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Mechanics Research Communications 50 (2013) 1– 7

Contents lists available at SciVerse ScienceDirect

Mechanics Research Communications

j o ur na l ho me pag e: www.elsev ier .com/ locate /mechrescom

mechanical model for predicting the long term behavior of reinforced polymeroncretes

alentino Paolo Berardi ∗, Geminiano Mancusiepartment of Civil Engineering, University of Salerno, Italy

r t i c l e i n f o

rticle history:eceived 25 September 2012eceived in revised form 31 October 2012vailable online 27 February 2013

eywords:olymer concretesreep behaviornalytical modelingumerical analysis

a b s t r a c t

Polymer concretes represent challenging materials in the Civil Engineering field, with them being charac-terized by a high value of the compressive strength and ultimate compressive strain, as well as by a goodchemical resistance when compared to that of traditional concretes. These innovative materials exhibita limited value of the strength in tension and therefore need to be reinforced with either steel or FRPpultruded internal bars. Moreover, their structural performance is strongly affected by the rheologicalbehavior of the resin as well as the internal bars, if made of FRP. In this paper, a mechanical model capa-ble of analytically evaluating the long term behavior of reinforced polymer concrete beams is presented,which accounts for the linear viscoelastic behavior of the constituent materials.

© 2013 Elsevier Ltd. All rights reserved.

. Introduction

Many innovative materials have been widely adopted for structural purposes over the last years in the rehabilitation and upgradingf existing structures under static and seismic conditions (Ascione et al., 2005; Ghobarah and El-Amoury, 2005; Bencardino et al., 2005;scione and Berardi, 2011; Herakovich, 2012).

Although these materials exhibit high structural performance, their use in the field of new constructions is still limited due to the needo investigate several critical aspects, such as those related to their reliability over time.

Within the context of new materials, polymer concretes can represent an interesting design option, with them being typically charac-erized by more than double the strength of Portland-cement concrete, a marked value of the ultimate compressive strain with respect toementitious concretes and, at the same time, a good chemical resistance to corrosive agents.

More specifically, these properties are due to the chemical composition of polymer concrete, which is made of natural aggregates (e.g.ilica sand or gravel) bound together via a thermoset resin.

Consequently, polymer concretes are particularly suitable for maritime constructions, tunnels, and prestressed concrete elements, asell as seismic applications, because their high ultimate compressive strain value makes it possible to reach a high section ductility, when

ompared to that of traditional cementitious concretes.On the other hand, due to the limited tensile strength values, these concretes have to be reinforced with either steel or FRP bars. In

he second case, GFRP or CFRP rods, often utilized in reinforced concrete members of new structures, can supply (Alsayed et al., 2000;azaqpur et al., 2004; Ascione et al., 2010).

From a design point of view, a very important aspect is represented by the rheological behavior of polymer concretes.In fact, it is well known that polymer-based materials may exhibit a marked deferred behavior (Maksimov and Plume, 2001; Petermann

nd Schulte, 2002; Ascione et al., 2008, 2011a, 2012; Sá et al., 2011) and, thus, the coupling of these innovative materials with traditionalnes may lead to a sensible stress migration toward the material characterized by a lower viscous flow (Ascione and Mancusi, 2007;rockiasamy et al., 2000; Muller et al., 2007; Chami et al., 2009; Ascione et al., 2011b; Zhang and Wang, 2011; Li et al., 2012; Mancusi et al.,012).

This phenomenon occurs even in the case of reinforced polymer concrete, where the different rheological properties of the constituentaterials may cause a significant increase of the stress state in the reinforcement over time, accompanied by an increase over time of the

isplacements of the structural member.

∗ Corresponding author. Tel.: +39 089964084; fax: +39 089968744.E-mail addresses: berardi@unisa.it (V.P. Berardi), g.mancusi@unisa.it (G. Mancusi).

093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.mechrescom.2013.02.001

2 V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7

u

va

a

oi

2

c

–––––––

r2

––

w

––––

wa

Fig. 1. Strain and curvature variation over time in the cross section.

These features need to be studied in greater detail because they may compromise the reliability and durability of the structural membersnder service conditions.

Within this context, the few experimental and theoretical studies available in literature on the topic have highlighted the relevantiscous behavior of polymer concretes (Khristova and Aniskevich, 1995; Aniskevich and Khristova, 1996; Maksimov et al., 1999; Griffithsnd Ball, 2000; Tavares et al., 2002; Aniskevich et al., 2003; Guedes et al., 2004; Lee, 2007).

These studies do not, however, allow to reach final conclusions, partly because the performed theoretical analyses are based onpproximated approaches (e.g. finite differences approach in time domain).

In this paper, the authors present a mechanical model capable of predicting, within the linear viscoelasticity, the long term behaviorf a polymer beam reinforced with either steel or FRP pultruded bars. The proposed model gives the analytical solution to the problemnstead of more approximated ones obtained through the afore-mentioned numerical approaches available in current literature.

. Mechanical model

The mechanical model schematizes the viscous behavior of a polymer concrete beam reinforced with either steel or FRP bars, underombined bending and axial load.

The basic assumptions are:

the generic plane cross-section remains plane after bending; a perfect adhesion exists between polymer concrete and rods; the external axial force, Next, and the bending moment, Mext, are constant over time; any material exhibits the same stiffness and strength in tensile as well as in compression; the behavior of polymer concrete is linear viscoelastic; no crack is present; the behavior of reinforcement bars is linear elastic.

In particular, the authors assume no creep contribution is due to the internal reinforcing bars. This hypothesis is valid for both steeleinforcements and GFRP/CFRP rods, as established in recently published papers (Maksimov and Plume, 2001; Petermann and Schulte,002), which have shown low creep strains for composite bars at room temperature.

In Fig. 1, some symbols have been introduced:

G* indicates the centroid of the transformed cross-section, assumed as origin of the x and y axes; εc and εb, denote, respectively, the instantaneous values of the axial strain in the polymer concrete and the reinforcing bars, accordingto the following relationships:{

εc(t, y) = �(t) + �(t)y = εce(t, y) + εcv(t, y)

εb(t, yb) = �(t) + �(t)yb = εbe(t)(1)

here:

�(t) and �(t) are the axial strain at G* and the cross-section curvature, respectively; yb is the ordinate of the centroid of the generic bars; εce(t, y) and εbe(t, yb) are the elastic polymer concrete and bar strain, respectively; εcv(t, y) is the viscous contribution to polymer concrete strain.

From Eq. (1), it is easy to obtain the following expressions of the stresses both in the polymer concrete as well as the bars:⎧⎪⎨ �c(t, y) = Ecεce(t, y) = Ec[(�(t) + �(t)y) − εcv(t, y)]

�bc(t) = Ebεbe(t, ybc) = Eb[(�(t) + �(t)ybc)] (2)

⎪⎩�bt(t) = Ebεbe(t, ybt) = Eb[(�(t) + �(t)ybt)]

here �c , �bc and �bt are the normal stresses in the polymer concrete and in both the top and bottom reinforcement bars, respectively; Ec

nd Eb are the initial Young’s moduli of the polymer concrete and the reinforcement bars, respectively.

e

U

T

w

–

–

E

I

i

T

V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7 3

Introducing the areas of polymer concrete, Ac, the top reinforcement rods, Abc, and the bottom reinforcement rods, Abt, the equilibriumquations of the cross-section can be expressed as:⎧⎪⎪⎨

⎪⎪⎩

∫Ac

�c(t, y)dAc +∫

Abc

�bc(t)dAbc +∫

Abt

�bt(t)dAbt = Next (translation)∫Ac

�c(t, y)y dAc +∫

Abc

�bc(t)ybc dAbc +∫

Abt

�bt(t)ybtdAbt = Mext (rotation about the x axis)(3)

sing the expressions (2), Eq. (3) can be written as follows:⎧⎪⎪⎨⎪⎪⎩

Ec

∫Ac

[(�(t) + �(t)y) − εcv(t, y)]dAc + Eb

∫Abc

[(�(t) + �(t)ybc)]dAbc + Eb

∫Abt

[(�(t) + �(t)ybt)]dAbt = Next

Ec

∫Ac

[(�(t) + �(t)y) − εcv(t, y)]y dAc + Eb

∫Abc

[(�(t) + �(t)ybc)]ybcdAbc + Eb

∫Ab

[(�(t) + �(t)ybt)]ybtdAbt = Mext

(4)

hese relations can be proposed as follows:⎧⎪⎪⎨⎪⎪⎩

(EcAc + EbAbc + EbtAbt)�(t) + (EcSc + EbSbc + EbSbt)�(t) − Ec

∫Ac

εcv(t, y)dAf = Next

(EcSc + EbSbc + EbSbt)�(t) + (EcIc + EbIbc + EbIbt)�(t) − Ec

∫Ac

εcv(t, y)ydAf = Mext

(5)

here:

Sc, Sbc and Sbt are the first moments of area about the x axis of the polymer concrete, the top reinforcement bars and the bottomreinforcement bars, respectively;

Ic, Ibc and Ibt are the moments of inertia about the x axis of the polymer concrete the top reinforcement bars and the bottom reinforcementbars, respectively.

Introducing the following symbols:

– nb = Eb

Ec

– A∗ = (Ac + nbAbc + nbAbt) (area of the transformed section)

– S∗ = (Sc + nbSbc + nbSbt) = 0 (first moment of area about thex axis of the transformed section)

– I∗ = (Ic + nbIbc + nbIbt) (moment of inertia about thex axis of the transformed section)

q. (5) become:⎧⎪⎪⎨⎪⎪⎩

EcA∗�(t) = Ec

∫Ac

εcv(t, y)dAc + Next

EcI∗�(t) = Ec

∫Ac

εcv(t, y)y dAc + Mext

(6)

n particular, the viscous deformation εcv(t, y) presents the following expression:

εcv(t, y) =∫ t

t0

�c(�, y)Ec

fc(�, t)d� (7)

n which:

– fc(�, t) = −Ec∂�c(�, t)

∂�

– �c(�, t) = 1Ec

(1 + ϕc(�, t)) (creep function)

– ϕc(�, t) (creep coefficient)

aking into account Eq. (3), the relation (7) becomes:

εcv(t, y) =∫ t

t0

[(�(�) + �(�)y) − εcv(�, y)]fc(�, t)d� (8)

Substituting Eq. (8) in Eq. (6), it results by simple algebra:⎧⎪⎪⎪ �(t) =(

Ac − 1)∫ t

�(�)fc(�, t)d� + Sc

∫ t

�(�)fc(�, t)d� + Next

(1 +

∫ t

fc(�, t)d�

)

⎨⎪⎪⎪⎩

A∗t0

A∗t0

EcA∗t0

�(t) =(

IcI∗

− 1)∫ t

t0

�(�)fc(�, t)d� + Sc

I∗

∫ t

t0

�(�)fc(�, t)d� + Mext

EcI∗

(1 +

∫ t

t0

fc(�, t)d�

) (9)

4 V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7

Table 1Reinforcement types.

ID b [mm]

I 8II 10III 12IV 16

Fig. 2. Polymer concrete beam strengthened with GFRP rods (dimensions of cross-section in mm).

Table 2Mechanical properties of materials.

EL

Ai

w�

3

s

t

e

Ec [GPa] fc [MPa] Eb [GPa] fb [MPa]

10.00 41.50 40.00 1000.00

q. (9) represents a coupled system of two Volterra integral equations in the unknowns �(t) and �(t), that can be solved by using theaplace transformation technique:⎧⎪⎪⎪⎨

⎪⎪⎪⎩L[�(t)] =

(Ac

A∗ − 1)

L

[∫ t

t0

�(�)fc(�, t)d�

]+ Sc

A∗ L

[∫ t

t0

�(�)fc(�, t)d�

]+ Next

EcA∗1s

+ Next

EcA∗ L

[∫ t

t0

fc (�, t) d�

]

L[�(t)] =(

IcI∗

− 1)

L

[∫ t

t0

�(�)fc(�, t)d�

]+ Sc

I∗L

[∫ t

t0

�(�)fc(�, t)d�

]+ Mext

EcI∗1s

+ Mext

EcI∗L

[∫ t

t0

f (�, t) d�

] (10)

ssuming t0 = 0 and being evidently �(t) = 0, �(t) = 0 for t < t0, fc(�,t) = 0 for t < �, the convolution theorem allows to rewrite the relation (10)n the following form as an algebraic equation system:⎧⎪⎪⎪⎨

⎪⎪⎪⎩�(s) =

(Af

A∗ − 1

)�(s)F(s) + Sc

A∗ �(s)F(s) + Next

EcA∗1s

+ Next

EcA∗F(s)

s

�(s) =(

IfI∗

− 1

)�(s)F(s) + Sc

I∗�(s)F(s) + Mext

EcI∗1s

+ Mext

EcI∗F(s)

s

(11)

here F(s), �(s) and �(s) represent the Laplace transforms of fc(�,t), �(t) and �(t), respectively.The inverse Laplace transforms of �(s) and(s) provide the solution of the viscous problem in the time domain.

. Case-study

A case-study concerning a polymer concrete beam strengthened with GFRP bars has been developed.Only balanced internal reinforcements have been accounted for, composed of two bars both at the top and the bottom of the cross-

ection.In Table 1, all the considered diameters, b, have been listed.The geometry, the boundary conditions and the external loads are shown in Fig. 2.Moreover, the mechanical properties of the polymer concrete and the GFRP bars are reported in Table 2.

The symbols Ec and Eb denote the longitudinal Young moduli of the polymer concrete and GFRP bars, respectively, while fc and fb are

he corresponding strengths.The long term behavior of the polymer concrete has been simulated via the Bruger–Kelvin viscoelastic model, obtained by combining

lastic springs and dashpots, as shown in Fig. 3.

Fig. 3. The Bruger–Kelvin model.

V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7 5

Table 3Rheological and mechanical properties of the GFA 45 polymer concrete.

E1 [GPa] E2 [GPa] E3 [GPa] �1 [GPa d] �2 [GPa d] �3 [GPa d]

7.38 0 0 5.47 × 102 0 0

d

o

i

t

r

Fig. 4. Curvature vs time.

The viscous properties of the polymer concrete have been assumed in accordance with the experiments reported in (Lee, 2007).Thus, the experimental data have been fitted by the ordinary least-squares method limiting the analysis to the secondary creep range,

ue to its main relevance when dealing with the service life of civil structures.In Table 3, the rheological properties considered in the present study are reported.The analysis was carried out for the different reinforcements indicated in Table 1 by considering the mid-span section, which is subject

nly to the bending moment: Mext = 20 kN m.More specifically, the axial strain at G* and the cross-section curvature were analytically evaluated in the time domain through the

nversion of the corresponding Laplace space functions obtained by the equation system (11).The instantaneous normal stresses in the concrete were evaluated through Eq. (3) considering the assumptions reported in §2, once

he instantaneous normal stresses in the reinforcement bars were calculated through Eq. (2).The values of the flexural curvature, the normal stresses at the top fiber of the beam, �cs, and the normal stresses in the bottom

einforcement bars, �bt , are plotted versus time in Figs. 4–6.

Fig. 5. Instantaneous stresses at the top fiber of the beam.

6 V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7

Fig. 6. Instantaneous stresses in the bottom reinforcement bars.

Table 4Instantaneous stresses in the polymer concrete.

Type of reinforcement �cs (0) [MPa] �cs (10 d) [MPa] �cs (100 d) [MPa] �cs (1000 d) [MPa] �cs (10,000 d) [MPa] ��cs/�cs (0) [%]

I −12.19 −11.77 −8.58 −0.37 0.00 −100.00II −11.87 −11.25 −6.96 −0.06 0.00 −100.00III −11.50 −10.67 −5.42 −0.01 0.00 −100.00IV −10.65 −9.42 −3.13 0.00 0.00 −100.00

Table 5Instantaneous stresses in the bottom reinforcement bars.

Type of reinforcement �bt (0) [MPa] �bt (10 d) [MPa] �bt (100 d) [MPa] �bt (1000 d) [MPa] �bt (10,000 d) [MPa] ��bt/�bt (0) [%]

I 50.20 84.54 344.99 1017.14 1047.07 1985.80

i

Tr

r

t

4

p

c

II 48.88 81.15 305.70 667.13 670.13 1270.97III 47.36 77.34 268.24 465.12 465.37 882.62IV 43.89 69.03 197.83 261.67 261.67 496.20

For example, for the type I reinforcement, the analytical expressions of the instantaneous curvature as well as the instantaneous stressesn the bottom rods and in the top fiber of concrete are given in the following:

�(t) = (2.7555 − 2.6234e−0.0035058 t )10−4

�cs(t) = 0.39961e0.00209464 t + 3.90621 × 10−6e0.00455312 t − 0.399614e0.00560044 t

−0.0327917e0.00560044 t − 2.77717 × 10−7e0.00805892 t + 0.032792e0.00910624 t+ 1.65411 × 10−11e0.00701159 t − 5.2188 × 10−7e0.00805892 t + 1.16369 × 10−7e0.00910624 t

−0.0327917e0.00560044 t − 2.77717 × 10−7e0.00805892 t + 0.032792e0.00910624 t

�bt(t) = 1047.07 + 996.865e−0.0035058 t + 0.0111087e−0.00104732 t

he main results of the analysis have been summarized in Tables 4 and 5. In particular, the percentage variations ��cs and ��bt , have beeneferred at the time instant t = 10,000 h.

It is important to highlight that the initial stresses in the polymer concrete are less than 40% of the corresponding failure strength. Thisequirement is pivotal in order to satisfy a basic assumption of the linear viscoelasticity theory.

Due to the symmetry of the problem with respect to the x axis, the instantaneous stresses in the bottom fiber of the beam, �ci, and inhe top bars, �bt , are given by the following relations:{

�ci = −�cs

�bc = −�bt

. Conclusion

In this paper, a mechanical model for predicting the long term behavior of a polymer concrete beam reinforced with either steel or FRP

ultruded bars is proposed.

The model makes it possible to study, within the context of linear viscoelasticity theory, reinforced polymer concrete beams underombined bending and axial load, starting from the rheological characterization of the polymer concrete.

The exact solution of the problem is obtained by using the Laplace transformation technique.

e

G

c

c

o

R

A

AAA

A

A

A

A

AA

A

AB

CGG

G

HKLL

MMM

M

PR

S

TZ

V.P. Berardi, G. Mancusi / Mechanics Research Communications 50 (2013) 1– 7 7

From a theoretical point of view, the numerical simulations highlight a high stress migration toward the reinforcement bars due to theffect of viscous flow in polymer concrete.

In detail, a relevant increase of the stresses within the GFRP rods, between about 400% and 1000%, is found.Moreover, the member failure is predicted in the case of the beam reinforced with 8 mm rebars for about 900 h, due to the stresses in

FRP rods being greater than the strength of the composite.In contrast, an elevated decrease in the compression and tensile peak stresses in the polymer concrete is observed and, thus, the

ontribution to the cross-section strength of the polymer concrete tends to be negligible over the time.In terms of deferred strains, the results highlight that the stiffness loss due to the creep phenomena may be relevant.Consequently, the observed stress and strain variations may significantly affect the stress verification, the deflection control and the

rack one at Serviceability Limit State of a member made of polymer concrete.The relevance of the results obtained by the authors suggests the urgent need to carry out an analysis of the time-depending behavior

f polymer concrete beams, in order to assess their durability and reliability over time.

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