-
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
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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
qs1 tk ; :::; qsn tk T
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
qs1 tk ; :::; qsn tk T
f n
qs1 tk ; :::; qsn tk T
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)
1.564 (outer layer)1.665 (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)
20.372 (outer layer)
13.804 (inner layer)
12.005 (inner layer)
21.787 (outer layer)
23.586 (outer layer)
14.778 (inner layer)
12.748 (inner layer)
22.637 (outer layer)
24.667 (outer 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)
10.434 (outer layer)
5.432 (inner layer)4.980 (inner layer)
10.219 (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)
0.409 (outer layer)0.443 (inner layer)
0.588 (outer layer)0.551 (inner layer)
0.592 (outer layer)0.591 (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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