 6 4 2 0

0

10

20

30

40

PCB2 proposed formula Induced by shrinkage only

5

Long-term deflection/mm

Long-term deflection /mm

creep model and Bank’s shear model in flexure [19], can be used to predict the structure deflections of the others due to the tension state of the GFRP profiles. The CEB-FIP MC2010 model is adopted for the analysis of the shrinkage and the creep of concrete here. Fig. 4 shows the comparison of the calculated long-term deflections with the experimental data. It can be seen that results of the proposed formula agree well with the test results, though there are some errors come from the effect of the variable environmental conditions. The deflections caused only by the shrinkage are calculated and presented in Fig. 4 as well. The terminal long-term deflections of PROT_125, Test 1, Test 2, PCB2, PCB4 and PCB5 caused by the concrete shrinkage are 0.57 mm, 0.23 mm, 0.27 mm, 0.36 mm, 0.19 mm and 0.37 mm, respectively, which are about 9.8%, 9.1%, 24.5%, 9.8%, 6.4% and 8.2% of their total long-term deflections. So, the long-term deflections due to concrete shrinkage could not be ignored, even when time of concrete under loaded condition is short.

4 3 2 1 0

50

0

20

40

60

(a) PROT_125 4.0

Long-term deflection/mm

Long-term deflection/mm

3.0 2.5 2.0 1.5 1.0 0.5 0.0

0

20

40

60

80

100

120

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

PCB4 proposed formula Induced by shrinkage only

0

20

40

60

(b) Test 1

1.0 0.5 0

20

40

60

PCB5 proposed formula Induced by shrinkage only

6

Long-term deflection/mm

Long-term deflection/mm

Test 2-experimental proposed formula Inducted by shrinkage only

1.5

0.0

100 120 140 160

(e) PCB4 7

3.0

2.0

80

Time/days

Time/days

2.5

100 120 140 160

(d) PCB2

Test 1-experimental proposed fomula Induced by shrinkage only

3.5

80

Time/days

Time/years

80

5 4 3 2 1 0

100

0

20

40

60

80

100 120 140 160

Time/years

Time/days

(c) Test 2

(f) PCB4

Fig. 4. Comparisons of test values and calculated values. 7

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formula in this paper with that of Timoshenko’s beam model (TBM) which has been verified in reference [19]. Although TBM are indeed simple to use by replacing the short-term modulus with the long-term modulus, the results obtained may be inaccurate since the shrinkage effect under sustained loading has been ignored. Moreover, the method cannot reveal the law of the long-term internal forces. Table 2 shows the differences of 5, 10, 30 and 50 years, where Δδ, δ0, Δδs and Δδcs are the long-term deflection, the instantaneous deflection, the shear deflection and the shrinkage deflection, respectively. As shown in Fig. 5 and Table 2, the long-term deflections based on the proposed formulas are much larger than the predictive values calculated by Timoshenko’s beam model because of the concrete shrinkage and effect of internal forces. Using the proposed formula in this paper, after 50 years these hybrid beams are expected to present mid-span long-term deflections of 22.71 mm, 5.41 mm, 4.52 mm, 10.29 mm, 8.75 mm and 11.23 mm for PROT_125, Test 1, Test 2, PCB2, PCB4 and PCB5, respectively. Yet, for these specimens, the long-term deflections computed with Timoshenko’s beam model are 18.91 mm, 4.87 mm, 3.17 mm, 7.54 mm, 7.09 mm and 7.81 mm, respectively, after the same period. After five years, shrinkage deflection of PROT_125, Test 1, Test 2, PCB2, PCB4

and PCB5, respectively, are about 12.21%, 14.90%, 12.94%, 6.72%, 6.71% and 6.70% of their total long-term deflections, as shown in Table 2. After that, the deflection increments induced by shrinkage were fairly slight because the shrinkage of concrete tends to be stabilizing. In addition, the shear deflections should not be overlooked. As can be seen, the values can reach 38.45%, 47.35%, 47.23%, 55.54% and 43.28% of the total long-term deflections at the age of 50 years, for Test 1, Test 2, PCB2, PCB4 and PCB5, respectively. 5. Example and analysis with partial interaction A new experimental study was conducted by Xue et al. [20] on the one-year long-term behavior of two GFRP-concrete hybrid beams with the effect of the interface slip. One hybrid beam (RCB) without external prestressed tendon was selected as referenced hybrid beam, which was not affected by external prestressed tendon. Span of the beam RCB is 4500 mm; height of the beam is 345 mm; width and height of concrete slab are 600 mm and 85 mm, respectively; height of the double-hole rectangular GFRP box beam is 260 mm and thickness of the wall is 10 mm. In addition, GFRP perfobond shear connectors were used in the 12

PROT_125

25

Long-term deflection/mm

Long-term deflection/mm

30

20 15 Proposed formula TBM

10 5 0

0

10

20

30

40

50

8 6 Poposed formula TBM

4 2 0

60

PCB2

10

0

10

20

Time/years

(a) PROT_125

4 Proposed formula TBM

3 2 1 0

10

20

30

40

50

0

10

20

30

40

(e) PCB4

50

60

12

Long-term deflection/mm

Long-term deflection/mm

2

(b) Test 1

3 2

Proposed formula TBM

1 10

Proposed formula TBM

4

Time/years

Test 2

0

6

Time/years

4

0

8

0

60

6 5

60

PCB4

5

0

50

(d) PCB2

Test 1

6

40

10

Long-term deflection/mm

Long-term deflection/mm

7

30 Time/years

20

30

40

50

10 8 6

2 0

60

Proposed formula TBM

4

0

10

20

30

Time/years

Time/years

(c) Test 2

(f) PCB5

Fig. 5. Predictions of long-term additional deflections with 50 years. 8

40

50

60

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Table 2 Predictions of long-term deflections. Specimens

Time elapsed (years)

Δδ (mm)

Δδ/δ0 (%)

Δδ (mm)

Δδ/δ0 (%)

Δδs

Δδs/Δδ (%)

Δδcs

Δδcs/Δδ (%)

PROT_125 [18]

5 10 30 50 5 10 30 50 5 10 30 50 5 10 30 50 5 10 30 50 5 10 30 50

14.17 16.21 20.35 22.71 4.16 4.5 5.08 5.41 3.3 3.62 4.2 4.52 6.85 7.74 9.39 10.29 5.47 6.3 7.87 8.75 7.76 8.67 10.32 11.23

112.91 129.16 162.15 181.27 16.12 17.44 19.69 20.97 21.81 23.93 27.76 29.87 46.60 52.65 63.88 70.00 37.59 43.30 54.09 60.14 53.33 59.59 70.93 77.18

10.46 12.36 16.31 18.94 3.48 3.8 4.43 4.87 2.26 2.47 2.88 3.17 4.73 5.4 6.8 7.54 4.27 4.94 6.35 7.09 5.04 5.7 7.08 7.81

83.35 98.49 129.96 150.92 13.49 14.73 17.17 18.88 14.94 16.33 19.04 20.95 32.18 36.73 46.26 51.29 29.35 33.95 43.64 48.73 34.64 39.18 48.66 53.68

0.54 0.66 0.92 1.08 1.01 1.25 1.77 2.08 1.06 1.31 1.83 2.14 2.35 2.93 4.14 4.86 2.35 2.93 4.14 4.86 2.35 2.93 4.14 4.86

3.81 4.07 4.52 4.75 24.28 27.78 34.84 38.45 32.12 36.19 43.57 47.35 34.31 37.86 44.09 47.23 42.96 46.51 52.60 55.54 30.28 33.79 40.12 43.28

1.73 1.78 1.82 1.83 0.62 0.66 0.69 0.7 0.81 0.86 0.89 0.9 0.46 0.52 0.64 0.69 0.38 0.43 0.54 0.6 0.52 0.58 0.7 0.76

12.21 10.98 8.94 8.04 14.90 14.67 13.58 12.94 24.55 23.76 21.19 19.91 6.72 6.72 6.82 6.71 6.95 6.83 6.86 6.86 6.70 6.69 6.78 6.77

Test 1 [19]

Test 2 [19]

PCB2 [21]

PCB4 [21]

PCB5 [21]

Fig. 6. Details of GFRP-concrete hybrid beams (Xue et al. [20]).

12

Table 3 Predictions of long-term deflections.

Deflection/mm

10

Time elapsed (years)

8 6

RCB

10

30

50

Total deflection

Δδ (mm) Δδ/δ0 (%)

4.50 66.37

5.10 75.22

5.99 88.34

6.42 94.69

Shear deflection

Δδs Δδs/Δδ (%) Δδcs Δδcs/Δδ (%)

0.85 18.89 2.13 47.33

1.05 20.59 2.35 46.07

1.46 24.37 2.52 42.01

1.70 26.48 2.56 39.88

Shrinkage deflection

Proposed formula

4

5

2 0

0

50

100

150

200

250

Time/days

300

350

where α is a modifying index, the average value of α is 1.33 × 10–2; Ap is the concrete wedge area; tp is the thickness of the FRP perforated plate (mm); and Ec is the elastic modulus of concrete. The horizontal shear stiffness per unit length (KL) is about 650 N/ mm2. The applied uniform load is 13.2 kN/m. The calculated values of instantaneous slip at the beam end and instantaneous deflection at the mid-span by Eq. (8) and Eq. (11) are 0.151 mm and 6.78 mm, respectively, with errors in 5% compared to the test results. According to the actual stress distribution on cross-section of RCB and the effect of interface slip, Bank’s model is applied.

400

Fig. 7. Comparisons of test values and calculated values.

hybrid beam. The detailing of the GFRP-concrete hybrid beam is shown in Fig. 6. The GFRP perfobond shear connectors have a 100 mm spacing and 30 mm diameter. According to the research on the shear behavior of GFRP perfobond shear connectors by Zou et al. [37] in 2016, the stiffness of the connectors can be calculated by Eq. (31):

k=

Ap t p2 Ec

5.1. Long-term deflection

(31)

During test period, average temperature and relative humidity are 9

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0.25

30 days 1 year 10 years 30 years 50 years 50 years inducted by shrinkage only

0.20

Long-term slip/mm

0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25

0

1000

2000 Span/mm

3000

4000

Shear at interface/kN

Fig. 8. Long-term interface slip along the span.

0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 -55 -60

30 days 1 year 10 years 30 years 50 years 50 years inducted by shrinkage only

0

1000

2000 Span/mm

3000

4000 Fig. 10. Effect of shear stiffness (KL) on long-term deflection.

Fig. 9. Long-term interface shear.

force as shown in Fig. 9.

18 °C and 75%, respectively, and the loading time of concrete is 90 days [20]. The test results and calculating results of the deflections of RCB at mid-span (long-term deflection and instantaneous deflection) are presented in Fig. 7. The time fluctuation of the deflections of RCB can be observed, which is caused by the changes of temperature and relative humidity. Although the proposed formulas did not take the changes of environmental conditions into account, the predicted results of longterm deflection generally are in accordance with the test results, as shown in Fig. 7. Table 3 shows the predictive values of the 50-year long-term deflections. The long-term deflection is 6.42 mm (94.69% of their instantaneous deflections), in which long-term deflections due to shear and concrete shrinkage are 26.48% and 39.88% of the total long-term deflection.

5.3. Effect of shear stiffness (KL) on long-term deflection Slip behavior at interface is influenced by not only shear stiffness of shear connectors but also degree of shear connection (longitudinal spacing of the connector). In order to take these two factors into account, a parameter KL = k/p(k and p are shear stiffness and longitudinal spacing of the connector, respectively) generally used in slip calculation of hybrid structures. Shear stiffness (KL) is a comprehensive reflection of connecting configuration. Fig. 10 shows the laws of long-term deflection (△f) and ratio of long-term deflection to instantaneous deflection (fe0) under different shear stiffness (KL) within 50 years. It is noted that as this parameter (KL) increases, △f and the ratio of △f to fe0 also increases. But the increment of deflection decreases gradually with KL increasing. When KL is greater than 650 MPa, values of △f tend to be constants. But the ratio of △f to fe0 still increases until KL rise to 1550 MPa. Obviously, the results indicate when the shear stiffness increases to a possible threshold value beyond which certain effects on the deflection can be negligible, similar to full interaction. For example, at the age of 50 years, the values of △f/fe0 are 1.02 for KL = 1550 MPa and 1.04 for KL → ∞, respectively, as shown in Fig. 10. In addition, Initial deflection is more susceptible to shear stiffness KL compared to long-term deflection.

5.2. Long-term interface slip and shear force Figs. 8 and 9 show the laws of long-term interface slip and shear force along the span of RCB at different time (30 days, 1 year, 10 years, 30 years and 50 years later). The long-term slip decreases gradually from end of the beam to mid span. The maximum slip occurs at the end of the beam, with no slip at the mid span. With the passage of time, the long-term slip and long-term shear force increase but the growth rates of them decrease gradually. Thirty years later, the long-term slip and the long-term shear tend to be constants: 0.18 mm maximum slip at the beam end and maximum shear force of 31.43 kN at the mid span. The significant effect of the concrete shrinkage on the long-term slip and the long-term shear were depicted in Figs. 8 and 9. After 50 years, the slip induced by the shrinkage at the beam end is 0.11 mm almost 61.1% of total long-term slip as shown in Fig. 7, and the shear force induced by the shrinkage at mid span is about 52.1% of the total long-term shear

6. Conclusions An effective calculation model is proposed to predict the long-term behavior of GFRP-concrete hybrid structures with partial shear connectors based on the age-adjusted effective modulus method. In this model, the time-dependent effects of the creep and the shrinkage of concrete, the creep of GFRP and the interface slip are considered. By the 10

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comparison of the results of calculation and that of the tests, the proposed model is proved to be accurate. Meanwhile, the following conclusions can be drawn:

theories and case study. 3rd asia-pacific conference on FRP in structures; 2012. [4] Canning L, Hodgson J, Karuna R, Luke S, Brown P. Progress of advanced composites for civil infrastructure. Proc Inst Civil Eng Struct Build 2007;160(6):307–15. [5] Knippers J, Pelke E, Gabler M, Berger D. Bridges with glass fibre-reinforced polymer decks: the road bridge in friedberg, germany. Struct Eng Int 2010;20(4). pp. 400–404(5). [6] Scott DW. Creep behavior of fiber-reinforced polymeric composites: a review of the technical literature. J Reinf Plast Compos 1995;14(6):588–617. [7] Sá MF, Gomes AM, Correia JR, Silvestre N. Creep behavior of pultruded GFRP elements – part 1: literature review and experimental study. Compos Struct 2011;93(10):2450–9. [8] Sá MF, Gomes AM, Correia JR, Silvestre N. Creep behavior of pultruded GFRP elements – part 2: analytical study. Compos Struct 2011;93(9):2409–18. [9] Canning L, Hollaway L, Thorne AM. 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[22] Fan J, Nie J, Quan L. Long-term behavior of composite beams under positive and negative bending (II) — analytical study. J Struct Eng 2010;136(7):858–65. [23] Tarantino AM, Dezi L. Creep effects in composite beams with flexible shear connectors. J Struct Eng 1992;118(8):2063–81. [24] Dezi L, Tarantino AM. Creep in composite continuous beams. I: theoretical treatment. J Struct Eng 1993;119(7):2095–111. [25] Jurkiewiez B, Buzon S, Sieffert JG. Incremental viscoelastic analysis of composite beams with partial interaction. Comput Struct 2005;83(21–22):1780–91. [26] Bradford MA, Manh HV, Gilbert RI. Numerical analysis of continuous composite beams under service loading. Adv Struct Eng 2002;5(1):1–12. [27] Bank LC. Composites for construction: structural design with FRP materials. New Jersey: John Wiley & Sons Inc; 2006. [28] Clarke JL, editor. Structural design of polymer composites – EuroComp design code and handbook. London: E&FN Spon; 1996. [29] Advisory Committee on Technical Recommendations for Construction. Guide for the design and construction of structures made of FRP pultruded elements. Rome: CNR; 2008. [30] Bažant ZP, Baweja S. Justification and refinements of model b3 for concrete creep and shrinkage 1. statistics and sensitivity. Mater Struct 1995;28(7):415–30. [31] Bažant ZP. Prediction of concrete creep effect using age-adjusted effective modulus method. ACI J 1972:212–7. [32] Euro-International committee for concrete, CEB-FIP model code 1990, EuroInternational committee for concrete, Lausanne; 1993. [33] International federation for structural concrete, CEB-FIP model code 2010, Ernst & Sohn, Berlin; 2013. [34] Hai N, Zatar W, Mutsuyoshi H. Hybrid frp–uhpfrc composite girders: part 2 – analytical approach. Compos Struct 2015;125:653–71. [35] Timoshenko S, Young DH, Weaver Jr W. Vibration problems in engineering. 4th ed. New York: John Wiley & Sons, Inc.; 1974. [36] Dávila P, Barros JO. 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(1) The proposed equations are simple and practical, which can be used to accurately predict the long-term behavior of the hybrid structures with considering the creep and the shrinkage of concrete, the creep of GFRP, as well as various levels of interaction. These equations are validated effective by the experimental data of three large-scale GFRP–concrete hybrid structures with full interaction based on an engineering prototype of a footbridge and a GFRP–concrete hybrid beam with partial interaction. It is turned out that the calculation results are in great agreement with the experimental results. (2) Although the formula of long-term deflection based on Timoshenko’s beam model is simple, the predicted results may be lower than the truth owing to the neglect of the concrete shrinkage and other factors. According to the analysis of three examples with full interaction based on the proposed calculation model, the longterm deflections induced by the concrete shrinkage are about 12.21%, 14.90%, 12.94%, 6.72%, 6.71% and 6.70% of their total long-term deflections at the age of 5 years. In addition, after 50 years, the long-term deflection of the example with partial interaction caused by the concrete shrinkage will reach 39.88% of its total long-term deflection. So the effect of the concrete shrinkage should not be ignored. (3) The long-term interface slip and long-term interface shear increase with the growth rate decreasing gradually as time goes by. Moreover, the significant effect of the concrete shrinkage on the above two has been found. (4) Taking the creep and the shrinkage of concrete and the creep of GFRP into account, the long-term deflection rises with the shear stiffness of shear connectors increasing. But when the shear stiffness increases to some extent, the deflection may tend to be stable, analogous to full interaction. (5) Under the sustained loading, time-dependent shear behavior of the shear connectors may make a difference (is likely to decrease) over time, which will have a direct impact on the long-term behavior of GFRP-concrete hybrid beams with partial interaction. So the timedependent shear behavior of the shear connectors is worth further studying. Acknowledgments This research was funded by the National Key Research and Development Plan of China (No. 2017YFC0703003) and Natural Science Foundation of China (No. 50978193). Moreover, this work was supported by the National Key Research and Development Plan of China (No. 2016YFC0700800). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2019.02.072. References [1] Ye LP, Feng P. Applications and development of fiber reinforced polymer in engineering structures. China Civil Eng J 2008;39:24–36. [2] Einde LVD, Zhao L, Seible F. Use of FRP composites in civil structural applications. Constr Build Mater 2003;17(6–7):389–403. [3] Feng P. All FRP and FRP-concrete hybrid components for bridges: experiments,

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