linear and non-linear creep models for a multi-layered concrete composite

Available online at www.sciencedirect.com

Original Research Article

Linear and Non-lineamulti-layered concre

R. Baleviciusn, G. Marciukaiti

Dept. of Reinforced Concrete and Masonry StruLithuania

a r t i c l e i n f o

Article history:

the lamination.

ual degradation of

pressive load. The

results show that the stress redistribution near the crack-tip under the nal period of a

pressive stress for

ins with the less

relieve the initial

its instantaneous

yers that possess

a r c h i v e s o f c i v i l a n d m e c h a n i c a l e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0Corresponding author. Fax: +370 527445225.E-mail addresses: [email protected], [email protected] (R. Baleviius).1644-9665/$ - see front matter & 2013 Politechnika Wroc"awska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved.http://dx.doi.org/10.1016/j.acme.2013.04.002

nconsiderably different creep and aging properties.

& 2013 Politechnika Wroc"awska. Published by Elsevier Urban & Partner Sp. z o.o. All rights

reserved.high level of sustained loading may lead to an additional required com

complete failure of the composite. Long-term failure primarily beg

deformable (stiffer) layers because the more-deformable layers can

stresses. Thus, the long-term strength of the composite can exceed

strength for early age composites or for composites composed of laof the layers. In particular, the evolution of vertical stress with time is d

the vertical strain and compatibility conditions in a direction parallel to

A fracture mechanics approach is also introduced to predict the grad

long-term strength for a multi-layered composite under a sustained comPoisson ratios of the layers are equal. This is valid even though the average value of the

vertical stress used to calculate the Volterra integral term is dependent on the Poisson ratio

ependent only onReceived 8 July 2012

Accepted 2 April 2013

Available online 10 April 2013

Keywords:

Non-linear creep

Volterra equation

Multi-layered quasi-brittle

composite

Long-term strengthr Creep models for ate composite

s

ctures, Vilnius Gediminas Technical University, Saultekio av. 11, Vilnius,

a b s t r a c t

One- and two-dimensional linear and nonlinear creep models for predicting the time-

dependent behavior of a concrete composite under compression are proposed. These

models use the analytical and iterative solutions of the Volterra integral equation. The

analytical approach is based on the age-adjusted effective modulus method, and the

nonlinear technique applies an iterative approach to the system of non-linear equations,

implying a generalization of the principle of superposition. Both models are validated in

this study.

It has been recently found that negative values of the aging coefcient can emerge in

early age multi-layered composites when the stress redistribution between the layers is

governed by the combination of considerably different creep strains and aging of the

layers.

In the plane-strain state, the two-dimensional creep analysis of multi-layered compo-

sites yields the same vertical stress-time history as that in a one-dimensional case if thejournal homepage: www.elsevier.com/locate/acme

a l1. Introduction

Recently, composite structures have been widely used instructural engineering applications. These applications areprimarily governed by economics; however, the thermalinsulating properties and the added strengthening of theexisting structures are still necessary. The strengthening isimportant for situations in which the structural elements aredamaged or an increase in load carrying capacity is required.Traditionally, building composite structures are multi-layered elements composed of completely different materi-als, providing a lightweight and economic alternative fortraditional building materials. Among others, the most pop-ular layered members are the three-layered wall structureswith external layers of a quasi-brittle composite, such asconcrete-based materials, and an internal layer designedaccording to the purpose of the structure. Currently, one ofthe greatest advantages of a composite structure is a shortproduction time at the stage when the concrete-based mate-rials for the layers are applied. The production of thesestructures is performed in a single technological cycle ratherthan casting one layer after another.

During the life of a structure, sustained loading is pre-dominant. Thus, a difference in material properties, espe-cially a time-variation in the physical characteristics of theconcrete layer, can affect the behavior of the composite as awhole. High-efciency multi-layered composite structuresare designed to account for the time-dependent stress redis-tribution between the layers. This design phase is difcultbecause the materials are often age and creep dependent,especially in the case of non-perfect bonds between thelayers. According to the studies reported in 1994 for theindustry in Great Britain [1], the cost of destruction resultingfrom material plasticization was evaluated to be approxi-mately 300 million per year, from which 10% is attributed todestruction induced by the creep process and especially tothe stress-relaxation effects.

When the compressive stress of a layer exceeds a certainlimit, the layer does not exhibit perfect linear creep behaviorand the creep strain is no longer proportional to the appliedstress. This condition can trigger an instability or evenelement failure [14].

It is also well known that, in the presence of creep and aperfect bond between the layers, a composite structureperfectly redistributes the initial stress over the layers tomaintain the sustained loading [5]. In particular, the higher-strain layer produces the relaxation, while the lower-strainlayer experiences a stress increase [5]. The delayed failurewould thus occur in the lower-strain material if the creepstrain ceases to be proportional to the acting stress. Forexample, in a composite frame of steel columns and compo-site beams, a signicant amount of moment redistributionfrom cracking, creep and shrinkage of concrete occurs [6].Experience combined with monitoring of three-layered con-crete composite structures has indicated that the cracking ofthe layers may be because there is insufcient bond strength

a r c h i v e s o f c i v i l a n d m e c h a n i cto accommodate the deformation of the layer [7], [8].It is analytically difcult to predict creep behavior in

a quasi-brittle multi-layered composite, which possessesage-variable properties in each of the layers. The best t ofthe creep strain to the experimental measurements requiresadopting mathematically complex functions that conse-quently complicate the derivation of an explicit solution fora time-varying stress history of the layers (cf. [9]-[12]). Theimplications of empirically based approaches are usuallyrestricted because of the lack of generality and the limitationsof the data.

Numerous methods [13] have been developed to predictthe instantaneous elastic properties of composite materials.However, fewer methods have been developed that enableone to evaluate the viscoelastic properties of compositematerials with aging.

Jones [14] modeled the viscoelastic properties of thecomposite using an operational approach. In this case, thestress-strain relations of the viscoelastic composite areretrieved from the elastic analysis as mappings. When thecreep strain is described by mathematically sophisticatedexpressions, derivation of the original stress-strain expres-sions from the obtained mappings appears to be extremelycomplicated.

The effective modulus method, proposed by Faber [15], isthe oldest and simplest approach that is applied to modelconcrete-based structures and that overcomes the aforemen-tioned difculties. This method is also well known, and it issuitable when the stress tends to be constant in time.However, for aging materials, this approach results in sig-nicant errors.

Baant [16] proposed the age-adjusted effective modulusmethod as an attractive renement of the latter approach,introducing the coefcient of aging. This method is theoreti-cally exact for any problem if the strains vary proportionallyto the creep coefcient. To nd the exact solution, the agingcoefcient must be computed in advance through the inver-sion of the Volterra integral equation for a given creepcoefcient [10]. This technique is generally used to modelthe time-dependent behavior of reinforced concrete struc-tures [17]. Additionally, in an approximate manner, thisapproach was applied to model the creep behavior of CFRP-strengthened reinforced concrete beams in [18].

Recently, an average stress-strain approach to creep ana-lysis of reinforced concrete elements has been proposed byBaleviius [10]. This approach produces the same values ofthe time-dependent stress-strain state as those determinedusing the well-known age-adjusted effective modulusmethod; however, the proposed approach does not requirethe introduction of the ctitious incremental restrainingactions, thus dramatically simplifying the computation.

A viscoelastic creep model for polymeric materials, suchas the adhesive Epidian 53: PAC100:80 at ambient tempera-ture, for different levels of uniaxial stress has successfullybeen developed using statistically estimated coefcients [11].In this case, because of all of the coefcients of the modiedBaileyNorton model were statistically signicant, the num-ber of statistically estimated parameters in the modiedBurger's model was reduced.

There are very few investigations of non-linear creeppredictions of concrete multi-layered composite with layers

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 473exhibiting age-variable properties found in the literature. Theproblems arising from the incompatibility of the layers

a two-layered composite. The composite comprised a high-

stress for high-strength cement-based materials [26].It is important to consider the range for the application of

the model (Fig. 1). The bond conditions between the layers areof primary importance in the performance of the entirestructure under sustained loading. When the bond strengthis sufciently high, the composite structure behaves mono-lithically, effectively mobilizing all layer strengths, and theultimate capacity of the entire composite increases. This is anecessary condition for a reliable repair of concrete-basedstructures, and it is also a necessary condition for theproposed model.

The common practice to obtain sufcient bond strength isto rst increase the roughness of the base layer by applying abonding agent or steel connectors (transverse reinforcement,staples, anchors, dispersed reinforcement, etc.), if required[7], [8], [27], [28].

In this model, a perfect bond between layers can beachieved when the production of the multi-layeredconcrete-based composite involves a single technologicalcycle, i.e., all layers are cast simultaneously. The result isthat the age for all the layers is the same, but the materialproperties and cross-sectional parameters of the layers canbe different. A single technological cycle for the production ofthe composite structure ensures strain compatibility betweenthe entire composite and each layer [24], and the bondstrength at the interface is higher than the strength of the

a lperformance concrete as the top layer and a normal strengthreinforced concrete as the bottom layer. Tests of this compo-site revealed a signicant reduction in the compressivestrains and measured deections. The authors concludedthat the observed positive effects resulted from the redis-tribution of stress between the concrete layers.

In the present study, one- and two-dimensional linear andnonlinear creep models of a multi-layered concrete compo-site are investigated. The analytical approach uses the age-adjusted effective modulus method, and the numerical tech-nique considers a generalization of the principle of super-position for the non-linear creep phenomenon. A fracturemechanics approach is also introduced to predict the gradualdegradation of the long-term strength of the multi-layeredcomposite under high levels of sustained loading.

2. Model formulation, basic assumptions andrange of application

Consider a time-dependent stress and strain state of themulti-layered composite element (Fig. 1). The element issubjected to a sustained loading. Introduce the followingassumptions:

A perfect bond exists between the layers until failure, There is no connement of the layers from Poisson'seffect,

The stress state is uniaxial, The second-order effects are negligible, Each layer obeys the linear/non-linear creep laws ofmaterial aging,

The instantaneous stress-strain state is linear and Post-failure conditions are ignored.

We also conne the model to creep analysis only. In theconcrete have been developed by Baant and Kim [20],Benboudjema et al. [21] and Pichler et al. [22].

Yamada et al. [23] were among the rst researchers toobserve the benecial effect of a two-layered compositestructure composed of layers of high-strength and normal-strength concrete. Later, apko et al. [24] and Sadowska-Buraczewska [25] conducted short-term experimental andnumerical investigations of beam samples constructed withloaof tcaslayding is missing in the literature because of the complexityhe problem. Some nonlinear multi-axial creep models forstra

A comprehensive prediction of the time-dependent stress-in state of a multi-layered composite under a triaxialsite behavior.

elements does not correspond to concrete multilayer compo-

beha two-layered composite, but the reinforcement usuallyaves elastically and the direct creep analysis of theseconascrete. In this case, the reinforced concrete may be treated

havdeformational properties are of primary concern in high risebuildings or concrete layer retrots [19]. Theoretical studies

e examined the nonlinear creep problems in reinforced

a r c h i v e s o f c i v i l a n d m e c h a n i c474e of linear analysis, the shrinkage (from drying of theers or from endogenous shrinkage) may be easilyevaluated separately, relying on the principle of superposi-tion. For the non-linear creep analysis, the principle ofsuperposition accounting for the shrinkage is violated andshrinkage strains should be incorporated into the straincompatibility equations. In some cases, even endogenousdrying can signicantly change the time-dependent state of

Fig. 1 Schematic of a multi-layered composite.

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0adjacent layers. Thus, longitudinal cracks will develop withinthe layers rather than at the interfaces [27], [29]. The second

case of monolithic behavior occurs when the age of the layersis different but the previously concreted layer is not com-pletely hardened during the production process. Then, amixed chemical and intermixture bond develops betweenthe previously concreted and new layers. In the contact zonebetween the layers, a seam with a unique chemical composi-tion and structure develops [30]. The elasticity modulus ofthis seam may be up to 5 times higher than that of the layers.

To ensure strain compatibility conditions between thelayers when the deformability characteristics of the layersare completely different, transverse reinforcement, staples,anchors and dispersed reinforcement can also be utilizedduring the casting cycle (Fig. 2). Recently, an innovative shearconnection with composite dowels has been studied byLorenc et al. [28].

The above model (Fig. 1) can also be effectively applied forstrengthening and retrotting the structures. As a simpleexample of such an application, a new layer can be wrapped

evaluated using a system of n+1 equations and n unknowns[35] as follows:

a r c h i v e s o f c i v i l a n d m e c h a n i c a laround a concrete column cross-section. This structuraltechnique can improve the strength, ductility and long-termperformance of the strengthened structure. Following thisstructural practice, wrapping a new layer around the old oneproduces the monolithic behavior between the old and newlayers. The bond between the cement mortar and concretelayers is strong when the previously formed layer is hydro-philic because the layer is moistened well with water andgrout. To ensure compatibility between the strains, the sur-face must be porous, with knobs and caves. The caves andknobs can be simply and effectively formed by embrocatingthe surface of the strengthening structure with sulfuric acid.Various investigations have shown that the strength of thebond between the layers depends on the shape, size andquantity of the roughness. It has been demonstrated that abond formed with cement mortar or concrete is strongerwhen the surface knobs and caves have an irregular orconical shape, and the bond is weaker when the knobs andcaves have a round shape [31], [32].

In most repairing or strengthening cases, the difference inthe layer ages usually prevails and the long-term strength ofFig. 2 Application scheme.i n

i 1si t Ai N

c t 1 t 0c t i t 0c t n t 0

; i 1;; n;

8>>>>>>>>>>>>>>>>>>>>>:

1

where si (t) and i(t) are the stress and strain at time t,respectively; Ai is the cross-sectional area of the i-th layer;c(t) is the strain of the entire composite at time t; and n is thetotal number of layers.

In the set of eq. (1), the rst equation represents the forceequilibrium between the internal forces and induced loadingN, while the other equations describe the strain compatibilitybetween the entire composite and each layer. Because thecreep strain increases with time, the stresses between thelayers redistribute to maintain the force equilibrium. Whenthe layer stress exceeds a certain limit, the system no longerexhibits perfect linear creep behavior and the non-linearcreep strain equations should be evaluated.

As shown in [36], although non-linear behavior is consid-ered, the principle of superposition for the creep strains stillholds. Arutyunyan [9] generalized the principle of super-the old structure compared with the long-term strength ofthe strengthened composite is unknown. Evaluation of theseproblems will be discussed below.

Finally, the model does not involve the evaluation ofsecond-order effects, i.e., geometrical non-linearity isexcluded from the current analysis. Therefore, the multi-layered composite should be non-slender, i.e., c c;lim (wherec and c;lim are the slenderness of the entire composite and alimit value of the slenderness, respectively). A short elementof a rectangular cross-section is non-slender if its length doesnot exceed approximately four times the minimal value ofthe cross-sectional width. In Fig. 2, the case of a high-lengththree-layered wall, but without the second-order effects,matching the above model (cf. Fig. 1) is also illustrated. Inthis illustration, the lateral walls can eliminate the second-order behavior of the three-layered wall; however, to considera three-layered wall separately, the transfer connectorsshould be constructed with a mild amount of reinforcement.A buckling analysis via FEM and approximate relations toaccount for the behavior of the exible ties of the three-layered wall were given in [33]. The FEM modeling andanalysis of critical loads of a three-layered plate with a softcore composed of foam with elastic and viscoelastic proper-ties is highlighted and discussed in a previous work [34].

3. The governing relations: a one-dimensionalmodel

In view of the above assumptions, the time-dependent stressand strain state of the n-th layered composite can be

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 475position by introducing the nonlinear creep strain. Followingthis approach, the i-th layer strain at t can be determined

in terms of a piecewise constant function as follows:

sustained load, it is a practically impossible task. However,

a las follows:

i t si t Ei t

Z tt0si

dd

1Ei

d

Z tt0F si

Ci t;

d; i 1;; n; 2

where si(t) and Ei(t) are the stress and elasticity modulus,respectively, of the i-th layer at t, C t; is the specic creep, t0is the time instant at the initial loading, and F si is thenon-linear stress function.

A non-linear dependency of creep strain and stress for the i-th layer for concrete is dened by the following functionproposed by Bondarenko and Bondarenko [37] and later speci-ed in NIIZHBs specications [38] with some additional factors:

F si si 1 visi f c;i

!m" #; 3

where the coefcient vi denes the increase in the creep strainof the i-th layer during the delayed failure and depends on thecompressive strength, f c;i is the material strength of the i-thlayer, and m4 is the factor used to evaluate the creep strainand stress nonlinearity. A theoretical and experimental analysisof F si for varying complexities was given in [39]. It wasfound that eq. (3) provides a reasonable t to the experimentaldata for plain concrete. This construction of the equation isappropriate for use in our derivation.

Combining (3) with (2), we obtain the following:

i t si t Ei t

Z tt0si

i t;

d|{z}crit;t0

viZ tt0

si m1f c;i m

Ci t;

d|{z}cri

si=f c;i ;t;t0

; 4

where i t; 1Ei Ci t; , i1,, n.This relation implies that the total strain of the layer is the

sum of the instantaneous strain, 1=Eit, resulting from thestress si t developed at t; the linear creep strain, cri t; t0 ,evaluating the effect of creep and aging; and the non-linearcreep strain, cri si=f c;i; t; t0

, reecting the afnity of the creep

corresponding to the different stress-strength levels.By substituting expression (4) into system (1), we obtain a

set of non-linear equations with unknowns si as follows:

i n

i 1si t Ai N

s1 t E1 t

R tt0s1 1 t; dv1

R tt0

s1 m1f 1 m

C1 t; d c t

si t Ei t

R tt0si i t; dvi

R tt0

si m1f i m

Ci t; d c t

sn t En t

R tt0sn n t; dvn

R tt0

sn m1f n m

Cn t; d c t

8>>>>>>>>>>>>>>>>>>>>>>>>>:

5

where i is the layer number, and there are n layers.Assuming that vi 0, we arrive at the linear formulation of

(4) as follows:

i n

i 1si t Ai N

s1 t E1 t

R tt0s1 1 t; d c t

si t Ei t

R tt0si i t; d c t

8>>>>>>>>>>>>>>>>>>>>

6

a r c h i v e s o f c i v i l a n d m e c h a n i c476sn t En t

R tt0sn n t; d c t

>>>:the analytical solution may be dened using the coefcientof aging.

Suppose the coefcient of aging t; t0 is known a priori fora given creep function. Then, for the case of linear creep, thesi t0 sj t0 Ei t0 Ej t0

; ij; i 1; 2; :::;n; 9

Finally, the entire composite instantaneous stress, strainand elasticity modulus, induced by load N, can be predictedusing the following formulae:

sc t0 N

i n

i 1Ai

: 10

c t0 N

i n

i 1Ei t0 Ai

: 11

Ec t0 i n

i 1Ei t0 Ai

i n

i 1Ai

; i 1;2;;n 12

3.2. Linear creep analysis: analytical approach

For practical applications, it is important to have the relation-ships for predicting the time-dependent stress-strain state ofthe multi-layered composite. Theoretically, an analyticalsolution of eq. (4) for the unknown stress history mayoccasionally be obtained by reducing the Volterra integralterm to a rst-order differential equation (with variablecoefcients) when a singular kernel is established as aproduct of functions of t and or the series of these products[9], [36]. However, this approach results in considerablemathematical complications, even for the pure relaxationtest, while, for the multi-layered composite subjected to aThus, the mathematical consideration of (1) for the multi-layered concrete compressive composite becomes the set oflinear (6) or non-linear (5) equations.

3.1. The instantaneous solution

For the instantaneous analysis, when t t0, the strain of thei-th layer is dened as follows:

i t0 si t0 Ei t0

: 7

The substitution of this equation into (1) results in theequation for the i-th layer stress as follows:

si t0 NEi t0

i n

i 1Ei t0 Ai

; i 1; 2;;n: 8

The i-th and j-th layer stresses of the composite interrelate

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0i-th integral eq. (4) can be rearranged into an algebraicequation following the Trost-Baant method [16], [40] as

a lfollows:

i t si t0 Ee;i t

si t Ee;i t

; i 1; 2;;n; 13

where

Ee;i t Ei t0

1 i t; t0 ; 14

Ee;i t Ei t0

1 i t; t0 i t; t0 ; 15

si t si t si t0 ; 16where Ee;i t and Ee;i t are the effective and age-adjustedeffective elasticity moduli, respectively, for the i-th layer;si t is the i-th layer stress increment between the initialstress and the stress developed at t; and i t; t0 is the creepcoefcient.

Eq. (13) describes the unknown stress increment, si t ;therefore, we rewrite the force equilibrium in (1) in theincremental form as follows:

i n

i 1si t Ai 0; i 1; 2;;n 17

A combination of eq. (17) with the strain compatibilityconditions results in the following equation:

i n

i 1c t

si t0 Ee;i t

Ee;i t Ai 0; 18

From eq. (18), the strain of the entire composite developedat t can be determined as follows:

c t i n

i 1si t0 AiEe;i t=Ee;it

i n

i 1Ee;i t Ai

; 19

When c t i t is dened, the combination of formula(19) with eq. (13) and subsequently with (16) provides theformula to predict the i-th layer stress at t as follows:

si t si t0 i t si t0 Ee;i t

Ee;i t : 20

A prediction of the creep coefcient for an entire compo-site, containing n layers, is calculated using the followingexpression:

c t; t0 c;cr t c;el t0

c t c t0

1; 21

where c;el t0 c t0 and c;cr t are the elastic, i.e., instanta-neous, and the creep strains of the composite.

3.3. Linear creep analysis: numerical approach

To validate the above relations, a numerical solution forsystem (6) should be considered. Potential solutions, suchas an exponential algorithm, may be found in the literature,e.g., [41]. This approach would require an additional conver-sion to the Maxwell/Kelvin chain. We chose to use a differentapproach, i.e., we rewrite system (6) into a recurrent form.

For the discretized time scale (t0, t1, , tj-1, tj, , tk), thestress relation s;i tj; tj1

si tj =2 sj tj1 =2 is a middle

a r c h i v e s o f c i v i l a n d m e c h a n i cvalue for stress, for [tj-1, tj], to nd si t0 , si t1 , , si tj1

,si tj

,, si tk . Application of the intermediate value theoremand the rst mean value theorem for integration yields thefollowing expression for the total strain of the i-th layer:

i tk si tk Ei tk

j k1

j 1s;i tj; tj1

i tk; tj1

i tk; tj

s;i tk; tk1 Cni tk; tk1 ; i 1;2; :::;n; k 1; 2;; k k; 22where Cni tk; tk1 1Ei tk1

1Ei tk Ci tk; tk1 is the pure specic

creep [12], [42], [43].Using the expression for stress, si tk , from (22) and

substituting into system (6), we obtain an equation for thestrain of the entire composite at t tk as follows:

c tk 2 i n

i 1

Ai2=Ei tk Cni tk; tk1

!1

N i n

i 1

j k1

j 1s;i tj; tj1

i tk; tj1

i tk; tj si tk1 Cni tk; tk1

2=Ei tk Cni tk; tk1 Ai

1CCCCA

0BBBB@i 1;2; :::;n; k 1; 2;; k k 23Relationship (23) represents a recurrent form for the

solution of system (6). For ti-ti-1-0, the numerical solutionapproaches the explicit solution [10]. Thus, the numericalanalysis may also be treated as an exact solution of (6).A number of variations for the numerical inversion of theVolterra integral can also be found in [16], [17], [44], [46], [47].

To apply the age-adjusted modulus method, the predic-tion of the coefcient of aging is required. Thus, we adopt anexplicit inversion of the Volterra equation in terms of theaging coefcient for the i-th layer following [10] as follows:

i tk; t0 s;i tk; t0 k tk si tk 1k tk si t0

si tk si t0 ; for Ec t Ec t0 ;

24where

ki tk Cni tk; t0 Ci tk; t0

; 25

s;i tk; t0

j n1

j 1s;i tj; tj1

i tk; tj1 s;i tk; tk1 Cni tk; tk1

!Cni tk; t0

;

26

i tk; tj1 i tk; tj1 i tk; tj ; k 1;2;; k k 27In these equations, an average value of stress, s;i tk; t0 ,

that satises the Volterra integral term, s;i t; t0 R tt0si

=i t; d=Cni t; t0 , in the entire time interval, t-t0, linksto the stress s;i tj; tj1

si tj =2 sj tj1 =2 for the discretetime interval [tj1, tj].

The numerical implementation of the above equations isachieved using a procedural programming concept with theFORTRAN 90 language and the Compaq compiler. Primarily, allparameters dening an instantaneous state were denedaccording to the relations described in Section 4. The creepanalysis was then implemented using a three-loop cycle.The rst (outer) loop runs rst over t instant, for t(t0, t1,,tj1,tj, , tk). The second (inner) loop runs for the prediction of allparameters for the i-th layer, i(1, 2,, n), according to Section 6.The third loop, working within the second loop, runs for the

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 477Volterra equation variable , discretized using vector t indicesas (t1, t2,, tj, tj+1,, tk-1). Within this loop, the summation

variables, such as the coefcient of aging, etc., were stored

in a one-dimensional structure.Assume that the layers of a multi-layered composite

a ldeform elastically, satisfying Hookes law. For the instanta-neous prediction, the strain tensor for each layer can bedened as follows:

ij Ksij 12G

sij; i; j 1;2;3; or i; jx; y; z 28

where 3s skk, sij sijsij is the deviatoric stress tensor, ij isthe Kronecker delta, K 12 =E is the bulk compliancemodulus,G E= 2 1 and E are the shear and elasticmoduli, respectively, and is the Poisson ratio of the layer.

4.1. An instantaneous solution

The imposed displacement constraints (Fig. 1), the kinematicconstraint, i.e., a uniform strain develop within the layer, thestrain compatibility, the layers strain and the entire compo-site strain in the z and y directions are the same, and theplane strain state are expressed mathematically as follows:zz;i zz;c 0; yy;i yy;c; xy;i zy;i 0; sxxjx x0 sxxjx xn 0;within the appropriate matrix of n k.Because a small integration step and long creep period are

required for the analysis, the numerical computation alwaysdemands high CPU time, much of which is wasted. To ndthe compromise between desirable computational perfor-mance and low articial damping, the numerical analysisshould be performed using an increasing time step, such asti

1016

pti1 [16]. Some analytical suggestions on the selec-

tion of a suitable time step for the numerical integration werealso provided in [48].

4. A two-dimensional model

In a two-dimensional multi-layered composite structure(Fig. 1), the normal strain, zz, is constrained by the nearbymaterial along the z axis and is small compared to the cross-sectional strains, xx and yy, if the structure length in the zdirection is substantially larger than in the other directions.A plane strain state is an acceptable approximation requiring anon-zero szz to maintain the constraint zz 0, thus reducingthe 3-D problem to a two-dimensional problem. Additionally,the stiffness of such a composite structure (Fig. 1) will begreater in the direction normal to the layers; however,because szz0, the two-dimensional structure is exposed toa smaller vertical strain, yy, in the entire composite than thatof termsj k1j 1 in eqs. (23) and (26) was executed. Whenthe third and second loops stop, a prediction of terms withinthe rst loop that depends on variable tk nishes the compu-tation of eqs. (23) and (26) for the composite strain c tk andfor the averaged stresses satisfying the Volterra integral,s;i tk; t0 . Additionally, the computation of stress si tk by eq.(22) with condition i tk c tk is also performed here. All ofthe stress and strain history as well as the other required

a r c h i v e s o f c i v i l a n d m e c h a n i c478uxxjx x0 uyyjy 0 0. Introducing these conditions in (28)together with the force equilibrium in the y direction attt0 results in the following:

i n

i 1syy;i t0 Ai N

yy;c t0 yy;i t0 yy;c t0 12i

syy;i t0 =Ei t0 0xx;i t0 i 1 i syy;i t0 =Ei t0

; i 1; :::;n;

8>>>>>>>:

29where yy;c t0 is the strain of the entire composite at time t0.

The solution of system (29) yields the following formulae:

sxx;i t0 0; 30

szz;i t0 isyy;i t0 ; 31

syy;i t0 syy;j t0 Ei t0 Ej t0

12j 12i ; ij; i 1;2;;n; 32

syy;i t0 NEi t0

12i

i n

i 1Eit0Ai=12i

; i 1;2;;n 33

yy;c t0 N

i n

i 1Eit0Ai=12i

: 34

The horizontal displacement of the free surface isobtained by integration of xx;i t0 along the x axis as follows:

uxxx xn

i n

i 1i 1 i

Z xi1xi

syy;i t0 =Ei t0 dx: 35

Comparison of eq. (33) for the two-dimensional modelwith (8) for the one-dimensional case indicates that thevertical stresses are different when the Poisson ratios of thelayers are not equal. When the Poisson ratios are the same,there is no difference in the vertical stresses of the one- ortwo-dimensional composite structures. The comparison ofthe vertical strains for the entire composites (34) and (11)shows that the strain in the two-dimensional model is stifferin the y direction because of the action of the constrainingstress szz;i t0 isyy;i t0 along the z axis. For example, usingthe Poisson ratio i 0:2 for the uncracked plain concrete forall layers, the strain yy;c t0 of the two-dimensional compositestructure is less than that of the one-dimensional structureby approximately 1.04 times. For different Poisson ratios ofthe layers, this difference depends on the term 12i

in the

summation term of (34).

4.2. Linear creep analysis

Applying the principle of superposition for the axial andtransverse strains, Arutyunyan [9] generalized Hookes law(28) for the linear creep problem as follows:

ij t K t s t ijsij t 2G t

Z tt0

sij

t; t; skk t ij

t;

d; 36

where K t 12 t =E t , G t E t = 2 1 t , t; E t; C t; , and and t; are the coefcients of transverseinstantaneous and creep strains (Poisson ratios), respectively.

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0Following the experimental investigations described in[36], t; is slightly lower than and the effect of different

theorem and the rst mean value theorem for the integrationof (4) results in the following:

a lvalues of these coefcients on a two-dimensional stressstrain state of the concrete specimens was minimal. Thus,it can reasonably be assumed that t; t; const,allowing the simplication of eq. (36). In particular, thisresults in t; t; , and the strain zz;i t from eq. (36)can be written in a more convenient form for the i-th layer asfollows:

zz;i t Sz;i t Ei t

Z tt0Sz;i

i t; d 0; 37

where

Sz;i szz;i sxx;i syy;i

: 38

Incorporating (31) and (32), we obtain szz;i isyy;i ,sxx;i t0 sxx;i 0, and thus, Sz;i 0. However, a zero ofSz;i can be directly obtained from (37). It shows that thevertical stress function, syy;i , is independent of condition(37) and depends only on the vertical strain, yy;i t , and thecompatibility conditions in a direction parallel to thelamination.

For the vertical strain equation, we rewrite eq. (36) asfollows:

yy;i t Sy;i t Ei t

Z tt0Sy;i

i t; d yy;c t ; 39

where

Sy;i syy;i sxx;i szz;i

: 40

Incorporating (31) and (32), we obtain Sy;i 12i

syy;i .Hence,

yy;i t syy;i t Ei t

Z tt0syy;i

i t; dsyy;i t Ei t

s;yy;i t; t0 Cni t; t0 yy;c t 12i : 41

The comparison of (41) and (6) indicates that the verticalstrain at time t of the entire composite in a two-dimensionalformulation resolves similarly, as shown in relation (22)(independent of (37)) for the one-dimensional case. Thus:

yy;c tk 2 i n

i 1

Ai12i

2=Ei tk Cni tk; tk1 !1

N i n

i 1

j k1

j 1s;yy;i tj; tj1

i tk; tj1

i tk; tj syy;i tk1 Cni tk; tk1

12i

2=Ei tk Cni tk; tk1 Ai

0BBBB@

1CCCCA;

i 1;2; :::;n; k 1; 2;; k k: 42where s;yy;i tj; tj1

syy;i tj =2 syy;j tj1 =2 is the average ver-tical stress for the discrete time interval [tj-1, tj].

An average value of stress, s;yy;i tk; t0 , satisfying theVolterra integral term is determined by relation (43) asfollows:

s;yy;i t; t0 12i

j n1

j 1s;yy;i tj; tj1

i tk; tj1 s;yy;i tk; tk1 Cni tk; tk1

!Cni tk; t0

43Substituting the strain, yy;c tk , calculated by formula (42)

a r c h i v e s o f c i v i l a n d m e c h a n i cinto eq. (41) and combining with relation (43), the verticalstress, syy;i t , at any t can be determined. When const fori tk si tk Ei tk

j k1

j 1s;i tj; tj1

i tk; tj1

si tk si tk1 2

Cni tk; tk1

vi j k1

j 1

s;i tj; tj1 m1

f ;i tj; tj1 m Ci tk; tj1

12vi

si tk si tk1 m1

f c;i tk f c;i tk1 m Ci tk; tk1 46

(for i1,, n; k1, , kk )where

Ci tk; tj1 Ci tk; tj1 Ci tk; tj ; 47

f ;i tj; tj1 f c;i tj =2 f c;i tj1 =2; 48

Cni tk; tk1 1

Ei tk1

1Ei tk

Ci tk; tk1 : 49all the layers, syy;i t in the two-dimensional model is thesame as that in the one-dimensional model.

Finally, the horizontal strain, xx;i, for the i-th layer can beintegrated using the vertical stress function, syy;i t , as follows:

xx;i t Sx;i t Ei t

Z tt0Sx;i

i t; dSx;i tk Ei tk

12j n

j 1Sx;i tj Sx;i tj1 i tk; tj1 ; 44

where

Sx;i sxx;i syy;i szz;i i 1 i syy;i 45

The stress-strain relations involving the nonlinear creepphenomena were presented in [45] for the two-dimensionalstate. These relations can be applied for the current multi-layered composite structure in a similar way. The analysisbelow applies to the modeling of one-dimensionalnonlinear creep.

5. Non-linear creep analysis: a one-dimensional model

An explicit solution of the system of nonlinear eq. (5) for thelayer stresses at t is a complicated task. However, an analy-tical solution is available for some cases if the parallel (ratherthan afne) creep curve assumption for material aging isadopted. Otherwise, explicit solutions for polynomials withdegrees greater than four do not exist. Finally, complicationssolving system (5) arise as the total number of layersincreases.

In this study, we adopt a numerical solution for the i-thlayer with an unknown stress history, si t0 , si t1 , , si tj1

,

si tj

,, si tk . Then, the application of an intermediate value

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 479By substituting i tk for i t in (1), system (5) can berearranged into the following set of non-linear functions,

a lf i , as follows:f 1 s1 tk ; s2 tk ; :::; sn tk 0f i si1 tk ; si tk 0f n sn1 tk ; sn tk 0

; i 2;n;

8>>>>>>>:

50

where:

f 1 s1 tk ;s2 tk ;;sn tk i n

i 1si t AiN; 51

f i si1 tk ; si tk i1 si1; tk i si; tk ; 52

where i si; tk i tk , using (46).Generally, system (50) involves n non-linear equations

with n unknowns. Strictly, system (50) with respect toeq. (46) represents a set of non-linear recurrence relationsdening the successive terms of a sequence s1 tk ; s2 tk ;

:::;

sn tk gTRn as nonlinear functions of the preceding terms.We can rearrange the nonlinear system (50) into a classical

Newtons type of system of linear equations over all t0, t1,, tk1, tk in t0tk. Assume that si : t; t0-R (for any i1, n)is a continuous differentiable function in [t, t0]. Assume thatf i s1; :::; sn : si si t0 ; si t Rn (i1, n) are also contin-uous differentiable functions. Let f 1=s1; :::; f 1=sn, ,f n=s1; :::; f n=sn be the derivatives of f 1 s1; :::; sn ,f n s1; :::; sn at the points f 1 s1 tk ; :::; sn tk , f n s1 tk ; :::; sn tk , while ds1=d, , dsn=d are thederivatives (i.e., the stress ux) over s1 , , sn at any point tk.

Then, derivatives of the complex functions exist such that

df1d

f 1s1

ds1 d ; ;

f 1sn

dsn d

df nd

f ns1

ds1 d ; ;

f nsn

dsn d

8>>>: 53

or,

df id

j n

j 1

f isj

dsj d

; i 1;n 54

In mathematical terms, from (54), a differential for func-tions f i s1; :::; sn is independent of d, resulting in thefollowing:

df i j n

j 1

f isj

dsj ;i 1;n: 55

The increment of f i s1; :::; sn can be dened as follows:f i f i s1 tk s1 tk ; :::; sn tk sn tk

f i s1 tk ; :::; sn tk ; i 1;n: 56Assuming df if i consider tk. Then, dsi tk si tk ,

si si tk , and from (55) and (56), we express the followingapproximate relationship for any f i:

f i s1 tk s1 tk ; :::; sn tk sn tk

f i s1 tk ; :::; sn tk j n

j 1

f isj tk

sj tk ; i 1;n 57

Relation (57) traditionally approximates the non-linear i-thfunction using a linear equation involving only a value of thisfunction and the sum of its rst derivative at the points. The

a r c h i v e s o f c i v i l a n d m e c h a n i c480application of a Taylor series has been comprehensively studiedin [46], [47] for creep analysis of one-layered pre-stressedcellular concrete wall panels. In these investigations, the con-cept of a Taylor expansion was implemented for obtaining thesolution of the linear Volterra integral for an unknown stressfunction and its derivatives as a set of linear algebraic equa-tions. Those studies found that to maintain a sufcient accu-racy in the approximation, when t-t0 increases, a minimum often Taylor series terms should be kept in the computation.Generally, instead of rst-order Taylor series terms in eq. (57),the implementation of the ten terms for each layer could beadopted because the high-order derivatives for f i s1; :::; sn canbe dened explicitly. However, this implementation would notbe reasonable in engineering computations because of theconversion of the linear equations into nonlinear ones at eachtime step for unknown si tk .

Thus, because the nonlinear problem is replaced by thelinear eq. (57), errors occur during the replacement. Hence,the solution is not exact, but it obviously may be denedwithin an allowed tolerance of tol units.

Let q1s1 tk ; :::; q1sn tk be a new approximation for thelayer stress, and let qs1 tk ; :::; qsn tk be the current approx-imation. The new approximation is calculated as follows:

q1si tk qsi tk qsi tk ; i 1;n; 58

where qs1 tk ; :::; qsn tk is the set of errors for the q-thapproximation or the unknown layers stress increments attime tk and iteration q.

Inserting (58) into (57) and replacing s1 tk ; :::; sn tk byqs1 tk ; :::; qsn tk , then substituting qs1 tk ; :::; qsn tk fors1 tk ; :::; sn tk , we obtain the following:

f iq1s1 tk ; :::; q1sn tk f i qs1 tk ; :::; qsn tk

j n

j 1

f isj tk

qsj tk ; i 1;n 59

When qs1 tk ; :::; qsn tk

-0, (59) becomes f iq1s1 tk ;

:::; q1sn tk -f i qs1 tk ; :::; qsn tk

, and then the stresses atthe q+1-th approximation are equal to the stresses at the q-thapproximation, i.e., the solution converges.

From (50), function f iq1s1 tk ; :::; q1sn tk 0. Summing

over n in (59), we classically arrive to the Newtonian methodfor the numerical solution of the system of non-linearequations as follows:

f 1s1 tk

qs1 tk ::: f 1sn tk qsn tk f 1 qs1 tk ; :::qsn tk

f n

s1 tk qs1 tk ::: f nsn tk

qsn tk f n qs1 tk ; :::qsn tk

8>:

60

In a classical way, system (60) may be convenientlyrewritten into a matrix of linear equations with an unknownvector of the layer stress increments qr tk

at iteration q

and any tk as follows:

J qs tk qr tk f qs tk ; 61

where

J qs tk f 1

qs1 tk ; :::; qsn tk T

f n


2664

3775 62

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0qr tk qs1 tk ; :::; qsn tk T 63

Fig. 5. The creep coefcient and specic creep, which arerequired for the above model, are interrelated as Ci(t, t0)

a lf qs tk f 1


f n


2664

3775 64

where J qs tk

is an n-by-n Jacobian matrix involving agradient of vector f qs tk

at points qs1 tk ; :::; qsn tk for the

iteration q at any tk.The Jacobian n-by-nmatrix with respect to eqs. (5) and (50)

for a multi-layered composite can thus be written as follows:

J qs tk

A1 A2 A3 An1 Anf 2

s1 tk f 2

s2 tk 0 0 0

0 f 3s2 tk f 3

s3 tk 0 0 0 0 0 f nsn1 tk

f nsn tk

266666664

377777775

65

The major difculty in this formulation is the explicitdenition required for all functions included in the Jacobian.The formulation of (50) with respect to (46) allows explicitdifferentiating over s1 tk ; :::; sn tk . Thus, all terms in theJacobian are generalized using the following formulae:

f isj tk

0B@ 1Ej tk 12Cnj tk; tk1

m 12

vj

qsj tk qsj tk1 mf c;i tk f c;i tk1 m Cj tk; tk1

1CA

f i=sztk 0, for zi & zj

; where z1,n; i2,n, j i1,i;

where 1 ij1 is provided for the sign conversion.Thus, it has been demonstrated that the nonlinear creep

problem discretized over all t0, t1, , tk-1, tk in t0-tk may berearranged for unknowns qs1 tk ; :::;qsn tk developed atthe current time instant tk. The layer stresses accumulatedfrom any preceding time tk1 may be treated as independentparameters for the current time instant tk, where the explicitJacobian functions depends only on two stresses, si tk1 andsi tk .

The adopted solution algorithm is as follows. At time tk,we replace the system of nonlinear eqs. (5052) with thelinear one (61) and determine the unknowns qs1 tk ; :::;qsn tk at iteration q using the following formula:

J qs tk 1 f qs tk qr tk : 66Then, the solution obtained for qr tk

is employed to

obtain a better approximation for q+1 as follows:

q1r tk qr tk qr tk 67

When q1r tk

qr tk

otol is satised, the iterationprocess is terminated, achieving convergence between itera-tions q and q+1. The iterative computation is repeated untilthe last time instant in the time series. The stress vector0r tk1

is used as the rst guess.As an illustration, snapshots from the nonlinear creep

analysis of system (50), which are described in the examplebelow, are shown in Fig. 3. Eq. (51), describing static equili-brium, is represented by the plane (Fig. 3a). The straincompatibility eq. (52) (Fig. 3b) yields a nonlinear surface. In

a r c h i v e s o f c i v i l a n d m e c h a n i cFig. 3c, the solution for r t at the intersection of thesurfaces is demonstrated.(t,t0)/Ei(t0).As shown in Fig. 5, the creep strain of the inner layer is

approximately 2.6 times higher than that of the outer one, cf.2 t; t0 2:023 and 1;3t; t0 0:767 (at t01, t30000 days).In this example, the values of i t; t0 are given for an effectivethickness of the layers, h1,390.9 mm, h2166.7mm. The rela-The iterative process runs well and converges at all timesconsidered. However, at time intervals near t0, instabilities inthe layer stress eld began to appear. To overcome theseinstabilities, a small initial time step, t1-t0 equal to 0.0001 day,provided a fairly accurate result.

6. Application and results: a one-dimensionalmodel

6.1. The linear analysis and its numerical verication

The proposed linear approach represents an explicit inver-sion of the Volterra integral equation without using anyempirical factors or simplications. Therefore, comparingthe results obtained using the proposed method to thosemeasured experimentally would be a verication of thefundamental Volterra equation for modeling a time-variable stress-strain state of concrete, but this comparisonis not a verication of the method. Errors arising from theapplication of the Volterra superposition and those emergingfrom the best t of creep curves to the experimental mea-surements are well-known [9], [12], [36]. Thus, a directcalculation of the time-dependent stress-strain values usingthe proposed approach and their comparison to those pro-duced using the numerical technique is the most efcientway to verify the analytical method.

Consider a three-layered composite wall structure (Fig. 4).Assume that the concrete composite structure is made ofconcrete with creep properties described by code EN 1992-1-12004 [49]. Assume that the outer layers have a high-strengthclass of compressive concrete, C90/105, and that the innerlayer is made of a low-strength compressive class concrete,C12/15. Hereafter, it will be denoted as a high/low/highstrength layered composite structure.

The cylinder mean compressive strength for each layer isequal to f cm;1 f cm;3 98 MPa and f cm;2 20 MPa. The devel-opment of the elasticity and strength moduli in time are alsodetermined by [49]: Ei t

cc;i t

pEi 28 and f cm;i t cc;i t

f cm;i 28 (where cc;i t is the coefcient evaluating the age ofthe concrete for i-th layer). Additionally, the elasticity moduliof the layers are related to the cylinder mean strength, f cm;i,

by the formula Ei 28 2:15104f cm;i=10

3q

.

The values for the creep coefcient, t; t0 , determined bycode [49] should be additionally multiplied by the factor cc t because of the code-based methodological peculiarities (see,Ghali et al. [50]).

Graphs showing the creep coefcient, t; t0 , used in thecurrent analysis for the inner and outer layers are shown in

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 481tive humidity is selected to be RH95%. For this selection,the examined parameters are considerably different to

a la r c h i v e s o f c i v i l a n d m e c h a n i c482demonstrate the important peculiarities that may arise in thedesign of this type of structure.

The thickness of the outer layers is 0.1 m (A1A30.1 m2),and that of the inner layer is 0.2 m (A20.2 m2). For the linearanalysis, the composite wall is subjected to a sufcientlylow sustained loading, equal to N1.0 MN, induced att01, 10, 28, 90 and 730 days. It represents an early-,normal-, and late-age concrete used to construct the layeredwall structure.

The stress-strain state of the layers and the entire com-posite at any time t is shown in Fig. 5. The coefcient of agingof the layers (Fig. 5b) is computed by eq. (24), evaluating theformulae (2527). The stress history (Fig. 5c) is derived fromeq. (22), expressing si tk for i tk c tk at any tk, for i1, 2, 3.Meanwhile, the strain of the entire composite c tk (Fig. 5d) isdetermined by formula (23).

The following conclusions can be drawn from the results. InFig. 5c, the layer possessing the higher strain tends to produce arelaxation, whereas the layers with a lower strain experience thestress increase to maintain the equilibrium between internal andexternal forces. If non-linear creep occurs, failure could primarilyoriginate from the layer that possesses a lower strain.

Fig. 3 Illustration of the solution of the system of non-linear eq. (equation; (b) non-linear surface of strain compatibility equationsunknown layer stresses r1 and r2.e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0As shown in Fig. 5b, the variation in the aging coefcient ofthe layers is different. This variation depends on the stresshistory and the aging of the layers. The descending branchesof the aging coefcient are governed by the material aging;however, for old concrete, the aging process is minimal, resultingin i t; t0 -1.

The values of the creep coefcient dened by (21) for theentire composite (denoted by bold lines) fall mainly withinthe values of the layers creep coefcients (Fig. 5a). In addition,the nal values of the entire composite, c 30000; 1 1:156 andc 30000; 10 1:126, are virtually negligible, indicating that thedifference in the layer age is mild.

This simulation can be used to validate the proposedanalytical approach. To this end, we select the loading timet01 day and the time of consideration t 3104 days. In thiscase, from Fig. 5(a, b), the creep and aging coefcientsare 1 t; t0 3 t; t0 0:767 and 1 t; t0 3 t; t0 0:014,respectively, for the outer layers. The values for these para-meters for the inner layer are t; t0 2:147and 2 t; t0 0:305, respectively. The elasticity moduli at the time ofloading are as follows: E1 t0 E3 t0 26907:61 MPa, E2 t0 15841:99 MPa.

50) for a three-layered structure: (a) plane of static equilibrium; and (c) intersection of plane and surface as a solution for

a la r c h i v e s o f c i v i l a n d m e c h a n i cAn instantaneous stress-strain state. The axial stiffness of thethree-layered wall structure is as follows:

i n

i 1Ei t0 Ai 226907:610:1 15841:990:2 8549:80 MN:

Thus, the layers stresses at t0 are calculated using (8) asfollows:

s1 t0 s3 t0 126907:618549:80

3:147 MPa;

s2 t0 115841:998549:80 1:853 MPa:

Using (1012), the instantaneous stress and elasticitymodulus for the entire composite are as follows:

sc t0 120:1 0:2 2:5 MPa;

Ec t0 8549:8020:1 0:2 21374:5 MPa:

The values for the stress and elasticity modulus for thethree-layered wall fall between their values for the layers.

From the assumption of the perfect bond between thelayers, the strains from eq. (11) and those from (7) should beequal. Thus:

1 t0 3 t0 3:14726907:61 1:17104;

c t0 1

8549:80 1:17104:

Linear creep analysis. To perform the creep analysis, thecoefcient of aging should be known in advance. Thus, when

Fig. 4 Schematic for the analysis of a three-layeredstructure.i t; t0 is determined, accounting for Ei t Ei t0 , then theanalytical predictions of the stress-strain state can be basedonly on the initial value of the elastic modulus, Ei t0 . Thisrelationship was proven in [10].

Using (1415), we can calculate the effective and the age-adjusted effective elasticity moduli for i-th layer. Thus:

Ee;1 t Ee;3 t 26907:611 0:767 15227:85 MPa;

Ee;2 t 15841:991 2:147 5034:0 MPa;

Ee;1 t Ee;3 t 26907:611 0:0140:767 27199:68 MPa;

Ee;2 t 15841:99

1 0:3052:147 9573:154 MPa:

The composite strain developed at time t is determinedfrom formula (19) as follows:

c t 23:1470:127199:68=15227:85 1:8530:29573:154=5034

227199:680:1 95730:2 ;

c t 2:4869104:The composite strain obtained using the numerical tech-

nique, c t 0:002487(see Fig. 5d), is coincident with theanalytical prediction.

When c t is known, the layer stresses may also bedetermined using (20) as follows:

s1 t s3 t 3:147 2:48691043:147

15227:85

27199:68;

s1 t s3 t 4:290 MPa, (cf., 4.289 from Fig. 5c);

s2 t 1:853 2:48691041:8535034

9573:154;

s2 t 0:710 MPa, (cf., 0.711 from Fig. 5c).These results show that the outer (less deformable) layer

is further compressed under sustained loading, while theinner layer (more deformable) produces the relaxation fromthe tensional stress increment developed with t.

The creep coefcient for the three-layered wall can bepredicted by relation (21) as follows:

c t; t0 2:48691:17 1 1:126, (cf. 1.126 from Fig. 5c).Thus, the exact value of the coefcient of aging yields the

same values of the stress-strain state of the entire composite asthose produced by the numerical inversion of the Volterraintegral.

According to these results, negative values for the coef-cient of aging occur if the time-dependent stress redistribu-tion between the layers is governed by the combination of theconsiderably different material aging and creep strains forthe case of the early age multi-layered composite.

For example, for reinforced concrete structures (where theaging coefcient is extensively investigated), ; t0 canrange primarily from 0.5 to 1. The code (EN 1992-1-1 2004)suggests an approximate nal value of ; t0 0:8 as asuitable case for predicting pre-stress loses due to creep. Inthis case, if the designer uses the concrete layered compositevalue of 0:8 in the analysis, sufcient errors occur inthe prediction of the high strength layer stress because the

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 483actual value 1;3 1; 0:014 (Fig. 5b) is very far from thatsuggested by the code (EN 1992-1-1 2004).

a l0

0.5

1

1.5

2

2.5

100 101 102 103 104 105

t, t0

(t, t

0)

2.147 (inner layer)

0.767 (outer layer)

1.126 (composite)

0.787 (outer layer)

2.203 (inner layer)

1.157 (composite)

1.973 (inner layer)

0.705 (outer layer)

1.040 (composite)

0.596 (outer layer)

1.667 (inner layer)

0.884 (composite)

1.163 (inner layer)

0.416 (outer layer)

0.626 (composite)

4.54.289 (outer layer)

a r c h i v e s o f c i v i l a n d m e c h a n i c484In contrast, as a well-known rule, if the loading age exceedsapproximately 90 days, the aging coefcients of the layers tendto ()0.8. Then, for most practical purposes requiring longcreep periods, tt0 (exceeding 360 days), an approximate value ofthe layers aging coefcient equal to 0.8 should be recommended(see, (Fig. 5, b)) for predicting the stress-strain state of the layeredcomposite using the proposed analytical method. Thus, 0:8 may be implemented for the analytical time-dependent analysis of a three-layered concrete structure fort0 90days.

6.2. Non-linear analysis

Consider the effect of non-linear creep on the time-dependent stress-strain state for the above three-layeredstructure. Now, suppose that this structure is made of layersof middle/low/middle strength concrete. The C25/30 strengthclass of concrete is used for the outer layers, and the innerlayer is constructed of low-strength C8/10 concrete. Assumethat the initial stress strength is 2 t0 0:94 and1;3 t0 0:58for the inner and for the outer layers,

100 101 102 103 104 105

t, t0

0.5

1

1.5

2

2.5

3

3.5

4

i(t

, t0)

0.711 (inner layer)

1.032 (inner layer)

3.968 (outer layer)

1.136 (inner layer)

3.864 (outer layer)

1.233 (inner layer)

3.767 (outer layer)

1.373 (inner layer)

3.627 (outer layer)

Fig. 5 Stress-strain state at time t (measured in days) for the entithe layers (plotted by regular curves) and the entire composite (boevolution with time; and (d) total strain evolution for the entire c100 101 102 103 104 105

t, t0

0.2

0

0.2

0.4

0.6

0.8

1

1.2

i(t,

t 0)

0.305 (inner layer)

0.014 (outer layer)

0.618 (inner layer)

0.532 (outer layer)

0.715 (inner layer)0.666 (outer layer)

0.805 (inner layer)0.780 (outer layer)

0.914 (outer layer)0.920 (inner layer)

2.6 x 104

0.0002487

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0respectively. The initial stress strength selected is near theinstantaneous failure state for the low-strength (inner) layer.

These levels were held constant over all t0 considered.Therefore, the magnitude of the sustained load shouldchange to keep the assumed stress strength constant fromthe instantaneous prediction, which illustrates the stressevolution at different t0 with unied initial stress conditions.The stress-strength ratio pertains to the mean value of thecompressive cylinder strength.

For the baseline material properties (strength, elasticitymoduli, creep strains), the code EN 1992-1-1:2004 model wasused. Following NIIZHB [49] specications, the factor m4 isrecommended. The coefcient vi, accounting for theincrease in the creep strain during the delayed failure, iscomputed as vi 44:47=f pr;i 28 , for f pr;i 28 38 MPa; other-wise, vi 1:22 (where f pr;i 28 is the compressive prismstrength). Thus, the mean value of the compressive prismstrength, which was used in the approach of [49], wasdivided by a factor approximately equal to 0.95 to obtainthe mean value of the compressive cylinder strength. Thegraphs dening the linear creep coefcient t; t0 used in the

100 101 102 103 104 105

t, t0

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4 c

(t) 0.0001605

0.0001395

0.0001220

0.0001006

re composite loaded at different ages t0: (a) creep coefcient forld curves); (b) coefcient of aging of the layers; (c) layers stressomposite.

t, t0

nd

a lcurrent analysis for the inner and outer layers are shown inFig. 6.

In Fig. 7, the stress-strain evolution for each layer isdepicted. The results accounting for non-linear creep areshown as a dashed line, and the values specifying the linearcreep prediction are shown as solid lines. The ages at whichthe loading is imposed were t02, 28, 180 and 740 days.

As shown in Fig. 7a, the nonlinear creep produces an

100 101 1020

0.5

1

1.5

2

2.5

3

(t, t

0)

Fig. 6 Creep coefcient of the layers (h091 a

a r c h i v e s o f c i v i l a n d m e c h a n i cincrease in stress relaxation for the more deformable layer(the inner layer) compared to the linear creep behavior. Thestresses in the less-deformable layer determined by the non-linear creep analysis are greater than those found in thelinear prediction.

In structural design, the decrease in a more-deformablelayers stress from non-linear creep may be considered acertain reserve of the linear law. However, the increase instress of the less-deformable layer may be a dangerous statethat leads to a delayed failure of the layer. To this end, thenext section considers the long-term strength or a delayedfailure prediction.

6.3. Long-term strength and delayed failure analysis

The most important factor for the designer is to predict themaximum load an entire multi-layered composite can with-stand under a long period of sustained compression imposedon the layered composite structure. This prediction is acomplicated task because the concrete materials havecracked, i.e., the fracture is related to the crack propagation,which is not completely brittle and usually possesses somedegree of ductility.

Models from fracture mechanics each claim partial suc-cess when performing the instantaneous fracture analyses(e.g., [51], [52], [53]). In general, when a closed form solutiondoes not exist, an FEM analysis can be adopted to study thesophisticated failure mechanisms of composite materials,including the material and geometrical non-linearities com-bined with the complex composite geometries. The FEManalysis of a compressive shear fracture test [54] has beenperformed, focusing on the near crack-tip behavior of a two-layered composite specimen with linear elastic isotropicproperties that contains a curved interface crack when itssurfaces come into contact and are able to dissipate energy

103 104 105

2.567 (inner layer)

1.821 (outer layer)

2.205 (inner layer)


1.181 (outer layer) 1.297 (inner layer)

0.920 (outer layer)

168 mm, RH95%) for the non-linear analysis.

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 485during the formation of new crack surfaces via the slidingfriction rule. By applying the loading process to the stifferlayer with the weaker layer supported and vice versa, atheoretical estimation of the mode II stress intensity factorswas obtained.

In concrete materials, the fracture process is complex; ifthe action of creep strain is dominant over the process ofhydration in cement paste, the cracks start to propagate in astable manner. Depending on the stress level induced, thecracks can interact with each other, and beyond a certainperiod after loading, the crack propagation may form anunstable pattern. In this case, the cracks coalesce, forming asingle continuous crack of a critical length within the layer. Itis also well known that a time-dependent decrease in thecompressive strength of the quasi-brittle materials occurswhen a non-linear creep strain dominates.

Wittmann and Zaitsev [55], [56] successfully determinedthe decrease in the long-term strength of concrete under ahigh sustained load. Relying on classical fracture mechanics,they analytically found the following function, which evalu-ates the state of the material under sustained loading:

t; t0 E t0 t0 E t t

s E t0 ~E t

ss t s t0

; 68

where t0=t Et=Et0f ct0=mt; t0f ct2 is the ratio ofthe effective surface energies at the time instant of initialloading and the time of consideration; ~E t is an operator that

a l15

20

25

i(t

, t0)

a r c h i v e s o f c i v i l a n d m e c h a n i c486converts the instantaneous plane-stress state into the statedeveloped near a crack-tip under linear creep action;m t; t0 11:2 is the factor dening the increase in strengthat time t from the action of the preceding stress [56]; andst=st0 is the stress ratio manifesting the time variation ofthe effective load near a crack-tip [56].

The rst term in function (68) evaluates the solidicationof the material due to the hydration of cement paste, thesecond term accounts for the damage to the material due tothe growing cracks, and the last term denes the relaxationor increase in stress near the crack-tip.

According to [56], an inversion of t; t0 yields the ratio ofthe materials long-term strength, f c t; t0 , to its strength at

100 101 1020

5

10

t, t0

100 101 1020.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 103

t, t0

c(t)

Fig. 7 Development of stress (a) and strain (b) in a three-layerelinear creep (solid lines) behavior.11.515 (inner layer)

19.076 (outer layer)


13.804 (inner layer)








e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0the loading time, f c t0 ; thus

u t; t0 t; t0 1 f c t; t0 f c t0

: 69

We adopt this approach to predict the long-term strengthof the composite. Arutyunyans [9] theorem states that thepresence of linear creep does not change the instantaneousequilibrium equations of the plane-stress state of the classi-cal theory of elasticity when external loading is applied.In this case, the creep strain only inuences the total strain.This theorem allows us to account for the linear creep strain,cri t; t0 , in eq. (68) by using the operator 1= ~E t , which isexpressed from the total strain, i t c t , dened in eq. (4)

103 104 105

9.982 (outer layer)


5.432 (inner layer)4.980 (inner layer)


103 104 105

0.0011814

0.0012564

0.0017687

0.0013689

0.0016788

0.0012252

0.00150600.0015132

d composite structure for non-linear creep (dashed lines) and

for the i-th layer stress at t as follows:

1= ~Ei t c t si t

1Ei t

s;i t; t0 si t

Cni t; t0 : 70

Numerically, this relation may be expressed using eq. (23).To include the effect of stress relaxation (or its increase) nearthe crack-tip of each layer, the ratio of s t =s t0 should becomputed from the non-linear solution of (47) because it simplyevaluates the effective load developed at time t (see [56]).

Following [55], [56], we can benecially introduce adamage index for the i-th layer as follows:

Mi t; t0 i t0 i t; t0 i t0

u;i t; t0 71

A value of Mi tni ; t0 1 means that the initial stress-

strength level, i t0 si t0 =f c;i t0 , causes the delayed failureof the i-th layer at time tni , where u;i t; t0 is the long-termstress-strength ratio. The stress dened at the failure time tniis the long-term strength of the layer f c t; t0 si tni

.

For Mi t; t0 o1, the applied stress-strength ratio i t0 is toolow and cannot cause layer failure at any t.

The theoretical prediction of the long-term strengthaccording to eq. (68) and failure time tn for the i-th layerdepends on the aging and creep properties, the abilities ofcement hydration to heal the crack-tip and the ability toredistribute the stress near the crack-tip. In particular, thestress relaxation produces growth of i t; t0 and, in turn,growth in the long-term strength. Meanwhile, the stressincrease near the crack-tip results in the long-term strengthreduction.

Other more complicated techniques based on viscous-elastic damage and rheological modeling may be found inthe literature for plain concrete. For example, a nonlinearcreep damage approach was implemented by Mazzotti andSavoia [57] for plain concrete under uniaxial compression.In their study, the creep strain was modeled using a modiedversion of the solidication theory with a damage indexbased on the positive strains. Recently, viscous-elasticdamage modeling has also been implemented by Verstryngeet al. [19] for masonry structure analysis. A multiscalemodeling of early age concrete behavior was reported in [22].

In Fig. 8, a theoretical long-term strength prediction isillustrated. The graphs shown in Fig. 8(a, b) are attributed to

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

i(t

)/fci

(t 0)

0.700 (outer layer)

1.387 (inner layer)

1.011 (outer layer)

0.688 (outer layer)

0.711 inner layer)

0.872 (inner layer)

0.718 (inner layer)

0.852 (inner layer)

0.725 (inner layer)

0.878 (inner layer)

)))

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Mi(t

,t 0)

0.692 (outer layer)

1.085 (outer layer)

0.816 (inner layer)

1.115 (outer layer)

0.843 (inner layer)

1.077 (outer layer)

0.827 (inner layer)

Outer layers failure

a r c h i v e s o f c i v i l a n d m e c h a n i c a l e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0 4870.5

0.6

100 101 102 103 104 105t, t0

100 101 102 103 104 105t, t0

i(t

)/fci

(t 0)

0.634 (outer layer)0.685 (outer layer)

0.614 (outer layer)0.633 (outer layer)

Outer layers failure

0

0.5

1

1.5

2

2.5

0.466 (outer layer)

0.369 (inner layer)

1.138 (outer layer)

0.432 (outer layer)

0.734 (outer layer

0.535 (inner layer)

1.208 (inner layer)

2.400 (inner layer)

0.707 (outer layer0.600 (inner layer)

1.089 (inner layer)

0.418 (outer layer)

0.719 (outer layer0.636 (inner layer)

1.076 (inner layer)

0.411 (outer layer)Fig. 8 Function i t; t0 , specifying the state of material under sua-b) composite composed of middle/low/middle strength concrete100 101 102 103 104 105t, t0

100 101 102 103 104 105t, t0

0.4

0.5

Mi(t

,t 0)

0.496 (inner layer)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.154 (inner layer)




0.572 (outer layer)stained loading and the damage index Mi t; t0 for each layer:; c-d) composite composed of high/low/high strength concrete.

layer allows the additional loading to be borne by the compo-

a lthe structure composed of layers that possess middle/low/middle strengths: f cm;1 33 MPa, f cm;2 16 MPa andf cm;3 33 MPa. The plots in Fig. 8(c, d) represent the long-term strength prediction of the high/low/high layer strengths.All material properties were given in the previous analysis.

The middle/low/middle strength composite structure(Fig. 8a, b) was loaded using an initial stress-strength level at2 t0 0:94 and 1;3 t0 0:58. This level was held constant forall ages of t0 considered. As shown in (Fig. 8a), the early agestructure (loaded at t02 days) has an extensive ability forstress redistribution over the layers from the development ofcreep strain, aging and cement hydration. In particular, theinner (low strength) layers stress-strength ratio, 2 t0 0:94,immediately falls to 0.8 after load application, and it nallyreduces to 0.688. In the interim, the outer layers stress-strength levels increase from the initial value of 1;3 t0 0:58to 1;3 t 0:7.

The limit value for the long-term stress-strength ratio (dottedbold lines in Fig. 8) should be used as aminimum for u t; t0 (see,[56]). Hence, the comparison of u;1;3 t; t0 0:940:7 andu;2 t; t0 140:688 (for t30000 and t02 days in Fig. 8a) indi-cates that long-term failure is not reached in all layers. Thedestruction index, M1;2;3 t; t0 o1, more visually shows this con-clusion for t02 days in Fig. 8b. In additional analyses, when theload was increased to almost instantaneous failure, i.e.,2 t0 0:98 and 1;3 t0 0:62, the long-term failure of the outerlayer was not obtained in any layer because of the aforemen-tioned stress redistribution.

The destruction factor, M1;3 t; t0 1 (horizontal line indi-cating the failure), shown in Fig. 8(b) occurs for the outerlayers loaded at ages t028, 180 and 740 days. The delayedlong-term failure of these layers occurs after a certain periodunder sustained loading (intersection point of horizontal linewith function Mi t; t0 in Fig. 8b). Theoretically, the failure ofthe outer layers occurs when the crack length increases to itscritical length, while the inner layer withstands the sustainedcompression because its stress relieves throughout. Thepractical conclusion from Fig. 8(b) is that the initial loadinglevel applied for all ages of t028, 180 and 740 days is too highand should be diminished to less than its ultimate values,i t0 i tn u;i t; t0 , to avoid long-term failure.

In the analysis of the high/low/high strength composite(Fig. 8c,d), the initial stress-strength levels were introducednear the instantaneous failure, 2 t0 0:98, for the inner layerand 1;3 t0 0:33 for the outer layers. The results shown inFig. 8(c, d) illustrate that no long-term failure occurs (i.e.,M1;2;3 t; t0 o1) at any t for any layer; consequently, no longterm failure occurs for the entire composite either.

Based on these results, after the nal period of the sus-tained loading, when the delayed failure was avoided, theapplied load could be increased to cause the failure of thecomposite structure. This is important when decisions regard-ing the structure strengthening should be given after theperiod of exploitation. The plots in Fig. 8(b, d) show that thelong-term strength of the composite can be greater than itsinstantaneous value when M1;2;3 t; t0 o1. As shown, the initialinstantaneous load value after the nal period of action maybe increased by at least 1020% to reachMi t; t0 1. This effect

a r c h i v e s o f c i v i l a n d m e c h a n i c488can be explained in the following way. The low-strength layerhas a large creep strain, while the high-strength material issite after the period of sustained loading. This feature will beconrmed below based on the experimental investigationsfound in the literature. Additionally, a low creeping of thehigh- strength concrete layers also governs the increasedpresence of crack propagation in comparison with the low-strength layer (cf., Fig. 8(a) and (c), using u;1;3 t; t0 ).

6.4. The experimental data

To verify the theoretical prediction of the long-term strengthor delayed failure time of a multi-layered composite element,experimental investigations are required. Unfortunately,experimental tests involving long-term strength tests oflayered composites under sustained compressive loadingare not currently available in the literature. However, anindirect qualitative verication can be discussed, relying onsome published experimental data.

Prokopovich and Zedgenidze [36] have summarized theavailable experimental data on reinforced concrete beamssubjected to sustained bending and concluded that thebeams having a high reinforcement ratio (more than 3%)and when subjected to a high sustained load (0.9 of theinstantaneous failure load), the beams did not collapse afterperiods of 648, 1048 and 390 days. These beams were ableto withstand a portion (1.121.42% of the instantaneousfailure load) of the additional loading to reach the failure.The authors related the increase in the load-carrying capacityof the RC beams to the increased abilities of concrete torelieve stress under high levels of sustained loading.

In the short-term experimental and numerical investiga-tions of two-layered composite beams involving high-performance concrete for the top layer and normal strengthreinforced concrete for the bottom layer, apko et al. [24] alsoconcluded that because of a redistribution of stress betweenthe layers, the composite beams withstood a higher load thenthose made of the normal strength concrete without layers.

Experimental evidence demonstrating that plain high-strength concrete is more prone to crack propagation andexhibits increased long-term strength in comparison withlow-strength concrete was also previously reported by Iravaniand MacGregor [58] and Smadi et al. [59]. The latter authorsdetermined that u t; t0 0:65 and u t; t0 0:8 for concretewith strengths of fc20 MPa and fc60 MPa, respectively. Theempirical relations found in the literature cannot capture theeffect of concrete strength on its long-term degradation.

These experimental results are an indirect qualitativeverication of the results presented in Section 9. In this case,a relevant test series should be conducted in the future toobtain a direct verication of the theoretical prediction of thelong-term strength, strain and the delayed failure time ofmulti-layered composite elements.

7. Conclusionscharacterized by a low creep strain (Fig. 5a). Thus, because ofthe layers strain compatibility, the stress in the low-strength

e n g i n e e r i n g 1 3 ( 2 0 1 3 ) 4 7 2 4 9 0Both linear and nonlinear creep models for predicting the time-dependent behavior of a concrete composite were proposed.

a lThe presented one- and two-dimensional models use theanalytical and iterative analyses of the Volterra integral equa-tion, implying the validity of the principle of superposition.

The analytical approach is based on the age-adjustedeffective modulus method. The model was validated througha direct calculation of the time-dependent stresses andstrains for a three-layered composite wall structure, compar-ing the theoretical values with those determined numerically.In particular, the analytical approach yields identical valuesfor the time-dependent stress strain state parameters. It hasrecently been found that negative values for the agingcoefcient can be obtained when the stress redistributionbetween the layers is governed by a combination of consider-ably different creep strains and aging of the layers for theearly age multi-layered composite.

Under the plane strain state, the two-dimensional creepanalysis of the multi-layered composite results in the samevertical stress-time history as that in a one-dimensional caseif the Poisson ratios of the layers are equal. This result holdseven though an average value of the vertical stress to satisfythe Volterra integral term is dependent on the Poisson ratioof the layers. In particular, the evolution of vertical stresswith time is dependent only on the vertical strain andcompatibility conditions in a direction parallel to thelamination.

The fracture mechanics approach has been adopted tostudy the gradual degradation of the materials and the long-term strength of the multi-layered composite under sus-tained compression. Particularly, it was dened that thelong-term failure mainly starts from the layers having lessdeformability because the more deformable layers can relievethe initial stresses. In particular, the stress redistribution nearthe crack-tips during the nal period of high sustainedloading may result in the need to apply additional compres-sive stress to reach failure of the composite. Therefore, thelong-term strength of the composite may exceed its instan-taneous strength for some cases, such as the early agecomposite or the composite made of layers possessing con-siderably different creep and aging properties. This feature isimportant in the design of layered wall structures to ensurethe proper selection of materials for the layers.

The proposed analysis and methods provide a basis forfurther investigations on the time-dependent behavior of aconcrete multi-layered compressive composite.

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