CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

A model for Textile Reinforced Concrete under imposed uniaxial deformations

Jens Hartig∗ and Ulrich Häußler-Combe∗

Institute of Concrete Structures, Technische Universität Dresden
01062 Dresden, Germany
e-mail: jens.hartig@tu-dresden.de, ulrich.haeussler-combe@tu-dresden.de

Abstract

This paper presents investigations regarding the uniaxial load-bearing behaviour of the novel cement-based composite Textile Rein-
forced Concrete under restrained imposed deformations. Imposed deformations might result from shrinkage and hydration heat of the
cementitious matrix and external temperature changes. As a consequence, bond degradation between matrix and reinforcement as well
as matrix cracks can occur, which can impair both the load-bearing capacity as well as serviceability. For modelling, a one-dimensional
finite element model is applied consisting of bar elements representing the load-bearing behaviour of either matrix or reinforcement
and bond elements to model the interaction in between. Imposed deformations are applied as additional strains to the bar elements. The
model provides results regarding the global load-bearing response of the composite, which can be directly compared with experimental
results. Additionally, information regarding local stress distributions are available, which can be usually not determined in experiments.
Thus, further insight in the load-bearing behaviour of the composite is obtained with the model.

Keywords: composites, concrete, cracks, finite element methods, numerical analysis, textiles

1. Introduction plication of a cement-based matrix by internal sources, e. g. hy-

dration heat and shrinkage. The effect of hydration heat can be
Textile Reinforced Concrete (TRC) is a composite of a fine- assessed as small as the thickness of TRC structural elements is
grained concrete matrix and a reinforcement of multi-filament usually small (in the range of a few centimetres) and, thus, ther-
yarns processed to textile structures, see e. g. Refs. [4] and [8]. mal gradients are small as well.
The reinforcement material can be chosen according to the de- Restrained imposed deformations might result in deteriora-
sired application of the composite. For structural applications, fi- tion of the composite at least for two reasons. On the one hand,
bres exhibiting sufficiently large Young’s modulus and high ten- it might lead to additional cracking of the matrix if the stabilised
sile strength such as alkali-resistant (AR) glass or carbon fibres cracking state is not already reached due to previous loading. On
are usually applied. Moreover, these fibre materials do not suffer the other hand, the bond between matrix and reinforcement might
from corrosion when embedded in the alkaline matrix, see Refs. be degraded. This can be especially critical in the case of thermal
[5, 6, 7], which allows for a considerable reduction of concrete loading if matrix and reinforcement have different thermal ex-
cover compared to conventional reinforced concrete and the de- pansion coefficients. Carbon fibres are known to exhibit negative
sign of thin structural elements, see Fig. 1. thermal expansion in axial direction at room temperature while
the matrix has positive thermal expansion. To investigate the in-
fluence of imposed deformations on the uniaxial load-bearing be-
haviour of TRC, the model presented in Ref. [11, 12, 13] is en-
hanced by imposed strains.
In the subsequent section, the material behaviour of the con-
stituents of TRC, the matrix and the reinforcement, as well as the
composite itself is described briefly. Afterwards, the model for
the simulation of the load-bearing behaviour of TRC is presented.
This is followed by computational results regarding the response
of the composite to restrained imposed deformations. At the end,
the findings are summarised and some conclusions are drawn.

1cm 2. Selected material properties

2.1. Fine-grained concrete

The matrix used for the composite under consideration is a
Figure 1: Textile reinforced concrete; photo from Ref. [11] fine-grained concrete with a maximum aggregate size of 1 mm
and a CEM III cement as binder, see e. g. Refs. [5] and [14]. It
A special load case is imposed deformation, which might is known that due to a relatively slow hydration speed charac-
be critical for the integrity of structural elements of TRC if re- teristic of CEM III cements also low hydration heat per time is
strained. Imposed deformations might be caused by external produced, see e. g. Ref. [20]. Additionally structural elements of
sources, e. g. external temperature changes, and due to the ap- TRC are usually thin and, thus, only small temperature gradients
∗
The authors gratefully acknowledge the financial support of this research from Deutsche Forschungsgemeinschaft DFG (German Research Foundation) within the Sonder-
forschungsbereich (Collaborative Research Center) 528 “Textile Reinforcement for Structural Strengthening and Retrofitting” at Technische Universität Dresden.
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

occur. Thus, the effect of hydration heat of the matrix is negli- ferable bond stresses over the yarn cross section are large. In
gible for TRC. Nevertheless, external temperature reduction will general, only the filaments in the so-called fill-in zone or sleeve
lead to contraction of the matrix, which can result also in con- zone, which is the outer layer of the yarn where matrix or coat-
crete cracks if restrained. The thermal expansion coefficient of ing intrudes continuously, are bonded initially by adhesion to the
cementitious matrices is αT,m ≈ 1.0 · 10−5 K−1 . matrix. The adhesive bond is limited by the bond strength. When
A further material property of the matrix, which results in the bond strength is exceeded, bond degradation occurs, which is
imposed deformations is shrinkage. As the matrix shows rela- associated with a successive transition to frictional load transfer.
tively large shrinkage, matrix cracks are supposed to occur if the For the load transfer in the core of the yarns where a negligible
shrinkage strains are restrained. The maximum shrinkage strains amount of matrix or coating material is present, frictional load
of the fine-grained concrete consisting of autogenous and drying transfer between the filaments is usually assumed.
shrinkage can be estimated with 0.2 %, which is an order of mag-
nitude larger than the ultimate tensile strain. 2.4. Uniaxial load-bearing behaviour
Under uniaxial tensile loading, the material behaviour of ce- The load-bearing behaviour of TRC due to tensile loading
mentitious matrices can be assumed linear elastic up to a ten- is usually determined with plate-shaped specimens. However,
sile strength fmt . According to Ref. [14], Young’s modulus experimental results under imposed deformations are, hitherto,
and tensile strength of the fine-grained concrete have values missing. An extensive experimental work on the tensile be-
of Em0 ≈ 28,500 N/mm2 and fmt ≈ 5 N/mm2 . After exceeding haviour of TRC was carried out by Jesse [14]. Selected results
fmt , brittle failure can be assumed in a first approach. Exper- are summarised in the following.
iments reveal, however, a certain post-cracking resistance also
referred to as tension softening, which depends on several prop- 500 mm
erties of the matrix. It is, e. g., known that concretes with large
rough aggregates show stronger post-cracking resistance than support measurement length 100 mm
concretes with small round aggregates, see e. g. Refs. [16] and
[20]. As the used fine-grained concrete has small round aggre-
gates, relatively small post-cracking resistance can be expected. 100 mm
Tension softening of the applied fine-grained concrete was, hith- F(u) F(u)
erto, not experimentally investigated. Investigations on a similar

8 mm
matrix presented in Ref. [3] revealed, however, a fracture energy
Gf ≈ 40 N/mm and crack width at complete crack face separa-
tion of wc ≈ 0.2 mm
2.2. Reinforcement
Figure 2: Test setup used by Jesse [14] for the determination of
The fibres typically used for TRC are AR glass and carbon the tensile behaviour of TRC
fibres. While AR glass has a positive thermal expansion coeffi-
cient of αT,g ≈ 5 · 10−6 K−1 similar to the matrix, carbon has a
The specimens, see Fig. 2, are usually applied with reinforce-
negative value of αT,c ≈ −1... − 5 · 10−7 K−1 in fibre direction
ment ratios in a range of 1 % up to 3 %, which preserves multiple
at room temperature, see Refs. [9, 17]. Shrinkage is not observed
cracking of the matrix. The specimens are attached to testing ma-
for both fibre materials.
chine by means of clamps. Loading is applied with displacement
The uniaxial tensile behaviour of both reinforcement mate-
(u) control. During loading, forces F are measured with a load
rials can be assumed as linear elastic up to failure reaching the
cell and relative displacements with extensometers on the surface
tensile strength. The reinforcement yarns show usually a certain
of the specimen over a measurement length of 0.2 m. The forces
waviness resulting from the production process. The waviness
are divided by the cross-sectional area of the specimen leading
leads to a delayed activation of the reinforcement and to larger de-
to a mean stress. The relative displacements are related to the
formations of the cracked composite compared to stretched fibres.
measurement length resulting in a mean strain.
The Young’s modulus of the carbon fibres is with approximately
210,000 N/mm2 considerably higher than the value of glass fibres
with approximately 80,000 N/mm2 , see Ref. [1]. Moreover, the multiple tension
exploitable tensile strength is usually larger for the carbon fibres cracking stabilised cracking stiffening
(1000-2500 N/mm2 ) than for the glass fibres (500-1500 N/mm2 ).
2.3. Composite uncracked
mean stress ¾

In the case of thermal loading, in TRC with AR glass rein-

forcement an almost compatible deformation of matrix and rein-
forcement is possible because the thermal expansion coefficients
are similar. Thus, bond deterioration due to thermal loading shall
be small. In contrast, the negative thermal expansion coefficient
of carbon fibres might lead to considerable bond deterioration due
to opposed deformations. Moreover, also matrix cracking might
be increased in case of temperature reduction. In the case of ma-
trix shrinkage, the reinforcement might restrain matrix deforma-
tions.
In general, the interface is stronger between AR glass fibres mean strain "
and concrete than between carbon fibres and concrete. However,
for both cases bond can be considerably improved with additional Figure 3: Typical mean stress-strain behaviour of TRC under uni-
coating, see e. g. Refs. [18, 19]. Moreover, different bond zones axial tensile loading; from [11]
usually exist over the cross section of a yarn due to incomplete
penetration with either matrix or coating, see e. g. Ref. [14]. Es- A respective mean stress-strain relation under uniaxial tensile
pecially without additional coating the differences of the trans- loading is shown in Fig. 3. It consists of three distinct state. The
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

Layer model Discretisation

layer matrix Transverse Longitudinal direction:
direction: loading: ¢T, ¢"imp,s
yarn

circum. c ¿ (s)mr
radial
direction direction r
(c) (r) ¿ (s)rr

¿ (s)rr
longitudinal
x)
direction ((x)
x node matrix reinforcement bond
r element element element
Constitutive relations
Tensile material law for matrix and reinforcement: Bond law ¿ (s)mr: Bond law ¿ (s)rr:
(smax, ¿max)
ft
bond
bond stress ¿

bond stress ¿
degradation
stress ¾

E
friction (smax, ¿max) = (sres, ¿res)
1
(sres, ¿res) friction
unloading unloading
strain " "t slip s slip s

Figure 4: Layer model and constitutive relations

uncracked state were the matrix bears most of the load. When the reinforcing yarns. Furthermore, transverse deformations due to
tensile strength of the matrix is exceeded, the first matrix crack Poisson’s ratio are neglected. For the reinforcement, it is assumed
occurs and the mean stress drops. The reinforcement is activated that all yarns behave approximately the same, which allows for
and bridges the crack. Distant to the crack the load is transferred representation of the entire system of a multitude of yarns by just
back to the matrix by means of bond mechanisms, which can be one yarn embedded in matrix. As the reinforcement consists of a
e. g. adhesion and friction. If a sufficient amount of reinforce- large number of single fibres, which are coupled discontinuously
ment is available and the bond is sufficiently strong, the load can by matrix or coating cross-linkages, the yarn cross section needs
be further increased. This leads also to further matrix cracks. At a to be modelled with more than one bar element chain. Therefore,
certain stage, crack spacing becomes too small to reach the matrix a layer model, see Fig. 4, is applied assuming that only in radial
tensile strength again, which is also associated with bond degra- direction r of the yarns differences in the load-bearing character-
dation. At this stage, the state of multiple cracking is finished istics occur while they are negligible in circumferential direction
and the stabilised cracking state starts. The stabilised cracking c. The layer model provides also the cross-sectional areas A,
state is primarily controlled by the material properties of the re- which are property of the bar elements. The bar element chains
inforcement and, thus, the slope of this state corresponds approx- are coupled at corresponding nodes with zero-thickness bond ele-
imately to the stiffness of the plain reinforcement. However, also ments according to the scheme shown in Fig. 4. The bond surface
the matrix participates in load-bearing between the cracks lead- areas S are also determined based on the layer model. Detailed
ing to tension stiffening, i. e. reduced strains, compared to the descriptions of the determination of the values of A and S accord-
stiffness corresponding to the reinforcement. ing to the layer model are given in Refs. [11, 13]. Boundary con-
For imposed loading, such a mean stress-strain relation can- ditions are given with prescribed displacements at the end nodes
not be established because the mean strain is always approxi- of the bar element chains. Loading can be applied as prescribed
mately zero. Therefor, it might be advantageous to establish some displacements, forces and imposed strains.
mean stress-imposed strain relations. Nevertheless, similar load-
bearing mechanisms as in the case of force-controlled loading in- 3.2. Bar elements
volving, e. g., matrix cracking and bond degradation will appear. The uniaxial material behaviour of the matrix and the rein-
forcement is modelled with two node bar elements as already
3. Model pointed out. The element stiffness matrix is given with
 
EA 1 −1
3.1. Geometrical characteristics Kbar = (1)
Lel −1 1
A model according to Refs. [11, 12, 13] based on the finite el-
where E is the Young’s modulus of the material, A is the cross-
ement method is applied, which consists of two types of elements.
sectional area and Lel is the bar element length.
The model is shown schematically in Fig. 4. One-dimensional
For both, matrix and reinforcement linear-elastic material be-
bar elements represent the uniaxial load-bearing behaviour of ma-
haviour according to the initial Young’s modulus E0 with limited
trix and reinforcement or parts of it. In longitudinal direction x,
tensile strength ft is assumed in a first approach
which is also the loading direction, a sufficient number of bar el- (
ements are arranged in series to represent the interaction between E0 (ε − εimp ) for 0 ≤ (ε − εimp ) ≤ Eft0
the constituents of the composite as well as multiple cracking of σ= (2)
0 for (ε − εimp ) > Eft0
the matrix appropriately. The matrix is modelled with one bar
element chain assuming that shear gradients are small in the ma- with stress σ, measurable strain ε and imposed strain εimp . If
trix due to small specimen thickness and small distances between εimp results from temperature changes ∆T , the respective im-
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

posed strain is εimp,T = αT ∆T with the thermal expansion bond stress value τres . Unloading is modelled based on the con-
coefficients αT of the materials given in Section 2. Imposed cept of plasticity, i. e. with a linear unloading path different to
strains due to shrinkage εimp,s are prescribed to the matrix el- the loading path and remaining deformations. For the filament-
ements only. The main portion of strains due to matrix shrinkage filament interaction between the core layers primarily frictional
occur at the early age of the matrix where also the material pa- load transfer is assumed. This is modelled in bond law τ (s)rr
rameters change considerably. Such time-dependent changes are with a bond strength τmax equal to the frictional bond stress τres .
not taken into account in these investigations and it is assumed in
a simplifying manner that material parameters are constant over 3.4. Numerical solutions
time. The cracking events in the matrix and the nonlinear bond laws
Tension softening of the matrix can be taken into account as lead to nonlinear systems of equations. Therefor, the load is ap-
implemented in Refs. [11, 12] applying the stress-crack width re- plied incrementally and an iterative solution is performed. As
lation by Remmel [16] and the crack band approach by Bažant & solution method, the BFGS approach [15], which is a Quasi-
Oh [2] for regularisation. The respective stress-strain relation is Newton method, combined with line search is used. Matrix
given for fmt /Em0 < (ε − εimp ) < fmt /Em0 + wc /Lel with: cracking events are limited to one per load step. When a crack
    c  occurred the system is recalculate on the same load step with a
Lel (ε − εimp ) − Efmtm0
respectively modified stiffness matrix.
σ = fmt1 exp −   +
w1
4. Simulations
  
Lel (ε − εimp ) − Efmt
fmt2 1 −
m0
. (3) 4.1. Specification of the model
wc
Exemplary simulations are carried out to show the abilities
of the model. Unfortunately, a validation of the results cannot be
Equation (3) contains a form parameter c, two parameters fmt1 performed, hitherto, as respective experimental data is not avail-
and fmt2 associated with the tensile strength fmt as well as two able. For the simulations, it is assumed that the specimens by
characteristic crack widths w1 and wc . For the determination of Jesse [14] as shown in Fig. 2 are loaded with imposed deforma-
the parameters and the derivative of Eq. 3, see Ref. [11]. Unload- tions in the free part of the specimen on a length of 300 mm. The
ing in the softening range is implemented based on the concept of length of the bar elements is defined with 0.2 mm, which results
damage mechanics, i. e. with reduced stiffness. The crack width in 1500 elements per bar element chain.
w of a cracked element can be determined as The concrete is modelled with one bar element chain, while
the reinforcement is represented by five bar element chains ac-
 
fct
w = ε − εimp − Lel . (4) cording to the layer model described in Section 3.1. The out-
Em0
ermost layer represents the filaments in the fill-in zone, which
assuming that one bar element does only represent one crack. are connected adhesively to the matrix and are supposed to share
their load uniformly or in other words globally. For the bond el-
3.3. Bond elements ements between the matrix and the outermost reinforcement bar
The interaction between the matrix and the reinforcement as element chain, bond law τ (s)mr with values τmax = 9 N/mm2 ,
well as between the reinforcement layers is modelled with zero- smax = 4·10−3 mm, τres = 3 N/mm2 and sres = 8·10−3 mm is ap-
thickness bond-link elements. The element stiffness matrix of the plied. Due to the weak load transfer between the filaments in the
bond-link elements is given with core of the yarns resulting primarily from friction, local load-
sharing depending on the radial position in the yarn has to be
expected. Thus, the core of the yarn is finer discretised with four
 
1 −1
Kbond = GS (5) layers. Between the reinforcement bar element chains, bond ele-
−1 1
ments with bond law τ (s)rr and values τmax = τres = 3 N/mm2
where G = dτ /ds is the bond modulus and S is the bond area. and smax = τres = 4·10−3 mm is applied. The bond law pa-
The constitutive relations of the bond elements are bond rameters were estimated in a calibration process performed in
stress-slip (τ -s) relations, see Ref. [13]. The slip s is the rel- Ref. [11].
ative displacement of the two nodes of a bond element. The The material properties of the matrix are applied as given in
τ -s relations are defined by supporting points (s, τ ). Between Section 2.1. Two types of simulations are carried out: neglect-
these supporting points, it is interpolated. Therfor, the so-called ing and considering matrix tension softening. For the soften-
PCHIP approach by Fritsch & Carlson [10], which uses piece- ing law Eq. (3), it is assumed that c = 1, fmt1 = 4.9 N/mm2 and
wise cubic Hermite polynomials for interpolation is applied. The fmt2 = 0.1 N/mm2 . The cross-sectional area of the matrix is de-
PCHIP approach ensures that the interpolation functions do not fined with 800 mm2 corresponding to Fig. 2. For the reinforce-
locally overshoot the values of the supporting points. Moreover, ment, where the cases of glass and carbon reinforcement are taken
the smooth transition at the supporting of the bond law reduces, into account, the material parameters are given in Section 2.2.
at least, numerical problems during computations at these points. For the carbon reinforcement, a thermal expansion coefficient of
Based on the PCHIP approach also the tangential bond modulus αT,c = −5 · 10−7 K−1 is applied. However, reinforcement fail-
G needed in Eq. (5) can be determined, see Ref. [11]. ure is not considered in the simulation. In a simplifying manner,
Due to the incomplete penetration of the reinforcement with it is assumed that the bond laws are identical for the glass-matrix
matrix or coating, different bond zones have to be taken into ac- interface and the carbon-matrix interface, which is not necessar-
count over the yarn cross section. Therefore, two different bond ily the case in reality as pointed out in Section 2.3. In the simu-
laws τ (s)mr and τ (s)rr are applied, see Fig. 4. lations, a reinforcement ratio of 2 % is applied. Glass and carbon
Bond law τ (s)mr is applied between the matrix and the layer yarns have usually different cross-sectional areas and, thus, bond-
representing the filaments in the fill-in zone of the yarn. Based surface areas. In the following, values of two typical yarn types
on the assumption of adhesive bond, τ increases initially with in- are used. For the glass reinforcement, the cross-sectional area of
creasing slip s. After exceeding the bond strength τmax , bond one yarn is assumed with 0.11 mm2 leading to 146 yarns. For
degradation with transition to friction is assumed. This is mod- the carbon reinforcement, a cross-sectional area of 0.45 mm2 is
elled with a decreasing course of the τ -s relation until a residual applied, which results in 36 yarns. For both yarn types, it is as-
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

sumed that 36 % of the cross-sectional correspond to filaments in crack occurs. However, the maximum mean stress levels de-
the fill-in zone and 64 % to the yarn core. This ratio corresponds crease with increasing imposed strains, which can be explained
to a constant thickness of the layers in the layer model. with bond degradation resulting in a reduction of tension stiffen-
Two loading cases are investigated. The first case is matrix ing. The lower limit is given in the case of completely destroyed
shrinkage where imposed strains of -0.1 % are applied incremen- bond where only the imposed reinforcement deformations are re-
tally decreasing to the matrix bar element chain. The second case strained.
is thermal loading where the temperature is reduced in the matrix The smallest reduction is observable for the case of thermal
and reinforcement elements incrementally to -100 K. For the ma- loading and glass reinforcement, which result from contraction
trix elements, this corresponds also to imposed strains of -0.1 %. of the reinforcement compensating the loss of tension stiffening.
This compensation does not occur in case of matrix shrinkage,
4.2. Results of the simulations which leads to a larger reduction of the mean stress. Even larger
In this section, the results of the simulations are presented is this effect in the case of carbon reinforcement where the rein-
and analysed. At first, mean stress-imposed strain relations as forcement expands with increasing temperature reduction. This
shown in Figs. 5 and 6 are analysed. The mean stress is deter- is also the reason why the decrease of the mean stress is larger
mined as the sum of the reaction forces at the nodes of one end in the case of thermal loading than for matrix shrinkage. Matrix
of the model divided by the cross-sectional area of the matrix. tension softening reduces the decrease of mean stress as a cer-
The imposed strain is either the applied matrix shrinkage strain tain portion of normal stress is transferred in the matrix over the
or the strain due to temperature reduction applied to matrix and crack, see Fig. 6. However this effect decreases with increasing
reinforcement. imposed strains because crack width increases and the stresses
transferred in the matrix decrease.
6.0
2.0
glass
5.0
glass
mean stress [N/ mm2]

crack width [10-2 mm] 1.5

4.0

3.0
1.0
2.0 carbon
carbon
0.5
1.0
thermal loading
matrix shrinkage thermal loading
0 matrix shrinkage
0 -0.02 -0.04 -0.06 -0.08 -0.1 0
imposed strain [%] 0 -0.02 -0.04 -0.06 -0.08 -0.1
imposed strain [%]
Figure 5: Mean stress-imposed strain relations neglecting matrix
tension softening Figure 7: Mean crack width-imposed strain relations neglecting
matrix tension softening

6.0
glass 2.0
5.0
mean stress [N/ mm2]

1.5
crack width [10-2 mm]

4.0
glass
3.0
carbon 1.0
2.0

1.0 0.5
thermal loading carbon
matrix shrinkage thermal loading
0 matrix shrinkage
0 -0.02 -0.04 -0.06 -0.08 -0.1 0
imposed strain [%] 0 -0.02 -0.04 -0.06 -0.08 -0.1
imposed strain [%]
Figure 6: Mean stress-imposed strain relations considering ma-
trix tension softening Figure 8: Mean crack width-imposed strain relations considering
matrix tension softening
It can be seen that the mean stress increases initially for all
considered cases with increasing imposed strain until the tensile Figures 7 and 8 show the mean crack widths as a function
strength of the matrix is reached for the first time. Afterwards the of the imposed strains for the different reinforcement and load-
mean stress drops until the reinforcement is activated to bridge ing cases neglecting and considering matrix tension softening. It
the crack. As a sufficient amount of reinforcement and bond is can be seen that the mean crack widths increase with increasing
available, the mean stress increases again until the next matrix imposed strains. The type of loading, i. e. matrix shrinkage or
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

normal stresses, normal stresses, bond stresses, bond stresses,

first crack final load step first crack final load step
6 600 10

reinforcement stress [N/mm²]

crack
glass reinforcement,

matrix stress [N/mm²]

5 500

bond stress [N/mm²]

thermal loading

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
6 600 10

reinforcement stress [N/mm²]

crack
glass reinforcement,

matrix stress [N/mm²]

5 500

bond stress [N/mm²]

matrix shrinkage

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
6 600 reinforcement stress [N/mm²] 10
carbon reinforcement,

crack
matrix stress [N/mm²]

5 500
bond stress [N/mm²]
thermal loading

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200
6 600 10
reinforcement stress [N/mm²]

crack
carbon reinforcement,

matrix stress [N/mm²]

5 500
bond stress [N/mm²]
matrix shrinkage

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
x [mm] x [mm] x [mm] x [mm]

Figure 9: Normal and bond stress distribution on a range of 0 ≤ x ≤ 0.05 m neglecting matrix tension softening

thermal loading has minor influence on the crack width. The ini- haviour of the composite can be explained by bond stress and
tial crack widths are smaller for carbon reinforcement because of normal stress distributions. Respective diagrams are shown in
the higher stiffness compared to the glass reinforcement, which Figs. 9 and 10 for the load step corresponding to the first crack
restrains crack opening. Due to the larger bond surface area more and the final load step. In the two left columns, normal stress dis-
cracks develop with glass reinforcement, which results in the ef- tributions in the matrix and the reinforcement layers on a range
fect that the stabilised cracking state is not reached during the of 0 ≤ x ≤ 0.05 m are shown for the state after the first ma-
range of loading. In contrast, in the simulations with carbon rein- trix crack and at the final load step. In the right two columns the
forcement the stabilised cracking state is reached in all cases. In respective bond stresses are shown.
the stabilised cracking state, the cracks open strongly leading to It can be seen that the stress transfer length are larger in case
increasing crack widths. of carbon reinforcement compared to glass reinforcement. This
Matrix tension softening initially restrains crack opening, can be explained by the higher stiffness of the carbon reinforce-
which leads to a reduction of crack widths. For carbon rein- ment compared to the glass reinforcement. In all cases, the re-
forcement this leads also to a shortening of the strain range where inforcement possesses compressive stresses between the cracks
cracking occurs but the numbers of the cracks are not influenced. in the final load step. This results from matrix contraction. The
For glass reinforcement also the numbers of cracks increase con- largest compressive stresses are observable for carbon reinforce-
sidering matrix tension softening. ment where these compressive stresses are slightly larger in the
Most of the aforementioned properties of the load-bearing be- case of thermal loading compared to matrix shrinkage due to ad-
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

normal stresses, normal stresses, bond stresses, bond stresses,

first crack final load step first crack final load step
6 600 10

reinforcement stress [N/mm²]

crack
glass reinforcement,

matrix stress [N/mm²]

5 500

bond stress [N/mm²]

thermal loading

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
6 600 10

reinforcement stress [N/mm²]

crack
glass reinforcement,

matrix stress [N/mm²]

5 500

bond stress [N/mm²]

matrix shrinkage

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2
6 600 reinforcement stress [N/mm²] 10
carbon reinforcement,

crack
matrix stress [N/mm²]

5 500
bond stress [N/mm²]
thermal loading

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
6 600 10
reinforcement stress [N/mm²]

crack
carbon reinforcement,

matrix stress [N/mm²]

5 500
bond stress [N/mm²]
matrix shrinkage

4 400 5
3 300
2 200 0
1 100
0 0 -5
sleeve layer
-1 innermost core layer -100
matrix
-2 -200 -10
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
x [mm] x [mm] x [mm] x [mm]

Figure 10: Normal and bond stress distribution on a range of 0 ≤ x ≤ 0.05 m considering matrix tension softening

ditional compressive stresses because of the negative thermal ex- reinforcement relatively large ranges of intact bond are observ-
pansion of carbon. In the case of glass reinforcement, the com- able. In contrast, the bond strength fronts have reached relatively
pressive stresses are smaller due to lower stiffness of the rein- large distances to the crack positions in the final load step for
forcement. Moreover, the compressive stresses are further re- carbon reinforcement, which indicates strong bond degradation.
duced in the case of thermal loading due to the contraction of Moreover, it can be seen that differences in the bond stress distri-
the reinforcement, which is only slightly smaller than the con- bution between the case of matrix shrinkage and thermal loading
traction of the matrix according to the applied thermal expansion are small for carbon reinforcement. Thus, the negative thermal
coefficients. expansion coefficient of carbon reinforcement seems to have a
The normal stress distributions are directly associated with subordinate influence. The reason is that the deformations of the
bond stress distributions. Bond stresses equal or close to zero carbon reinforcement are more than one order of magnitude lower
indicate intact bond in case of monotonic loading without fibre than those of the matrix.
pull-out phenomena. Intact bond is observable at the end of the In the case considering matrix tension softening, it can be
stress transfer length (e.g. in the bond stress distributions for the seen that especially in the case of the first crack considerable nor-
case of the first crack) and in the centre between two cracks (e.g. mal stresses are transferred in the matrix, which results in reduced
in the bond stress distributions for the final load step). Degraded reinforcement and bond stresses. However, this effect reduces
bond exists in the range between the crack position and the po- with increasing loading. Although in the final load step still stress
sition where the bond stress reaches the bond strength. For glass is transferred over the matrix in the crack the normal stress dis-
CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

tributions of the reinforcement and bond stresses differ not much ten Glasfaser-Multifilamentgarnen, Dissertation, Technis-
from the cases neglecting matrix tension softening. Only in the che Universität Dresden, Dresden, 2009.
case of glass reinforcement more cracks develop and, thus, crack
spacing is decreased. [6] Butler, M., Mechtcherine, V. and Hempel, S., Experimental
investigations on the durability of fibre-matrix interfaces in
textile-reinforced concrete, Cem. Concr. Compos., 31, pp.
5. Summary and conclusions
221-231, 2009.
In this paper, a numerical model within the framework of the [7] Butler, M., Mechtcherine, V. and Hempel, S., Durability of
Finite Element Method for the simulation of the uniaxial tensile textile reinforced concrete made with AR glass fibre: ef-
load-bearing behaviour of Textile Reinforced Concrete exposed fect of the matrix composition, Mater. Struct., 43, pp. 1351-
to imposed deformations was presented and results of exemplary 1368, 2010.
simulations were shown. Despite a relatively simple geometri-
cal representation of the composite, the model gains complexity [8] Curbach, M. and Jesse, F. (Eds.), Textilbeton - Theorie
by nonlinear formulations of the constitutive relations, such as und Praxis: Proceedings of the 4. Kolloquium zu textilbe-
limited tensile strength resulting in matrix cracks, matrix tension wehrten Tragwerken (CTRS4) und zur 1. Anwendertagung,
softening and nonlinear bond laws. Technische Universität Dresden, Dresden, 2009.
The simulations covered the simulation of the load-bearing
behaviour of tensile bars exposed to matrix shrinkage and thermal [9] Ehlig, D., Jesse, F. and Curbach, M., Textilbeton verstärkte
loading applied as imposed deformations. The deformations were Platten unter Brandbelastung, Beton- und Stahlbetonbau,
restrained by fixation of the bar ends. The presented results of the 105, pp. 102-110, 2010.
model are stress-imposed strain relations, crack width-imposed [10] Fritsch, F.N. and Carlson, R.E., Monotone Piecewise Cubic
strain relations and stress distributions between matrix and rein- Interpolation, SIAM J. Numer. Anal., 17, pp. 238-246, 1980.
forcement. It was shown that due to the imposed strains matrix
cracking occurred, which also led to bond degradation. The most [11] Hartig, J.U., Numerical investigations on the uniaxial ten-
detrimental effect on bond was observed in the case of carbon re- sile behaviour of Textile Reinforced Concrete, Dissertation,
inforcement. However, the bond degradation resulted primarily Technische Universität Dresden, Dresden, 2011.
from the high stiffness of the reinforcement. The negative ther-
mal expansion coefficient of the carbon fibres, which leads to an [12] Hartig, J. and Häußler-Combe, U., A model for Tex-
expansion of the reinforcement while the matrix contracts, has tile Reinforced Concrete exposed to uniaxial tensile load-
due to its low absolute value compared to the matrix a subordi- ing, Proceedings of 18th International Conference in Com-
nate influence on the load-bearing behaviour. Accordingly, it is of puter Methods in Mechanics (CMM 2009), Kuczma, M.,
secondary interest whether the imposed deformation results from Wilmański, K., Szajna, W. (Eds.), The University of Zielona
matrix shrinkage or thermal loading. In contrast, matrix shrink- Góra Press, Zielona Góra, pp. 203-204, 2009.
age has in case of glass reinforcement a more detrimental effect
than thermal loading where the contraction of the reinforcement [13] Hartig, J., Häußler-Combe, U. and Schicktanz, K., Influ-
leads to more compatible deformations of matrix and reinforce- ence of bond properties on the tensile behaviour of Textile
ment. Reinforced Concrete, Cem. Concr. Compos., 30, pp. 898-
In summary, restrained imposed deformations lead to a stiff- 906, 2008.
ness reduction of the composite due to matrix cracking and to
degradation of bond. In the considered range of imposed loading, [14] Jesse, F., Tragverhalten von Filamentgarnen in zement-
the reinforcement stresses were considerably below characteris- gebundener Matrix, Dissertation, Technische Universität
tic tensile strength values. Nevertheless, for additional external Dresden, Dresden, 2004.
loading less load-bearing reserves remain. Even though stress [15] Matthies, H. and Strang, G., The solution of non-linear fi-
gradients might be reduced due to creep of the matrix, the re- nite element equations, Int. J. Numer. Methods Eng., 14, pp.
duced composite stiffness and the degraded bond remains. Bond 1613-1626, 1979.
degradation might also necessitate the increase of the anchorage
length of the reinforcement. [16] Remmel, G., Zum Zug- und Schubtragverhalten von
Bauteilen aus hochfestem Beton, DAfStb Heft 444, Beuth,
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